{"id":1469,"date":"2018-12-11T13:24:17","date_gmt":"2018-12-11T18:24:17","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-linear-inequalities-2\/"},"modified":"2019-05-22T13:59:38","modified_gmt":"2019-05-22T17:59:38","slug":"solve-linear-inequalities-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-linear-inequalities-2\/","title":{"raw":"Solve Linear Inequalities","rendered":"Solve Linear Inequalities"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Graph inequalities on the number line<\/li>\r\n \t<li>Solve linear inequalities<\/li>\r\n \t<li>Translate words to an inequality and solve<\/li>\r\n \t<li>Solve applications with linear inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834327442\" class=\"be-prepared\">\r\n<p id=\"fs-id1167835311227\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol id=\"fs-id1167835233718\" type=\"1\">\r\n \t<li>Translate from algebra to English: \\(15&gt;x.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167829937221\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Translate to an algebraic expression: 15 is less than <em data-effect=\"italics\">x<\/em>.\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836513192\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831985734\">\r\n<h3 data-type=\"title\">Graph Inequalities on the Number Line<\/h3>\r\n<p id=\"fs-id1167832152811\">What number would make the inequality \\(x&gt;3\\) true? Are you thinking, \u201c<em data-effect=\"italics\">x<\/em> could be four\u201d? That\u2019s correct, but <em data-effect=\"italics\">x<\/em> could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality \\(x&gt;3.\\)<\/p>\r\n<p id=\"fs-id1167834472568\">We show all the solutions to the inequality \\(x&gt;3\\) on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.<\/p>\r\n<p id=\"fs-id1167835359606\">We can also represent inequalities using <strong data-effect=\"bold\">interval notation<\/strong>. There is no upper end to the solution to this inequality. In interval notation, we express \\(x&gt;3\\) as \\(\\left(3,\\infty \\right).\\) The symbol \\(\\infty \\) is read as \u201c<strong data-effect=\"bold\">infinity<\/strong>.\u201d It is not an actual number.<\/p>\r\n<p id=\"fs-id1167832143991\"><a href=\"#CNX_IntAlg_Figure_02_05_001\" class=\"autogenerated-content\">(Figure)<\/a> shows both the number line and the interval notation.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_05_001\">\r\n<div class=\"bc-figcaption figcaption\">The inequality \\(x&gt;3\\) is graphed on this number line and written in interval notation.<\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835380309\" data-alt=\"The figure shows the inquality, x is greater than 3, graphed on a number line from negative 5 to 5. There is shading that starts at 3 and extends to numbers to its right. The solution for the inequality is written in interval notation. It is the interval from 3 to infinity, not including 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the inquality, x is greater than 3, graphed on a number line from negative 5 to 5. There is shading that starts at 3 and extends to numbers to its right. The solution for the inequality is written in interval notation. It is the interval from 3 to infinity, not including 3.\" \/><\/span>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167835532500\">We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.<\/p>\r\n<p id=\"fs-id1167832116621\">The inequality \\(x\\le 1\\) means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at \\(x=1.\\) We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See <a href=\"#CNX_IntAlg_Figure_02_05_002\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\r\n<p id=\"fs-id1167835369795\">There is no lower end to those numbers. We write \\(x\\le 1\\) in interval notation as \\(\\left(\\text{\u2212}\\infty ,1\\right].\\) The symbol \\(\\text{\u2212}\\infty \\) is read as \u201cnegative infinity.\u201d <a href=\"#CNX_IntAlg_Figure_02_05_002\" class=\"autogenerated-content\">(Figure)<\/a> shows both the number line and interval notation.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_05_002\">\r\n<div class=\"bc-figcaption figcaption\">The inequality \\(x\\le 1\\) is graphed on this number line and written in interval notation.<\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835364648\" data-alt=\"The figure shows the inquality, x is less than or equal to l, graphed on a number line from negative 5 to 5. There is shading that starts at 1 and extends to numbers to its left. The solution for the inequality is written in interval notation. It is the interval from negative infinity to one, including 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the inquality, x is less than or equal to l, graphed on a number line from negative 5 to 5. There is shading that starts at 1 and extends to numbers to its left. The solution for the inequality is written in interval notation. It is the interval from negative infinity to one, including 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832067399\">\r\n<div data-type=\"title\">Inequalities, Number Lines, and Interval Notation<\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835304648\" data-alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\" \/><\/span>\r\n\r\n<\/div>\r\n<p id=\"fs-id1171791696312\">The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835334972\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835305891\">\r\n<div data-type=\"problem\" id=\"fs-id1167834120797\">\r\n<p id=\"fs-id1167834061620\">Graph each inequality on the number line and write in interval notation.<\/p>\r\n<p id=\"fs-id1167831871394\"><span class=\"token\">\u24d0<\/span>\\(x\\ge -3\\)<span class=\"token\">\u24d1<\/span>\\(x&lt;2.5\\)<span class=\"token\">\u24d2<\/span>\\(x\\le -\\frac{3}{5}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831071507\">\r\n<p id=\"fs-id1167835419516\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167835233973\" class=\"unnumbered unstyled\" summary=\"Shade to the right of negative 3 and put a left bracket at negative 3. The solution of the inequality x is greater than negative 3 is indicated on a number line with a left bracket at a and shading to the right. Write in intervak notation. The solution in interval notation is the interval from negative 3 to infinity enclosed within a left bracket and right parenthesis.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834252412\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Shade to the right of \\(-3,\\) and put a bracket at \\(-3.\\)<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831112379\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835375599\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_004c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167834306926\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167832060163\" class=\"unnumbered unstyled\" summary=\"Shade to the left of negative 2.5 and put a parenthesis at negative 2.5. The solution of the inequality x is less than 2.5 is indicated on a number line with a right bracket at 2.5 and shading to the left. Write in interval notation. The solution in interval notation is the interval from negative infinity to 2.5 enclosed within parentheses.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830962240\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Shade to the left of 2.5 and put a parenthesis at 2.5.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835322275\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_005b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830770108\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_005c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167834535692\" class=\"unnumbered unstyled\" summary=\"Shade to the left of negative three-fifths and put a right bracket at negative three-fifths. The solution of the inequality x is less than or equal to negative three-fifths is indicated on a number line with a right bracket at negative three-fifths and shading to the left. Write in interval notation. The solution in interval notation is the interval from negative infinity to negative three-fifths enclosed within a left parentheses and right bracket.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832057978\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Shade to the left of \\(-\\frac{3}{5},\\) and put a bracket at \\(-\\frac{3}{5}.\\)<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327153\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_006b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834061478\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_006c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835283921\">\r\n<div data-type=\"problem\" id=\"fs-id1167835595498\">\r\n<p id=\"fs-id1167835268064\">Graph each inequality on the number line and write in interval notation: <span class=\"token\">\u24d0<\/span> \\(x&gt;2\\) <span class=\"token\">\u24d1<\/span> \\(x\\le -1.5\\) <span class=\"token\">\u24d2<\/span> \\(x\\ge \\frac{3}{4}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835337188\">\r\n\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832054934\" data-alt=\"The graph of the inequality x is greater than 2 is indicated on a number line with a left parenthesis at 2 and shading to the right. The solution in interval notation is the interval from 2 to infinity enclosed within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_301_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is greater than 2 is indicated on a number line with a left parenthesis at 2 and shading to the right. The solution in interval notation is the interval from 2 to infinity enclosed within parentheses.\" \/><\/span>\r\n<p id=\"fs-id1167834156696\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831892896\" data-alt=\"The graph of the inequality x is less than or equal to negative 1.5 is indicated on a number line with a right bracket at negative 1.5 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 1.5 enclosed within a left parenthesis and right bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_302_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is less than or equal to negative 1.5 is indicated on a number line with a right bracket at negative 1.5 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 1.5 enclosed within a left parenthesis and right bracket.\" \/><\/span>\r\n<p id=\"fs-id1167826983806\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835310783\" data-alt=\"The graph of the inequality x is greater than or rqual to three-fourths is indicated on a number line with a left bracket at three-fourths and shading to the right. The solution in interval notation is the interval from three-fourths to infinity enclosed within a left bracket and left parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_303_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is greater than or rqual to three-fourths is indicated on a number line with a left bracket at three-fourths and shading to the right. The solution in interval notation is the interval from three-fourths to infinity enclosed within a left bracket and left parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835348452\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835419193\">\r\n<div data-type=\"problem\" id=\"fs-id1167832052461\">\r\n<p id=\"fs-id1167832053172\">Graph each inequality on the number line and write in interval notation: <span class=\"token\">\u24d0<\/span> \\(x\\le -4\\) <span class=\"token\">\u24d1<\/span>\\(x\\ge 0.5\\) <span class=\"token\">\u24d2<\/span> \\(x&lt;-\\frac{2}{3}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834413671\">\r\n<p id=\"fs-id1167834228113\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835319438\" data-alt=\"The graph of the inequality x is less than or equal to negative 4 is indicated on a number line with a right bracket at negative 4 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 4 enclosed within an left parenthesis and right bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_304_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is less than or equal to negative 4 is indicated on a number line with a right bracket at negative 4 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 4 enclosed within an left parenthesis and right bracket.\" \/><\/span>\r\n<p id=\"fs-id1167835368792\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835376724\" data-alt=\"The graph of the inequality x is greater than or equal to 0.5 is indicated on a number line with a left bracket at 0.5 and shading to the right. The solution in interval notation is the interval from 0.5 to infinity enclosed within a left bracket and right parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_305_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is greater than or equal to 0.5 is indicated on a number line with a left bracket at 0.5 and shading to the right. The solution in interval notation is the interval from 0.5 to infinity enclosed within a left bracket and right parenthesis.\" \/><\/span>\r\n<p id=\"fs-id1167835340509\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835375263\" data-alt=\"The graph of the inequality x is less than negative two-thirds is indicated on a number line with a right parenthesis at negative two-thirds and shading to the left. The solution in interval notation is the interval from negative infinity to negative two-thirds enclosed within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_306_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is less than negative two-thirds is indicated on a number line with a right parenthesis at negative two-thirds and shading to the left. The solution in interval notation is the interval from negative infinity to negative two-thirds enclosed within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835371142\">What numbers are greater than two but less than five? Are you thinking say, \\(2.5,3,3\\frac{2}{3},4,4,99?\\) We can represent all the numbers between two and five with the inequality \\(2&lt;x&lt;5.\\) We can show \\(2&lt;x&lt;5\\) on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See <a href=\"#CNX_IntAlg_Figure_02_05_007\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_05_007\"><span data-type=\"media\" id=\"fs-id1167835358311\" data-alt=\"The graph of the inequality 2 is less than x which is less than 5 shows open circles a 2 and 5 and shading in between.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_007_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality 2 is less than x which is less than 5 shows open circles a 2 and 5 and shading in between.\" \/><\/span><\/div>\r\n<div data-type=\"example\" id=\"fs-id1167835524181\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835356443\">\r\n<div data-type=\"problem\" id=\"fs-id1167835356057\">\r\n<p id=\"fs-id1167834252251\">Graph each inequality on the number line and write in interval notation.<\/p>\r\n<p id=\"fs-id1167830991263\"><span class=\"token\">\u24d0<\/span>\\(-3&lt;x&lt;4\\)<span class=\"token\">\u24d1<\/span>\\(-6\\le x&lt;-1\\)<span class=\"token\">\u24d2<\/span>\\(0\\le x\\le 2.5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835197472\">\r\n<p id=\"fs-id1167835321054\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167835351265\" class=\"unnumbered unstyled\" summary=\"Negative 3 is less than x which is less than 4. There open circles at negative 3 and 4 and shading between negative 3 and 4 on the number line. Put parenthese at negative 3 and 4. Write in interval notation.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Shade between \\(-3\\) and 4.\r\n<div data-type=\"newline\"><\/div>\r\nPut a parentheses at \\(-3\\) and 4.<\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835200506\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834219734\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167827943690\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167834201950\" class=\"unnumbered unstyled\" summary=\"Negative 6 is less than or equal to x which is less than negative 1. There is closed circle at negative 6 and an open circle at negative 1 and shading between negative 6 and negative 1 on the number line. Put a bracket at negative 6 and a parenthesis at 1. Write in interval notation.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834346774\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Shade between \\(-6\\) and \u22121.\r\n<div data-type=\"newline\"><\/div>\r\nPut a bracket at \\(-6,\\) and\r\n<div data-type=\"newline\"><\/div>\r\na parenthesis at \u22121.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835324999\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835333898\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167835622135\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167835236159\" class=\"unnumbered unstyled\" summary=\"0 is less than or equal to x which is less than or equal to 2.5. There are closed circles at 0 and 2.5 and shading between negative 0 and 2.5 on the number line. Put brackets at 0 and 2.5. Write in interval notation.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831920485\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Shade between 0 and 2.5.\r\n<div data-type=\"newline\"><\/div>\r\nPut a bracket at 0 and at 2.5.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835330839\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831880568\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835327661\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835370620\">\r\n<div data-type=\"problem\" id=\"fs-id1167835364899\">\r\n<p id=\"fs-id1167832060132\">Graph each inequality on the number line and write in interval notation:<\/p>\r\n<p id=\"fs-id1167835344258\"><span class=\"token\">\u24d0<\/span>\\(-2&lt;x&lt;1\\)<span class=\"token\">\u24d1<\/span>\\(-5\\le x&lt;-4\\)<span class=\"token\">\u24d2<\/span>\\(1\\le x\\le 4.25\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831890694\">\r\n<p id=\"fs-id1167835381590\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831040708\" data-alt=\"Negative 2 is less x which is less than 1. There are open circles at negative 2 and 1 and shading between negative 2 and 1 on the number line. Put parentheses at negative 2 and 1. Write in interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_307_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 2 is less x which is less than 1. There are open circles at negative 2 and 1 and shading between negative 2 and 1 on the number line. Put parentheses at negative 2 and 1. Write in interval notation.\" \/><\/span>\r\n<p id=\"fs-id1167835321144\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835327878\" data-alt=\"Negative 5 is less than or equal to x which is less than negative 4. There is a closed circle at negative 6 and an open circle at negative 4 and shading between negative 5 and negative 4 on the number line. Put a bracket at negative 5 and a parenthesis at negative 4. Write in interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_308_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 5 is less than or equal to x which is less than negative 4. There is a closed circle at negative 6 and an open circle at negative 4 and shading between negative 5 and negative 4 on the number line. Put a bracket at negative 5 and a parenthesis at negative 4. Write in interval notation.\" \/><\/span>\r\n<p id=\"fs-id1167831076658\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835346063\" data-alt=\"1 is less than or equal to x which is less than 4.25. There is closed circle at 1 and a closed circle at 4.25 and shading between 1 and 4.25 on the number line. Put brackets at 1 and 4.25. Write in interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_309_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"1 is less than or equal to x which is less than 4.25. There is closed circle at 1 and a closed circle at 4.25 and shading between 1 and 4.25 on the number line. Put brackets at 1 and 4.25. Write in interval notation.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834193336\">\r\n<div data-type=\"problem\" id=\"fs-id1167834382409\">\r\n<p id=\"fs-id1167835330830\">Graph each inequality on the number line and write in interval notation:<\/p>\r\n<p id=\"fs-id1171789787515\"><span class=\"token\">\u24d0<\/span>\\(-6&lt;x&lt;2\\)<span class=\"token\">\u24d1<\/span>\\(-3\\le x&lt;-1\\)<span class=\"token\">\u24d2<\/span>\\(2.5\\le x\\le 6\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835307740\">\r\n<p id=\"fs-id1167831949153\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167830700636\" data-alt=\"Negative 6 is less than x which is less than 2. There is an open circle at negative 6 and an open circle at 2 and shading between negative 6 and 2 on the number line. Put parentheses at negative 6 and 2. Write in interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_310_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 6 is less than x which is less than 2. There is an open circle at negative 6 and an open circle at 2 and shading between negative 6 and 2 on the number line. Put parentheses at negative 6 and 2. Write in interval notation.\" \/><\/span>\r\n<p id=\"fs-id1167834138113\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835371362\" data-alt=\"Negative 3 is less than or equal to x which is less than negative 1. There is a closed circle at negative 3 and an open circle at negative 1 and shading between negative 3 and negative 1 on the number line. Put a bracket at negative 3 and a parenthesis at negative 1. Write in interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_311_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 3 is less than or equal to x which is less than negative 1. There is a closed circle at negative 3 and an open circle at negative 1 and shading between negative 3 and negative 1 on the number line. Put a bracket at negative 3 and a parenthesis at negative 1. Write in interval notation.\" \/><\/span>\r\n<p id=\"fs-id1167835510464\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" data-alt=\"2.5 is less than or equal to x which is less thanor equal to 6. There is a closed circle at 2.5 and a closed circle at 6 and shading between 2.5 and 6 on the number line. Put brackets at 2.5 and 6. Write in interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_312_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"2.5 is less than or equal to x which is less thanor equal to 6. There is a closed circle at 2.5 and a closed circle at 6 and shading between 2.5 and 6 on the number line. Put brackets at 2.5 and 6. Write in interval notation.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834346402\">\r\n<h3 data-type=\"title\">Solve Linear Inequalities<\/h3>\r\n<p id=\"fs-id1167834104913\">A linear inequality is much like a linear equation\u2014but the equal sign is replaced with an inequality sign. A <strong data-effect=\"bold\">linear inequality<\/strong> is an inequality in one variable that can be written in one of the forms, \\(ax+b&lt;c,\\)\\(ax+b\\le c,\\)\\(ax+b&gt;c,\\) or \\(ax+b\\ge c.\\)<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167834138378\">\r\n<div data-type=\"title\">Linear Inequality<\/div>\r\n<p id=\"fs-id1167826978602\">A linear inequality is an inequality in one variable that can be written in one of the following forms where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and \\(a\\ne 0\\):<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167835303848\" class=\"unnumbered\" data-label=\"\">\\(ax+b&lt;c,\\phantom{\\rule{2em}{0ex}}ax+b\\le c,\\phantom{\\rule{2em}{0ex}}ax+b&gt;c,\\phantom{\\rule{2em}{0ex}}ax+b\\ge c.\\)<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834061592\">When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.<\/p>\r\n<p id=\"fs-id1167832212157\">We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167831895186\" data-alt=\"Negative 4 is less than 2. Negative 4 minus 5 is less than 2 minus 5. Negative 9 is less than negative 3, which is true. Negative 4 is less than 2. Negative 4 plus 7 is less than 2 plus 7. 3 is less than 9, which is true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_011_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 4 is less than 2. Negative 4 minus 5 is less than 2 minus 5. Negative 9 is less than negative 3, which is true. Negative 4 is less than 2. Negative 4 plus 7 is less than 2 plus 7. 3 is less than 9, which is true.\" \/><\/span>\r\n<p id=\"fs-id1167835387130\">Notice that the inequality sign stayed the same.<\/p>\r\n<p id=\"fs-id1167835422282\">This leads us to the Addition and Subtraction Properties of Inequality.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167835324958\">\r\n<div data-type=\"title\">Addition and Subtraction Property of Inequality<\/div>\r\n<p id=\"fs-id1167835309171\">For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c,<\/em> if \\(a&lt;b,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\)<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167834394945\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}a+c&lt;b+c\\hfill &amp; &amp; &amp; a-c&lt;b-c\\hfill \\\\ a+c&gt;b+c\\hfill &amp; &amp; &amp; a-c&gt;b-c\\hfill \\end{array}\\)<\/div>\r\n<p id=\"fs-id1167834537743\">We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167835349726\">What happens to an inequality when we divide or multiply both sides by a constant?<\/p>\r\n<p id=\"fs-id1167835305967\">Let\u2019s first multiply and divide both sides by a positive number.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167828421190\" data-alt=\"10 is less than 15. 10 times 5 is less than 15 times 5. 50 is less than 75 is true. 10 is less than 15. 10 divided by 5 is less than 15 divided by 5. 2 is less than 3 is true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_012_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"10 is less than 15. 10 times 5 is less than 15 times 5. 50 is less than 75 is true. 10 is less than 15. 10 divided by 5 is less than 15 divided by 5. 2 is less than 3 is true.\" \/><\/span>\r\n<p id=\"fs-id1167835334176\">The inequality signs stayed the same.<\/p>\r\n<p id=\"fs-id1167835375431\">Does the inequality stay the same when we divide or multiply by a negative number?<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167835331131\" data-alt=\"10 is less than 15 10 times negative 5 is blank 15 times negative 5? Negative 50 is blank negative 75. Negative 50 is greater than negative 75. 10 is less than 15. 10 divided by negative 5 is blank 15 divided by negative 5. Negative 2 is blank negative 3. Negative 2 is blank negative 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_013_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"10 is less than 15 10 times negative 5 is blank 15 times negative 5? Negative 50 is blank negative 75. Negative 50 is greater than negative 75. 10 is less than 15. 10 divided by negative 5 is blank 15 divided by negative 5. Negative 2 is blank negative 3. Negative 2 is blank negative 3.\" \/><\/span>\r\n<p id=\"fs-id1167835324788\">Notice that when we filled in the inequality signs, the inequality signs reversed their direction.<\/p>\r\n<p id=\"fs-id1167834085015\">When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.<\/p>\r\n<p id=\"fs-id1167831893684\">This gives us the Multiplication and Division Property of Inequality.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167834301124\">\r\n<div data-type=\"title\">Multiplication and Division Property of Inequality<\/div>\r\n<p id=\"fs-id1167834423555\">For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>,<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167834446398\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\text{multiply or divide by a positive}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&lt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&gt;0,\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&lt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&lt;\\frac{b}{c}.\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&gt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&gt;0,\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&gt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&gt;\\frac{b}{c}.\\hfill \\\\ \\\\ \\text{multiply or divide by a negative}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&lt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&lt;0,\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&gt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&gt;\\frac{b}{c}.\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&gt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&lt;0,\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&lt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&lt;\\frac{b}{c}.\\hfill \\end{array}\\)<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834308106\">When we divide or multiply an inequality by <em data-effect=\"italics\">a<\/em>:<\/p>\r\n\r\n<ul id=\"fs-id1167831933992\" data-bullet-style=\"bullet\">\r\n \t<li>positive number, the inequality stays the same.<\/li>\r\n \t<li>negative number, the inequality reverses.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1167835375489\">Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.<\/p>\r\n\r\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(x&gt;a\\phantom{\\rule{0.2em}{0ex}}\\text{has the same meaning as}\\phantom{\\rule{0.2em}{0ex}}a&lt;x\\)<\/div>\r\n<p id=\"fs-id1167834114104\">Think about it as \u201cIf Xander is taller than Andy, then Andy is shorter than Xander.\u201d<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835419716\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826781531\">\r\n<div data-type=\"problem\" id=\"fs-id1167835237688\">\r\n<p id=\"fs-id1167831943976\">Solve each inequality. Graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n<p id=\"fs-id1167835312278\"><span class=\"token\">\u24d0<\/span>\\(x-\\frac{3}{8}\\le \\frac{3}{4}\\)<span class=\"token\">\u24d1<\/span>\\(9y&lt;\\text{\u200b}\\text{\u200b}54\\)<span class=\"token\">\u24d2<\/span>\\(-15&lt;\\frac{3}{5}z\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834501314\">\r\n<p id=\"fs-id1167835390468\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167828434586\" class=\"unnumbered unstyled\" summary=\"sides of the inequality. x minus three-eighths plus three-eighths is less than or equal to three-fourths plus three-eighths. Simplify. x is less than nine-eighths. Graph the solution on a number line with a right bracket at nine-eighths with shading to the left. Write the solution, negative infinity to nine-eighths, within a parenthesis and bracket.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835331608\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Add \\(\\frac{3}{8}\\) to both sides of the inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835301636\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835381345\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835377147\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835595108\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167835333643\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167834324727\" class=\"unnumbered unstyled\" summary=\"9 y is less than 54. Divide both sides of the inequality by 9. Because 9 is less than 0, the inequality stays the same. 9 y divided by 9 is less than 54 divided by 9. Simplify. y is less than 6. Graph the solution on the number line with a right parenthesis at 6 with shading to the left. Write the solution, negative infinity to 6, within parentheses.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834463151\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Divide both sides of the inequality by 9; since\r\n<div data-type=\"newline\"><\/div>\r\n9 is positive, the inequality stays the same.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835366990\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831107033\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835365733\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834506066\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167835332678\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167835363141\" class=\"unnumbered unstyled\" summary=\"Negative 15 is less than three-fifths z . Multiply both sides of the inequality by five-thirds. Five-thirds times negative 15 is less than five-thirds times three-fifths z. Since five-thirds is greater than 0, the inequality stays the same. Simplify. Negative 25 is less than z. Rewrite with the variable on the left. Z is greater than negative 25. Graph the solution on the number line with a left parenthesis at negative 25 with shading to the right. Write the solution in interval notation. It is negative 25 to infinity within parentheses.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834191317\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Multiply both sides of the inequality by \\(\\frac{5}{3}.\\)\r\n<div data-type=\"newline\"><\/div>\r\nSince \\(\\frac{5}{3}\\) is positive, the inequality stays the same.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834439012\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835371351\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Rewrite with the variable on the left.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826978025\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835415285\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835342789\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834301181\">\r\n<div data-type=\"problem\" id=\"fs-id1167835595340\">\r\n<p id=\"fs-id1167832051471\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\r\n<p id=\"fs-id1171791609623\"><span class=\"token\">\u24d0<\/span>\\(p-\\frac{3}{4}\\ge \\frac{1}{6}\\)<span class=\"token\">\u24d1<\/span>\\(9c&gt;72\\)<span class=\"token\">\u24d2<\/span>\\(24\\le \\frac{3}{8}m\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835181898\">\r\n<p id=\"fs-id1167827957280\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167830757790\" data-alt=\"p is less than eleven-twelfths. The solution on the number line has a right bracket at eleven-twelfths with shading to the right. The solution in interval notation is, eleven-twelfths to infinity within a bracket and parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_313_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"p is less than eleven-twelfths. The solution on the number line has a right bracket at eleven-twelfths with shading to the right. The solution in interval notation is, eleven-twelfths to infinity within a bracket and parenthesis.\" \/><\/span>\r\n<p id=\"fs-id1167831239596\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834459154\" data-alt=\"c is less than 8. The solution on the number line has a left bracket at 8 with shading to the right. The solution in interval notation is, 8 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_314_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"c is less than 8. The solution on the number line has a left bracket at 8 with shading to the right. The solution in interval notation is, 8 to infinity within parentheses.\" \/><\/span>\r\n<p id=\"fs-id1167832153827\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835262243\" data-alt=\"m is greater than or equal to 8. The solution on the number line has a right bracket at 64 with shading to the right. The solution in interval notation is, 64 to infinity within a bracket and parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_315_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"m is greater than or equal to 8. The solution on the number line has a right bracket at 64 with shading to the right. The solution in interval notation is, 64 to infinity within a bracket and parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831112374\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834156710\">\r\n<div data-type=\"problem\" id=\"fs-id1167835236457\">\r\n<p id=\"fs-id1167832134181\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\r\n<p id=\"fs-id1171791492048\"><span class=\"token\">\u24d0<\/span>\\(r-\\frac{1}{3}\\le \\frac{7}{12}\\)<span class=\"token\">\u24d1<\/span>\\(12d\\le \\text{\u200b}60\\)<span class=\"token\">\u24d2<\/span>\\(-24&lt;\\frac{4}{3}n\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826857110\">\r\n<p id=\"fs-id1167830836919\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835318449\" data-alt=\"r is less than or equal to eleven-twelfths. The solution on the number line has a left bracket at eleven-twelfths with shading to the left. The solution in interval notation is negative infinity to eleven-twelfths within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_316_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"r is less than or equal to eleven-twelfths. The solution on the number line has a left bracket at eleven-twelfths with shading to the left. The solution in interval notation is negative infinity to eleven-twelfths within a parenthesis and a bracket.\" \/><\/span>\r\n<p id=\"fs-id1167832060027\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832053973\" data-alt=\"c is less than or equal to 5. The solution on the number line has a right bracket at 5 with shading to the left. The solution in interval notation is negative infinity to 5 within a parentheses and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_317_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"c is less than or equal to 5. The solution on the number line has a right bracket at 5 with shading to the left. The solution in interval notation is negative infinity to 5 within a parentheses and a bracket.\" \/><\/span>\r\n<p id=\"fs-id1167835595823\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831887927\" data-alt=\"n is greater than negative 18. The solution on the number line has a left parenthesis at negative 18 with shading to the right. The solution in interval notation is negative 18 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_318_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"n is greater than negative 18. The solution on the number line has a left parenthesis at negative 18 with shading to the right. The solution in interval notation is negative 18 to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835421968\">Be careful when you multiply or divide by a negative number\u2014remember to reverse the inequality sign.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835356688\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835356690\">\r\n<div data-type=\"problem\" id=\"fs-id1167835376561\">\r\n<p id=\"fs-id1167835376563\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n<p id=\"fs-id1167834190410\"><span class=\"token\">\u24d0<\/span>\\(-13m\\ge 65\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{n}{-2}\\ge 8\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835519089\">\r\n<p id=\"fs-id1167828433884\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167835377381\" class=\"unnumbered unstyled\" summary=\"Negative 13 m is greater than or equal to 65. Divided both sides of the inequality by negative 13. Negative 13 m divided y negative 13 is less than or equal to 65 divided by negative 13. Since negative 13 is less than 0, the inequality reverses. Simplify. m is less than or equal to negative 5. Graph the solution on the number line with a right bracket at negative 5 with shading to the left. Write the solution in interval notation. It is negative infinity to negative 5 within a parenthesis and bracket.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834523712\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Divide both sides of the inequality by \\(-13.\\)\r\n<div data-type=\"newline\"><\/div>\r\nSince \\(-13\\) is a negative, the inequality reverses.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832075837\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835326056\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835347022\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835340006\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167834423509\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167835339996\" class=\"unnumbered unstyled\" summary=\"n divided by negative 2 is greater than or equal to 8. Multiply both sides of the inequality by negative 2. Negative times the quantity n divided by negative 2 is less thann or equal to negative 2 times 8 Since negative 2 is less than 0, the inequality reverses. Simplify. n is less than or equal to negative 16. Graph the solution on the number line with a right bracket at negative 16 with shading to the left. Write the solution in interval notation. It is negative infinity to negative 16 within a parenthesis and bracket.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835253796\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Multiply both sides of the inequality by \\(-2.\\)\r\n<div data-type=\"newline\"><\/div>\r\nSince \\(-2\\) is a negative, the inequality reverses.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835309044\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830694037\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832060542\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835238691\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835366231\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835366234\">\r\n<div data-type=\"problem\" id=\"fs-id1167826857188\">\r\n<p id=\"fs-id1167826857190\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\r\n<p id=\"fs-id1171791461128\"><span class=\"token\">\u24d0<\/span>\\(-8q&lt;32\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{k}{-12}\\le 15.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835349959\">\r\n<p id=\"fs-id1167835349961\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832043282\" data-alt=\"q is greater than or equal to negative 4. The solution on the number line has a left parenthesis at negative 4 with shading to the right. The solution in interval notation is negative 4 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_319_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"q is greater than or equal to negative 4. The solution on the number line has a left parenthesis at negative 4 with shading to the right. The solution in interval notation is negative 4 to infinity within parentheses.\" \/><\/span>\r\n<p id=\"fs-id1167834190452\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" data-alt=\"k is greater than or equal to negative 180. The solution on the number line has a left bracket at negative 180 with shading to the right. The solution in interval notation is negative 180 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_320_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"k is greater than or equal to negative 180. The solution on the number line has a left bracket at negative 180 with shading to the right. The solution in interval notation is negative 180 to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835510891\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167828447049\">\r\n<div data-type=\"problem\" id=\"fs-id1167828447052\">\r\n<p id=\"fs-id1167835327276\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\r\n<p id=\"fs-id1171789604716\"><span class=\"token\">\u24d0<\/span>\\(-7r\\le \\text{\u200b}-70\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{u}{-4}\\ge -16.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835479773\">\r\n<p id=\"fs-id1167835167078\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835239491\" data-alt=\"r is greater than or equal to 10. The solution on the number line has a left bracket at 10 with shading to the right. The solution in interval notation is 10 to infinity within a bracket and parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_321_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"r is greater than or equal to 10. The solution on the number line has a left bracket at 10 with shading to the right. The solution in interval notation is 10 to infinity within a bracket and parenthesis.\" \/><\/span>\r\n<p id=\"fs-id1167832153412\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834494894\" data-alt=\"u is less than or equal to 64. The solution on the number line has a right bracket at 64 with shading to the left. The solution in interval notation is negative infinity to 64 within parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_322_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"u is less than or equal to 64. The solution on the number line has a right bracket at 64 with shading to the left. The solution in interval notation is negative infinity to 64 within parenthesis and a bracket.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834058969\">Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but make sure to pay close attention when we multiply or divide to isolate the variable.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835324646\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835324648\">\r\n<div data-type=\"problem\" id=\"fs-id1167834505676\">\r\n<p id=\"fs-id1167834505679\">Solve the inequality \\(6y\\le 11y+17,\\) graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835363956\">\r\n<table id=\"fs-id1167835338034\" class=\"unnumbered unstyled\" summary=\"6 y is less than or equal to 11 y plus 17. Subtract 11 y from both sides to collect the variables on the left. 6 y minus 11 y is less than or equal to 11 y minus 11 y plus 17. Simplify. Negative 5 y is less than or equal to 17. Divided both sides of the inequality by negative 5 and reverse the inequality. Negative 5 y is less than negative 5 is greater than or equal to 17 divided by negative 5. Simplify. y is greater than or equal to 17 divided by negative 5. Graph the solution on the number line with a left bracket at negative 17 divided by 5 with shading to the right. Write the solution in interval notation. It is negative 17 divided by 5 to infinity within brackets.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830924155\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Subtract \\(11y\\) from both sides to collect\r\n<div data-type=\"newline\"><\/div>\r\nthe variables on the left.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834123782\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835334058\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Divide both sides of the inequality by \\(-5,\\)\r\n<div data-type=\"newline\"><\/div>\r\nand reverse the inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826804444\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167828411035\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831871398\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834505663\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835321328\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832153759\">\r\n<div data-type=\"problem\" id=\"fs-id1167832153761\">\r\n<p id=\"fs-id1167832153764\">Solve the inequality, graph the solution on the number line, and write the solution in interval notation: \\(3q\\ge 7q-23.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834346527\"><span data-type=\"media\" id=\"fs-id1167834346529\" data-alt=\"q is less than or equal to 23 divided by 4. The solution on the number line has a right bracket at 23 divided by 4 with shading to the left. The solution in interval notation is negative infinity to 23 divided by 4 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_323_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"q is less than or equal to 23 divided by 4. The solution on the number line has a right bracket at 23 divided by 4 with shading to the left. The solution in interval notation is negative infinity to 23 divided by 4 within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835362248\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835416783\">\r\n<div data-type=\"problem\" id=\"fs-id1167835416786\">\r\n<p id=\"fs-id1167835416788\">Solve the inequality, graph the solution on the number line, and write the solution in interval notation: \\(6x&lt;10x+19.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834531312\"><span data-type=\"media\" id=\"fs-id1167834531314\" data-alt=\"x is greater than negative 19 divided by 4. The solution on the number line has a left parenthesis at negative 19 divided by 4 with shading to the right. The solution in interval notation is negative 19 divided by 4 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_324_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is greater than negative 19 divided by 4. The solution on the number line has a left parenthesis at negative 19 divided by 4 with shading to the right. The solution in interval notation is negative 19 divided by 4 to infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834396533\">When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don\u2019t have to multiply or divide by a negative\u2014which means we don\u2019t have to remember to reverse the inequality sign.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167834184909\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834382372\">\r\n<div data-type=\"problem\" id=\"fs-id1167834382374\">\r\n<p id=\"fs-id1167834382376\">Solve the inequality \\(8p+3\\left(p-12\\right)&gt;7p-28,\\) graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791694265\">\r\n<table id=\"fs-id1167835356650\" class=\"unnumbered unstyled can-break\" style=\"height: 446px\" summary=\"8 p plus 3 times the quantity p minus 12 is greater tgab 7 p minus 28. Simplify each side as much as possible. Distribute. The result is 8 p plus 3 p minus 36 is greater than 7 p minus 28. Combine like terms. The result is 11 p minus 36 is greater than 7 p minus 28. Subtract 7 p from both sides to collect the variables on the left, since 11 is greater than 7. Simplify. 4 p minus 36 is greater than negative 28. Add 36 to both sides to collect the constants on the right. 4 p minus 36 plus 36 is greater than negative 28 plus 36. Simplify. 4 p is greater than 8. Divided both sides of the inequality by 4. The inequality stays the same. 4 p divided by 4 is greater than 8 divided by 4. Simplify. p is greater than 2. Graph the solution on the number line with a left paarenthesis at 2 with shading to the right. Write the solution in interval notation. It 2 to infinity within parentheses.\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\"><\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(8p+3\\left(p-12\\right)&gt;7p-28\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify each side as much as possible.<\/td>\r\n<td style=\"height: 15px;width: 367px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Distribute.<\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.7em}{0ex}}8p+3p-36&gt;7p-28\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Combine like terms.<\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2.4em}{0ex}}11p-36&gt;7p-28\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 106px\">\r\n<td style=\"height: 106px;width: 237.533px\" data-align=\"left\">Subtract \\(7p\\) from both sides to collect the\r\n<div data-type=\"newline\"><\/div>\r\nvariables on the left, since \\(11&gt;7.\\)<\/td>\r\n<td style=\"height: 106px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.16em}{0ex}}11p-36-7p&gt;7p-28-7p\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify.<\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2.93em}{0ex}}4p-36&gt;-28\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 90px\">\r\n<td style=\"height: 90px;width: 237.533px\" data-align=\"left\">Add 36 to both sides to collect the\r\n<div data-type=\"newline\"><\/div>\r\nconstants on the right.<\/td>\r\n<td style=\"height: 90px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.7em}{0ex}}4p-36+36&gt;-28+36\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify.<\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{5em}{0ex}}4p&gt;8\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 90px\">\r\n<td style=\"height: 90px;width: 237.533px\" data-align=\"left\">Divide both sides of the inequality by\r\n<div data-type=\"newline\"><\/div>\r\n4; the inequality stays the same.<\/td>\r\n<td style=\"height: 90px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4.9em}{0ex}}\\frac{4p}{4}&gt;\\text{\u200b}\\frac{8}{4}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify.<\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{5.55em}{0ex}}p&gt;2\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 40px\">\r\n<td style=\"height: 40px;width: 237.533px\" data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td style=\"height: 40px;width: 367px\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345850\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_020a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"center\">\\(\\left(2,\\infty \\right)\\phantom{\\rule{0.5em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835172025\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834422978\">\r\n<div data-type=\"problem\" id=\"fs-id1167834422980\">\r\n<p id=\"fs-id1167834422982\">Solve the inequality \\(9y+2\\left(y+6\\right)&gt;5y-24\\), graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830700828\"><span data-type=\"media\" id=\"fs-id1167830700832\" data-alt=\"y is greater than negative 6. The solution on the number line has a left parenthesis at negative 6 with shading to the right. The solution in interval notation is negative 6 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_325_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"y is greater than negative 6. The solution on the number line has a left parenthesis at negative 6 with shading to the right. The solution in interval notation is negative 6 to infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831920268\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831920272\">\r\n<div data-type=\"problem\" id=\"fs-id1167831920274\">\r\n<p id=\"fs-id1167831920276\">Solve the inequality \\(6u+8\\left(u-1\\right)&gt;10u+32\\), graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835367653\"><span data-type=\"media\" id=\"fs-id1167835367656\" data-alt=\"u is greater than negative 10. The solution on the number line has a left parenthesis at 10 with shading to the right. The solution in interval notation is 10 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_326_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"u is greater than negative 10. The solution on the number line has a left parenthesis at 10 with shading to the right. The solution in interval notation is 10 to infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167831894185\">Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167831872446\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831872448\">\r\n<div data-type=\"problem\" id=\"fs-id1167831872451\">\r\n<p id=\"fs-id1167834060914\">Solve the inequality \\(8x-2\\left(5-x\\right)&lt;4\\left(x+9\\right)+6x,\\) graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791366521\">\r\n<table id=\"fs-id1167835595614\" class=\"unnumbered unstyled can-break\" summary=\"Simplify each side of 8 x minus 2 times the quantity 5 minus x is less than the product of 4 and the quantity x plus 9 plus 6 x. Distribute. 8 x minus 10 plus 2 x is less than 4 x plus 36 plus 6 x. Combine like terms. 10 x minus 10 is less than 10 x plus 36. Subtract 10 x from both sides to collect the variables on the left. 10 x minus 10 minus 10 x is less than 10 x plus 36 minus 10 x. Simplify. The result is negative 10 is less than 36. The x\u2019s are gone, and we havea true inequality. The inequality is an identity. Graph the solution all real numbers on the number line. The number line is shaded for all values. Write the solution in interval notation. It is negative infinity to infinity within parentheses.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Simplify each side as much as possible.<\/td>\r\n<td style=\"width: 183px\">\\(\\phantom{\\rule{4em}{0ex}}\\)<\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">\\(8x-2\\left(5-x\\right)&lt;4\\left(x+9\\right)+6x\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Distribute.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">\\(8x-10+2x&lt;4x+36+6x\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Combine like terms.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">\\(10x-10&lt;10x+36\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Subtract 10<em data-effect=\"italics\">x<\/em> from both sides to collect\r\n<div data-type=\"newline\"><\/div>\r\nthe variables on the left.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">\\(10x-10-10x&lt;10x+36-10x\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Simplify.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">\\(-10&lt;36\\phantom{\\rule{.5em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>\u2019s are gone, and we have a true\r\n<div data-type=\"newline\"><\/div>\r\nstatement.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">The inequality is an identity.\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is all real numbers.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835639989\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_021a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184px\" data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td style=\"width: 183px\"><\/td>\r\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832076309\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832076312\">\r\n<div data-type=\"problem\" id=\"fs-id1167835347614\">\r\n<p id=\"fs-id1167835347616\">Solve the inequality \\(4b-3\\left(3-b\\right)&gt;5\\left(b-6\\right)+2b\\), graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835343911\"><span data-type=\"media\" id=\"fs-id1167835343914\" data-alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_327_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835343967\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831025106\">\r\n<div data-type=\"problem\" id=\"fs-id1167831025108\">\r\n<p id=\"fs-id1167831025111\">Solve the inequality \\(9h-7\\left(2-h\\right)&lt;8\\left(h+11\\right)+8h\\), graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834059504\"><span data-type=\"media\" id=\"fs-id1167834059508\" data-alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_328_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1166401420698\">We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835609996\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835609999\">\r\n<div data-type=\"problem\" id=\"fs-id1167835610001\">\r\n<p id=\"fs-id1167835610003\">Solve the inequality \\(\\frac{1}{3}a-\\frac{1}{8}a&gt;\\frac{5}{24}a\\text{\u200b}+\\frac{3}{4},\\) graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835345971\">\r\n<table id=\"fs-id1167835345973\" class=\"unnumbered unstyled can-break\" summary=\"One-third a minus one-eighth a is greater than five twenty-fourths a plus three-fourths. Multiply both sides by the L C D, 24, to clear the fractions. 24 times the quantity One-third a minus one-eighth a is greater than 24 times the quantity five twenty-fourths a plus three-fourth. Simplify. 8 a minus 3 a is greater than 5 a plus 18. Combine like terms. Subtract 5 a from both sides to collect the variables on the left. 5 a minus 5 a is greater than 5 a minus 5 a plus 18. Simplify. 0 is less than 18. The statement is false. The inequality is a contradiction. There is no solution. As a result, there is no graph on the number line or interval notation.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834213893\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Multiply both sides by the LCD, 24,\r\n<div data-type=\"newline\"><\/div>\r\nto clear the fractions.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835423366\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830963400\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Combine like terms.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834429828\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Subtract \\(5a\\) from both sides to collect the\r\n<div data-type=\"newline\"><\/div>\r\nvariables on the left.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835368315\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830704524\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">The statement is false.<\/td>\r\n<td data-align=\"left\">The inequality is a contradiction.\r\n<div data-type=\"newline\"><\/div>\r\nThere is no solution.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832041553\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\r\n<td data-align=\"left\">There is no solution.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835345876\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835345879\">\r\n<div data-type=\"problem\" id=\"fs-id1167835345881\">\r\n<p id=\"fs-id1167834473392\">Solve the inequality \\(\\frac{1}{4}x-\\frac{1}{12}x&gt;\\frac{1}{6}x+\\frac{7}{8}\\), graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834097799\"><span data-type=\"media\" id=\"fs-id1167834097803\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_329_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834186297\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834186300\">\r\n<div data-type=\"problem\" id=\"fs-id1167835422250\">\r\n<p id=\"fs-id1167835422252\">Solve the inequality \\(\\frac{2}{5}z-\\frac{1}{3}z&lt;\\frac{1}{15}z\\text{\u200b}-\\frac{3}{5}\\), graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834394525\"><span data-type=\"media\" id=\"fs-id1167834196537\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_330_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831896625\">\r\n<h3 data-type=\"title\">Translate to an Inequality and Solve<\/h3>\r\n<p id=\"fs-id1167831896630\">To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like \u201cmore than\u201d and \u201cless than.\u201d But others are not as obvious. <a href=\"#fs-id1167826983841\" class=\"autogenerated-content\">(Figure)<\/a> shows some common phrases that indicate inequalities.<\/p>\r\n\r\n<table id=\"fs-id1167826983841\" summary=\"This table has four columns and two rows. The first row is a header row and it labels each column, \u201cgreater than,\u201d \u201cgreater than or equal to,\u201d and \u201cless than,\u201d and \u201cless than or equal to.\u201d In column 1, the phrases are \u201cis greater than,\u201d \u201cis more than,\u201d \u201cis larget than,\u201d and \u201cexceeds.\u201d. In column two, the phrases are \u201cis greater than or equal to,\u201d \u201cis at least,\u201d \u201cis no less than,\u201d and \u201cis the minimum.\u201d In column 3, the phrases are \u201cis less than,\u201d \u201cis smaller than,\u201d \u201chas fewer than,\u201d and \u201cis lower than.\u201d In column 4 the phrases are \u201cis less than or equal to,\u201d \u201cis at most,\u201d \u201cis no more than,\u201d and \u201cis the maximum.\u201d\" data-label=\"\">\r\n<thead>\r\n<tr>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(&gt;\\)<\/th>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(\\ge \\)<\/th>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(&lt;\\)<\/th>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(\\le \\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"middle\" data-align=\"left\">is greater than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis more than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis larger than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nexceeds<\/td>\r\n<td data-valign=\"middle\" data-align=\"left\">is greater than or equal to\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis at least\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis no less than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis the minimum<\/td>\r\n<td data-valign=\"middle\" data-align=\"left\">is less than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis smaller than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nhas fewer than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis lower than<\/td>\r\n<td data-valign=\"middle\" data-align=\"left\">is less than or equal to\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis at most\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis no more than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis the maximum<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-type=\"example\" id=\"fs-id1167835330450\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835330452\">\r\n<div data-type=\"problem\" id=\"fs-id1167835330454\">\r\n<p id=\"fs-id1167835328637\">Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167834539042\" class=\"unnumbered\" data-label=\"\">\\(\\text{Twenty-seven less than}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{is at least 48.}\\)<\/div>\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835326571\">\r\n<table id=\"fs-id1167835326573\" class=\"unnumbered unstyled\" summary=\"or equal to 48. Solve by adding 27 to both sides. x minus 27 plus 27 is greater than or equal to 48 plus 27. Simplify. The result is x is greater than or equal to 75. Graph the solution on the number line with a left bracket at 75 with shading to the right. Write the solution in interval notation. It is 75 to infinity within a bracket and parenthesis.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835489337\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Translate.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830757362\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Solve\u2014add 27 to both sides.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835333039\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Simplify.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834061828\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Graph on the number line.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835345148\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">Write in interval notation.<\/td>\r\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832065810\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835229537\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835229541\">\r\n<div data-type=\"problem\" id=\"fs-id1167835229543\">\r\n<p id=\"fs-id1167835229545\">Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n<p id=\"fs-id1167831887724\">Nineteen less than <em data-effect=\"italics\">p<\/em> is no less than 47.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830703809\"><span data-type=\"media\" id=\"fs-id1167830703813\" data-alt=\"p minus 19 is greater than or equal to 47. Its solution is p is greater than or equal to 66. The solution on the number line has a left bracket at 66 with shading to the right. The solution in interval notation is 66 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_331_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"p minus 19 is greater than or equal to 47. Its solution is p is greater than or equal to 66. The solution on the number line has a left bracket at 66 with shading to the right. The solution in interval notation is 66 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826869988\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834132735\">\r\n<div data-type=\"problem\" id=\"fs-id1167834132737\">\r\n<p id=\"fs-id1167834132739\">Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n<p id=\"fs-id1167830699113\">Four more than <em data-effect=\"italics\">a<\/em> is at most 15.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830699122\"><span data-type=\"media\" id=\"fs-id1167832058199\" data-alt=\"a plus 4 is less than or equal to 15. Its solution is a is less than or equal to 11. The solution on the number line has a right bracket at 11with shading to the left. The solution in interval notation is negative infinity to 11 within a parenthesis and bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_332_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"a plus 4 is less than or equal to 15. Its solution is a is less than or equal to 11. The solution on the number line has a right bracket at 11with shading to the left. The solution in interval notation is negative infinity to 11 within a parenthesis and bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831893517\">\r\n<h3 data-type=\"title\">Solve Applications with Linear Inequalities<\/h3>\r\n<p id=\"fs-id1167834213968\">Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.<\/p>\r\n<p id=\"fs-id1167834213973\">We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.<\/p>\r\n<p id=\"fs-id1167835532881\">Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835532887\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835532889\">\r\n<div data-type=\"problem\" id=\"fs-id1167835623451\">\r\n<p id=\"fs-id1167835623453\">Dawn won a mini-grant of ?4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost ?254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171784074417\">\r\n<p id=\"fs-id1167834535668\">\\(\\begin{array}{cccc}\\mathbf{\\text{Step 1. Read}}\\phantom{\\rule{0.2em}{0ex}}\\text{the problem.}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2. Identify}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill &amp; &amp; &amp; \\text{the maximum number of tablets Dawn can buy}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 3. Name}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Choose a variable to represent that}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{quantity.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\text{Let}\\phantom{\\rule{0.2em}{0ex}}n=\\text{the number of tablets.}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 4. Translate}}\\text{. Write a sentence that gives the}\\hfill \\\\ \\text{information to find it.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Translate into an inequality.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{?254.12 times the number of tablets is}\\hfill \\\\ \\text{no more than ?4,000.}\\hfill \\\\ \\hfill \\phantom{\\rule{0.3em}{0ex}}254.12n\\le 4000\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 5. Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the inequality.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{But}\\phantom{\\rule{0.2em}{0ex}}n\\phantom{\\rule{0.2em}{0ex}}\\text{must be a whole number of}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{tablets, so round to 15.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\begin{array}{c}n\\le 15.74\\hfill \\\\ n\\le 15\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 6. Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer in the problem}\\hfill \\\\ \\text{and make sure it makes sense.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Rounding down the price to ?250, 15}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{tablets would cost ?3,750, while 16}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{tablets would be ?4,000. So a}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{maximum of 15 tablets at ?254.12}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{seems reasonable.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 7. Answer}}\\phantom{\\rule{0.2em}{0ex}}\\text{the question with a complete sentence.}\\hfill &amp; &amp; &amp; \\text{Dawn can buy a maximum of 15 tablets.}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831887544\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831887548\">\r\n<div data-type=\"problem\" id=\"fs-id1167831887551\">\r\n<p id=\"fs-id1167831887553\">Angie has ?20 to spend on juice boxes for her son\u2019s preschool picnic. Each pack of juice boxes costs ?2.63. What is the maximum number of packs she can buy?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835341023\">\r\n<p id=\"fs-id1167835341026\">Angie can buy 7 packs of juice.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826814163\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826814167\">\r\n<div data-type=\"problem\" id=\"fs-id1167826814169\">\r\n<p id=\"fs-id1167826814171\">Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost ?42.75 per person for dinner, including tip and tax. His budget for the party is ?500. What is the maximum number of people Daniel can have at the party?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831955653\">\r\n<p id=\"fs-id1167831955656\">Daniel can have 11 people at the party.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167834065806\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834065808\">\r\n<div data-type=\"problem\" id=\"fs-id1167834065810\">\r\n<p id=\"fs-id1167834065812\">Taleisha\u2019s phone plan costs her ?28.80 a month plus ?0.20 per text message. How many text messages can she send\/receive and keep her monthly phone bill no more than ?50?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791453866\">\r\n<p id=\"fs-id1167835173750\">\\(\\begin{array}{cccc}\\mathbf{\\text{Step 1. Read}}\\phantom{\\rule{0.2em}{0ex}}\\text{the problem.}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2. Identify}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill &amp; &amp; &amp; \\text{the number of text messages Taleisha can make}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 3. Name}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Choose a variable to represent that}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{quantity.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\text{Let}\\phantom{\\rule{0.2em}{0ex}}t=\\text{the number of text messages.}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 4. Translate}}\\phantom{\\rule{0.2em}{0ex}}\\text{Write a sentence that}\\hfill \\\\ \\text{gives the information to find it.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Translate into an inequality.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{?28.80 plus ?0.20 times the number of}\\hfill \\\\ \\text{text messages is less than or equal to ?50.}\\hfill \\\\ 28.80+0.20t\\le 50\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 5. Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the inequality.}\\hfill \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{4em}{0ex}}0.2t\\le 21.2\\hfill \\\\ \\phantom{\\rule{5.3em}{0ex}}t\\le 106\\phantom{\\rule{0.2em}{0ex}}\\text{text messages}\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 6. Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer in the problem}\\hfill \\\\ \\text{and make sure it makes sense.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Yes,}\\phantom{\\rule{0.2em}{0ex}}28.80+0.20\\left(106\\right)=50.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\mathbf{\\text{Step 7. Write}}\\phantom{\\rule{0.2em}{0ex}}\\text{a sentence that answers the question.}\\hfill \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Taleisha can send\/receive no more than}\\hfill \\\\ \\text{106 text messages to keep her bill no}\\hfill \\\\ \\text{more than ?50.}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167828420745\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167828420749\">\r\n<div data-type=\"problem\" id=\"fs-id1167828420751\">\r\n<p id=\"fs-id1167828420753\">Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges ?75 a week plus ?0.25 a mile. How many miles can they travel during the week and still keep within their ?200 budget?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835194797\">\r\n\r\nSergio and Lizeth can travel no more than 500 miles.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834300216\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834300220\">\r\n<div data-type=\"problem\" id=\"fs-id1167834300222\">\r\n<p id=\"fs-id1167834300224\">Rameen\u2019s heating bill is ?5.42 per month plus ?1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of ?87.50.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831191514\">\r\n<p id=\"fs-id1167831191516\">Rameen can use no more than 76 therms.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167826814017\">Profit is the money that remains when the costs have been subtracted from the revenue. In the next example, we will find the number of jobs a small businesswoman needs to do every month in order to make a certain amount of profit.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167826814022\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826814024\">\r\n<div data-type=\"problem\" id=\"fs-id1167835352861\">\r\n<p id=\"fs-id1167835352864\">Felicity has a calligraphy business. She charges ?2.50 per wedding invitation. Her monthly expenses are ?650. How many invitations must she write to earn a profit of at least ?2,800 per month?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791497893\">\r\n<p id=\"fs-id1167826940798\">\\(\\begin{array}{cccc}\\mathbf{\\text{Step 1. Read}}\\phantom{\\rule{0.2em}{0ex}}\\text{the problem.}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2. Identify}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill &amp; &amp; &amp; \\text{the number of invitations Felicity needs to write}\\hfill \\\\ \\mathbf{\\text{Step 3. Name}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill &amp; &amp; &amp; \\text{Let}\\phantom{\\rule{0.2em}{0ex}}j=\\text{the number of invitations.}\\hfill \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Choose a variable to represent it.}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\mathbf{\\text{Step 4. Translate.}}\\phantom{\\rule{0.2em}{0ex}}\\text{Write a sentence that}\\hfill \\\\ \\text{gives the information to find it.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{Translate into an inequality.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{?2.50 times the number of invitations}\\hfill \\\\ \\text{minus ?650 is at least ?2,800.}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}2.50j-650\\ge 2,800\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 5. Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the inequality.}\\hfill \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{4.2em}{0ex}}2.5j\\ge 3,450\\hfill \\\\ \\phantom{\\rule{5.4em}{0ex}}j\\ge 1,380\\phantom{\\rule{0.2em}{0ex}}\\text{invitations}\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 6. Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer in the problem}\\hfill \\\\ \\text{and make sure it makes sense.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{1em}{0ex}}\\text{If Felicity wrote 1400 invitations, her}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{profit would be}\\phantom{\\rule{0.2em}{0ex}}2.50\\left(1400\\right)-650,\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{?2,850. This is more than ?2800.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 7. Write}}\\phantom{\\rule{0.2em}{0ex}}\\text{a sentence that answers the question.}\\hfill &amp; &amp; &amp; \\text{Felicity must write at least 1,380 invitations.}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826997011\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826997015\">\r\n<div data-type=\"problem\" id=\"fs-id1167826997017\">\r\n<p id=\"fs-id1167835370722\">Caleb has a pet sitting business. He charges ?32 per hour. His monthly expenses are ?2,272. How many hours must he work in order to earn a profit of at least ?800 per month?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835370728\">\r\n<p id=\"fs-id1167835370730\">Caleb must work at least 96 hours.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832076580\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832076584\">\r\n<div data-type=\"problem\" id=\"fs-id1167832076586\">\r\n<p id=\"fs-id1167832015429\">Elliot has a landscape maintenance business. His monthly expenses are ?1,100. If he charges ?60 per job, how many jobs must he do to earn a profit of at least ?4,000 a month?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832015435\">\r\n<p id=\"fs-id1167832015437\">Elliot must work at least 85 jobs.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167827988113\">There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167827988118\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834099288\">\r\n<div data-type=\"problem\" id=\"fs-id1167834099290\">\r\n<p id=\"fs-id1167834099292\">Malik is planning a six-day summer vacation trip. He has ?840 in savings, and he earns ?45 per hour for tutoring. The trip will cost him ?525 for airfare, ?780 for food and sightseeing, and ?95 per night for the hotel. How many hours must he tutor to have enough money to pay for the trip?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791551409\">\r\n<p id=\"fs-id1167834194392\">\\(\\begin{array}{cccc}\\mathbf{\\text{Step 1. Read}}\\phantom{\\rule{0.2em}{0ex}}\\text{the problem.}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2. Identify}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill &amp; &amp; &amp; \\text{the number of hours Malik must tutor}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 3. Name}}\\phantom{\\rule{0.2em}{0ex}}\\text{what you are looking for.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{Choose a variable to represent that}\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{quantity.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\text{Let}\\phantom{\\rule{0.2em}{0ex}}h=\\text{the number of hours.}\\hfill \\\\ \\begin{array}{c}\\begin{array}{c}\\mathbf{\\text{Step 4. Translate.}}\\phantom{\\rule{0.2em}{0ex}}\\text{Write a sentence that}\\hfill \\\\ \\text{gives the information to find it.}\\hfill \\end{array}\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{The expenses must be less than or equal to}\\hfill \\\\ \\text{the income. The cost of airfare plus the}\\hfill \\\\ \\text{cost of food and sightseeing and the hotel}\\hfill \\\\ \\text{bill must be less than the savings plus the}\\hfill \\\\ \\text{amount earned tutoring.}\\hfill \\end{array}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{Translate into an inequality.}\\hfill &amp; &amp; &amp; 525+780+95\\left(6\\right)\\le 840+45h\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 5. Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the inequality.}\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\begin{array}{c}1,875\\le 840+45h\\hfill \\\\ 1,035\\le 45h\\hfill \\\\ \\phantom{\\rule{1.3em}{0ex}}23\\le h\\hfill \\\\ \\phantom{\\rule{1.8em}{0ex}}h\\ge 23\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 6. Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer in the problem}\\hfill \\\\ \\text{and make sure it makes sense.}\\hfill \\\\ \\text{We substitute 23 into the inequality.}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}1,875\\le 840+45h\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}1,875\\le 840+45\\left(23\\right)\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}1,875\\le 1875\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 7. Write}}\\phantom{\\rule{0.2em}{0ex}}\\text{a sentence that answers the question.}\\hfill &amp; &amp; &amp; \\text{Malik must tutor at least 23 hours.}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835370593\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835370597\">\r\n<div data-type=\"problem\" id=\"fs-id1167835370599\">\r\n<p id=\"fs-id1167835370602\">Brenda\u2019s best friend is having a destination wedding and the event will last three days. Brenda has ?500 in savings and can earn ?15 an hour babysitting. She expects to pay ?350 airfare, ?375 for food and entertainment and ?60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835327683\">\r\n<p id=\"fs-id1167831921533\">Brenda must babysit at least 27 hours.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831921539\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831921544\">\r\n<div data-type=\"problem\" id=\"fs-id1167831239022\">\r\n<p id=\"fs-id1167831239025\">Josue wants to go on a 10-night road trip with friends next spring. It will cost him ?180 for gas, ?450 for food, and ?49 per night to share a motel room. He has ?520 in savings and can earn ?30 per driveway shoveling snow. How many driveways must he shovel to have enough money to pay for the trip?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831239032\">\r\n<p id=\"fs-id1167835377790\">Josue must shovel at least 20 driveways.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835377798\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1167834060088\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Inequalities, Number Lines, and Interval Notation<\/strong>\r\n<div data-type=\"newline\"><\/div>\r\n\\(x&gt;a\\phantom{\\rule{3em}{0ex}}x\\ge a\\phantom{\\rule{3em}{0ex}}x&lt;a\\phantom{\\rule{3em}{0ex}}x\\le a\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835342769\" data-alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_024_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\" \/><\/span><\/li>\r\n \t<li><strong data-effect=\"bold\">Linear Inequality<\/strong>\r\n<ul id=\"fs-id1167831920234\" data-bullet-style=\"open-circle\">\r\n \t<li>A <strong data-effect=\"bold\">linear inequality<\/strong> is an inequality in one variable that can be written in one of the following forms where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and \\(a\\ne 0:\\)\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"equation\" id=\"fs-id1167831884342\" class=\"unnumbered\" data-label=\"\">\\(ax+b&lt;c,\\phantom{\\rule{3em}{0ex}}ax+b\\le c,\\phantom{\\rule{3em}{0ex}}ax+b&gt;c,\\phantom{\\rule{3em}{0ex}}ax+b\\ge c.\\)<\/div><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Addition and Subtraction Property of Inequality<\/strong>\r\n<ul id=\"fs-id1167831922537\" data-bullet-style=\"open-circle\">\r\n \t<li>For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>, if \\(a&lt;b,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"equation\" id=\"fs-id1167834194116\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}a+c&lt;b+c\\hfill &amp; &amp; &amp; a-c&lt;b-c\\hfill \\\\ a+c&gt;b+c\\hfill &amp; &amp; &amp; a-c&gt;b-c\\hfill \\end{array}\\)<\/div><\/li>\r\n \t<li>We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Multiplication and Division Property of Inequality<\/strong>\r\n<ul id=\"fs-id1167826997478\" data-bullet-style=\"open-circle\">\r\n \t<li>For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>,\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{c}\\text{multiply or divide by a}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{positive}}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&lt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&gt;0,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&lt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&lt;\\frac{b}{c}.\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&gt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&gt;0,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&gt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&gt;\\frac{b}{c}.\\hfill \\\\ \\text{multiply or divide by a}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{negative}}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&lt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&lt;0,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&gt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&gt;\\frac{b}{c}.\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{if}\\phantom{\\rule{0.2em}{0ex}}a&gt;b\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c&lt;0,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}ac&lt;bc\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{c}&lt;\\frac{b}{c}.\\hfill \\end{array}\\)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Phrases that indicate inequalities<\/strong>\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167834516117\" class=\"unnumbered\" summary=\"This table has four columns and two rows. The first row is a header row and it labels each column, \u201cgreater than,\u201d \u201cgreater than or equal to,\u201d and \u201cless than,\u201d and \u201cless than or equal to.\u201d In column 1, the phrases are \u201cis greater than,\u201d \u201cis more than,\u201d \u201cis larget than,\u201d and \u201cexceeds.\u201d. In column two, the phrases are \u201cis greater than or equal to,\u201d \u201cis at least,\u201d \u201cis no less than,\u201d and \u201cis the minimum.\u201d In column 3, the phrases are \u201cis less than,\u201d \u201cis smaller than,\u201d \u201chas fewer than,\u201d and \u201cis lower than.\u201d In column 4 the phrases are \u201cis less than or equal to,\u201d \u201cis at most,\u201d \u201cis no more than,\u201d and \u201cis the maximum.\u201d\" data-label=\"\">\r\n<thead>\r\n<tr>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(&gt;\\)<\/th>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(\\ge \\)<\/th>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(&lt;\\)<\/th>\r\n<th data-valign=\"middle\" data-align=\"center\">\\(\\le \\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"middle\" data-align=\"left\">is greater than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis more than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis larger than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nexceeds<\/td>\r\n<td data-valign=\"middle\" data-align=\"left\">is greater than or equal to\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis at least\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis no less than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis the minimum<\/td>\r\n<td data-valign=\"middle\" data-align=\"left\">is less than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis smaller than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nhas fewer than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis lower than<\/td>\r\n<td data-valign=\"middle\" data-align=\"left\">is less than or equal to\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis at most\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis no more than\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nis the maximum<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832055404\">\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167832055407\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<p id=\"fs-id1167832055414\"><strong data-effect=\"bold\">Graph Inequalities on the Number Line<\/strong><\/p>\r\n<p id=\"fs-id1167832059729\">In the following exercises, graph each inequality on the number line and write in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167832059733\">\r\n<div data-type=\"problem\" id=\"fs-id1167832059735\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(x&lt;-2\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(x\\ge -3.5\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(x\\le \\frac{2}{3}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832058612\">\r\n<div data-type=\"problem\" id=\"fs-id1167832058614\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(x&gt;3\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(x\\le -0.5\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(x\\ge \\frac{1}{3}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832152484\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831239691\" data-alt=\"The solution for x is greater than 3 on a number line has a left bracket 3 with shading to the right. The solution in interval notation is 3 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_336_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is greater than 3 on a number line has a left bracket 3 with shading to the right. The solution in interval notation is 3 to infinity within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834130205\" data-alt=\"The solution for x is less than or equal to negative 0.5 on a number line has a right bracket at negative 0.5 with shading to the left. The solution in interval notation is negative infinity to negative 0.5 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_337_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is less than or equal to negative 0.5 on a number line has a right bracket at negative 0.5 with shading to the left. The solution in interval notation is negative infinity to negative 0.5 within a parenthesis and a bracket.\" \/><\/span>\r\n\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835306019\" data-alt=\"The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_338_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835318324\">\r\n<div data-type=\"problem\" id=\"fs-id1167835318326\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(x\\ge -4\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(x&lt;2.5\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(x&gt;-\\frac{3}{2}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835415275\">\r\n<div data-type=\"problem\" id=\"fs-id1167835415277\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(x\\le 5\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(x\\ge -1.5\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(x&lt;-\\frac{7}{3}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834501202\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835366746\" data-alt=\"The solution for x is less than or equal to 5 on a number line has a right bracket with shading to the left. The solution in interval notation is negative infinity to 5 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_342_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is less than or equal to 5 on a number line has a right bracket with shading to the left. The solution in interval notation is negative infinity to 5 within a parenthesis and a bracket.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831106373\" data-alt=\"The solution for x is greater than or equal to negative 1.5 on a number line has a left bracket with shading to the right. The solution in interval notation is negative 1.5 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_343_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is greater than or equal to negative 1.5 on a number line has a left bracket with shading to the right. The solution in interval notation is negative 1.5 to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834133022\" data-alt=\"The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_344_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834505878\">\r\n<div data-type=\"problem\" id=\"fs-id1167834505880\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-5&lt;x&lt;2\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(-3\\le x&lt;-1\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(0\\le x\\le 1.5\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834097969\">\r\n<div data-type=\"problem\" id=\"fs-id1167834097971\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-2&lt;x&lt;0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(-5\\le x&lt;-3\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(0\\le x\\le 3.5\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834523866\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834064750\" data-alt=\"Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_348_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834238997\" data-alt=\"Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_349_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832053454\" data-alt=\"0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_350_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834279403\">\r\n<div data-type=\"problem\" id=\"fs-id1167834279405\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-1&lt;x&lt;3\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(-3&lt;x\\le -2\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(-1.25\\le x\\le 0\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835387177\">\r\n<div data-type=\"problem\" id=\"fs-id1167835387179\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-4&lt;x&lt;2\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(-5&lt;x\\le -2\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(-3.75\\le x\\le 0\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834583310\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834408545\" data-alt=\"Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_354_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835376533\" data-alt=\"Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_355_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834064673\" data-alt=\"Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_356_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835311117\"><strong data-effect=\"bold\">Solve Linear Inequalities<\/strong><\/p>\r\n<p id=\"fs-id1167835311123\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167831922822\">\r\n<div data-type=\"problem\" id=\"fs-id1167831922824\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(a+\\frac{3}{4}\\ge \\frac{7}{10}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(8x&gt;72\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(20&gt;\\frac{2}{5}h\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835282992\">\r\n<div data-type=\"problem\" id=\"fs-id1167835229616\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(b+\\frac{7}{8}\\ge \\frac{1}{6}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(6y&lt;48\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(40&lt;\\frac{5}{8}k\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835363560\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831239561\" data-alt=\"The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_360_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834120655\" data-alt=\"The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_361_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831882629\" data-alt=\"The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_362_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835308740\">\r\n<div data-type=\"problem\" id=\"fs-id1167835308742\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(f-\\frac{13}{20}&lt;-\\frac{5}{12}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(9t\\ge -27\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(\\frac{7}{6}j\\ge 42\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834185458\">\r\n<div data-type=\"problem\" id=\"fs-id1167834372403\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(g-\\frac{11}{12}&lt;-\\frac{5}{18}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(7s&lt;-28\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(\\frac{9}{4}g\\le 36\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834533585\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832015591\" data-alt=\"The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_366_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834134703\" data-alt=\"The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_367_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835306108\" data-alt=\"The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_368_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835234273\">\r\n<div data-type=\"problem\" id=\"fs-id1167835234276\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-5u\\ge 65\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\frac{a}{-3}\\le 9\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831846739\">\r\n<div data-type=\"problem\" id=\"fs-id1167831846741\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-8v\\le 96\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\frac{b}{-10}\\ge 30\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834064605\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167831890647\" data-alt=\"The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_371_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835330571\" data-alt=\"The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_372_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832226771\">\r\n<div data-type=\"problem\" id=\"fs-id1167832226773\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-9c&lt;126\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(-25&lt;\\frac{p}{-5}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835517775\">\r\n<div data-type=\"problem\" id=\"fs-id1167835517777\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(-7d&gt;105\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(-18&gt;\\frac{q}{-6}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835305580\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167830699874\" data-alt=\"The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_375_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834395948\" data-alt=\"The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_376_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835357194\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167835357198\">\r\n<div data-type=\"problem\" id=\"fs-id1167835357200\">\r\n<p id=\"fs-id1167835433698\">\\(4v\\ge 9v-40\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835512207\">\r\n<div data-type=\"problem\" id=\"fs-id1167835512210\">\r\n<p id=\"fs-id1167835512212\">\\(5u\\le 8u-21\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835329406\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835329410\" data-alt=\"The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_378_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167835575754\">\r\n<p id=\"fs-id1167835575756\">\\(13q&lt;7q-29\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835512806\">\r\n<div data-type=\"problem\" id=\"fs-id1167835512808\">\r\n<p id=\"fs-id1167835512810\">\\(9p&gt;14p-18\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835302412\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834132718\" data-alt=\"The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_380_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831115257\">\r\n<div data-type=\"problem\" id=\"fs-id1167831115259\">\r\n<p id=\"fs-id1167831115261\">\\(12x+3\\left(x+7\\right)&gt;10x-24\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826849467\">\r\n<div data-type=\"problem\" id=\"fs-id1167826849469\">\r\n<p id=\"fs-id1167826849471\">\\(9y+5\\left(y+3\\right)&lt;4y-35\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826781047\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167826781051\" data-alt=\"The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_382_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835479515\">\r\n<div data-type=\"problem\" id=\"fs-id1167835479517\">\r\n<p id=\"fs-id1167835479519\">\\(6h-4\\left(h-1\\right)\\le 7h-11\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835361714\">\r\n<div data-type=\"problem\" id=\"fs-id1167835361716\">\r\n<p id=\"fs-id1167835362972\">\\(4k-\\left(k-2\\right)\\ge 7k-26\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830837665\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834428919\" data-alt=\"The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_384_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832151015\">\r\n<div data-type=\"problem\" id=\"fs-id1167832151017\">\r\n<p id=\"fs-id1167832151020\">\\(8m-2\\left(14-m\\right)\\ge \\text{\u200b}7\\left(m-4\\right)+3m\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831832843\">\r\n<div data-type=\"problem\" id=\"fs-id1167831832845\">\r\n<p id=\"fs-id1167831832848\">\\(6n-12\\left(3-n\\right)\\le 9\\left(n-4\\right)+9n\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826997719\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167826997723\" data-alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_386_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834133774\">\r\n<div data-type=\"problem\" id=\"fs-id1167834094539\">\r\n<p id=\"fs-id1167834094542\">\\(\\frac{3}{4}b-\\frac{1}{3}b&lt;\\frac{5}{12}b-\\frac{1}{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835262304\">\r\n<div data-type=\"problem\" id=\"fs-id1167835262306\">\r\n<p id=\"fs-id1167831883461\">\\(9u+5\\left(2u-5\\right)\\ge 12\\left(u-1\\right)+7u\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835379518\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835379522\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_388_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835530439\">\r\n<div data-type=\"problem\" id=\"fs-id1167835530442\">\r\n<p id=\"fs-id1167835530444\">\\(\\frac{2}{3}g-\\frac{1}{2}\\left(g-14\\right)\\le \\frac{1}{6}\\left(g+42\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831117186\">\r\n<div data-type=\"problem\" id=\"fs-id1167831117188\">\r\n<p id=\"fs-id1167831117190\">\\(\\frac{4}{5}h-\\frac{2}{3}\\left(h-9\\right)\\ge \\frac{1}{15}\\left(2h+90\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834228156\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834228160\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_390_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834184410\">\r\n<div data-type=\"problem\" id=\"fs-id1167834184412\">\r\n<p id=\"fs-id1167834279140\">\\(\\frac{5}{6}a-\\frac{1}{4}a&gt;\\frac{7}{12}a+\\frac{2}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834228204\">\r\n<div data-type=\"problem\" id=\"fs-id1167834228206\">\r\n<p id=\"fs-id1167834228208\">\\(12v+3\\left(4v-1\\right)\\le 19\\left(v-2\\right)+5v\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835514619\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835514623\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_392_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835519277\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167835519281\">\r\n<div data-type=\"problem\" id=\"fs-id1167835519283\">\r\n<p id=\"fs-id1167831887178\">\\(15k\\le -40\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832214484\">\r\n<div data-type=\"problem\" id=\"fs-id1167835215265\">\r\n<p id=\"fs-id1167835215267\">\\(35k\\ge -77\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835186339\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835186343\" data-alt=\"The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_394_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834184690\">\r\n<div data-type=\"problem\" id=\"fs-id1167834184692\">\r\n<p id=\"fs-id1167835303474\">\\(23p-2\\left(6-5p\\right)&gt;3\\left(11p-4\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835622040\">\r\n<div data-type=\"problem\" id=\"fs-id1167835622042\">\r\n<p id=\"fs-id1167835622044\">\\(18q-4\\left(10-3q\\right)&lt;5\\left(6q-8\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834346562\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834346566\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_396_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826996576\">\r\n<div data-type=\"problem\" id=\"fs-id1167826996578\">\r\n<p id=\"fs-id1167834228852\">\\(-\\frac{9}{4}x\\ge -\\frac{5}{12}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835381056\">\r\n<div data-type=\"problem\" id=\"fs-id1167835381058\">\r\n<p id=\"fs-id1167835381060\">\\(-\\frac{21}{8}y\\le -\\frac{15}{28}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167828434538\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167828434543\" data-alt=\"The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_398_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835288340\">\r\n<div data-type=\"problem\" id=\"fs-id1167835288342\">\r\n<p id=\"fs-id1167835288344\">\\(c+34&lt;-99\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830836892\">\r\n<div data-type=\"problem\" id=\"fs-id1167832031145\">\r\n<p id=\"fs-id1167832031147\">\\(d+29&gt;-61\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832043391\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832043395\" data-alt=\"The solution is g is greater than negative 90. The solution on a number line has a left parenthesis at negative 90 with shading to the right. The solution in interval notation is negative 90 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_400_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is g is greater than negative 90. The solution on a number line has a left parenthesis at negative 90 with shading to the right. The solution in interval notation is negative 90 to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834252306\">\r\n<div data-type=\"problem\" id=\"fs-id1167834252309\">\r\n<p id=\"fs-id1167834252311\">\\(\\frac{m}{18}\\ge -4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835349143\">\r\n<div data-type=\"problem\" id=\"fs-id1167835349145\">\r\n<p id=\"fs-id1167835349147\">\\(\\frac{n}{13}\\le -6\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835517862\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835421713\" data-alt=\"The solution is n is less than or equal to negative 78. The solution on a number line has a right bracket at negative 78 with shading to the left. The solution in interval notation is negative infinity to negative 78 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_402_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is n is less than or equal to negative 78. The solution on a number line has a right bracket at negative 78 with shading to the left. The solution in interval notation is negative infinity to negative 78 within a parenthesis and a bracket.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167832053963\"><strong data-effect=\"bold\">Translate to an Inequality and Solve<\/strong><\/p>\r\n<p id=\"fs-id1167832053969\">In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167830961003\">\r\n<div data-type=\"problem\" id=\"fs-id1167830961005\">\r\n<p id=\"fs-id1167830961008\">Three more than <em data-effect=\"italics\">h<\/em> is no less than 25.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834431153\">\r\n<div data-type=\"problem\" id=\"fs-id1167831116560\">\r\n<p id=\"fs-id1167831116562\">Six more than <em data-effect=\"italics\">k<\/em> exceeds 25.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831116571\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835417487\" data-alt=\"The inequality is k plus 6 is greater than 25. Its solution is k is greater than 19. The solution on a number line has a left parenthesis at 19 with shading to the right. The solution in interval notation is 19 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_404_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is k plus 6 is greater than 25. Its solution is k is greater than 19. The solution on a number line has a left parenthesis at 19 with shading to the right. The solution in interval notation is 19 to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831970324\">\r\n<div data-type=\"problem\" id=\"fs-id1167831970326\">\r\n<p id=\"fs-id1167831970329\">Ten less than <em data-effect=\"italics\">w<\/em> is at least 39.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826996665\">\r\n<div data-type=\"problem\" id=\"fs-id1167826996667\">\r\n<p id=\"fs-id1167826996669\">Twelve less than <em data-effect=\"italics\">x<\/em> is no less than 21.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835356207\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835356212\" data-alt=\"The inequality is x minus 12 is greater than or equal to 21. Its solution is x is greater than or equal to 33. The solution on a number line has a left bracket at 33 with shading to the right. The solution in interval notation is 33 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_406_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is x minus 12 is greater than or equal to 21. Its solution is x is greater than or equal to 33. The solution on a number line has a left bracket at 33 with shading to the right. The solution in interval notation is 33 to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835327956\">\r\n<div data-type=\"problem\" id=\"fs-id1167835327958\">\r\n<p id=\"fs-id1167835327960\">Negative five times <em data-effect=\"italics\">r<\/em> is no more than 95.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830865920\">\r\n<div data-type=\"problem\" id=\"fs-id1167830865922\">\r\n<p id=\"fs-id1167830865924\">Negative two times <em data-effect=\"italics\">s<\/em> is lower than 56.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835323758\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835323763\" data-alt=\"The inequality is negative 2 s is less than 56. Its solution is s is greater than negative 28. The solution on a number line has a left parenthesis at negative 28 with shading to the right. The solution in interval notation is negative 28 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_408_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is negative 2 s is less than 56. Its solution is s is greater than negative 28. The solution on a number line has a left parenthesis at negative 28 with shading to the right. The solution in interval notation is negative 28 to infinity within parentheses.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835483755\">\r\n<div data-type=\"problem\" id=\"fs-id1167835483757\">\r\n<p id=\"fs-id1167834279498\">Nineteen less than <em data-effect=\"italics\">b<\/em> is at most \\(-22.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835305054\">\r\n<div data-type=\"problem\" id=\"fs-id1167835254296\">\r\n<p id=\"fs-id1167835254298\">Fifteen less than <em data-effect=\"italics\">a<\/em> is at least \\(-7.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835352382\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167835352386\" data-alt=\"The inequality is a minus 15 is greater than or equal to negative 7. Its solution is a is greater than or equal to 8. The solution on a number line has a left bracket at 8 with shading to the right. The solution in interval notation is 8 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_410_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a minus 15 is greater than or equal to negative 7. Its solution is a is greater than or equal to 8. The solution on a number line has a left bracket at 8 with shading to the right. The solution in interval notation is 8 to infinity within a bracket and a parenthesis.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834063416\"><strong data-effect=\"bold\">Solve Applications with Linear Inequalities<\/strong><\/p>\r\n<p id=\"fs-id1167834346719\">In the following exercises, solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167834346722\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167834346724\">\r\n<p id=\"fs-id1167834120757\">Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830704639\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167830704641\">\r\n<p id=\"fs-id1167830704643\">The elevator in Yehire\u2019s apartment building has a sign that says the maximum weight is 2100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835331980\">\r\n<p id=\"fs-id1167835331982\">A maximum of 14 people can safely ride in the elevator.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835331987\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835331990\">\r\n<p id=\"fs-id1167834179717\">Andre is looking at apartments with three of his friends. They want the monthly rent to be no more than ?2,360. If the roommates split the rent evenly among the four of them, what is the maximum rent each will pay?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831894139\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167831894141\">\r\n<p id=\"fs-id1167831894143\">Arleen got a ?20 gift card for the coffee shop. Her favorite iced drink costs ?3.79. What is the maximum number of drinks she can buy with the gift card?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834246891\">\r\n<p id=\"fs-id1167834246893\">five drinks<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835375582\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835375584\">\r\n<p id=\"fs-id1167835375586\">Teegan likes to play golf. He has budgeted ?60 next month for the driving range. It costs him ?10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831894612\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167831894614\">\r\n<p id=\"fs-id1167831894616\">Ryan charges his neighbors ?17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least ?1,500?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835512300\">\r\n<p id=\"fs-id1167835512303\">86 cars<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834422606\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167834422608\">\r\n<p id=\"fs-id1167834422610\">Keshad gets paid ?2,400 per month plus 6% of his sales. His brother earns ?3,300 per month. For what amount of total sales will Keshad\u2019s monthly pay be higher than his brother\u2019s monthly pay?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835345550\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835345552\">\r\n<p id=\"fs-id1167835345555\">Kimuyen needs to earn ?4,150 per month in order to pay all her expenses. Her job pays her ?3,475 per month plus 4% of her total sales. What is the minimum Kimuyen\u2019s total sales must be in order for her to pay all her expenses?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826997350\">\r\n<p id=\"fs-id1167826997352\">?16,875<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826997357\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167826997360\">\r\n<p id=\"fs-id1167826997362\">Andre has been offered an entry-level job. The company offered him ?48,000 per year plus 3.5% of his total sales. Andre knows that the average pay for this job is ?62,000. What would Andre\u2019s total sales need to be for his pay to be at least as high as the average pay for this job?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835367892\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835367894\">\r\n<p id=\"fs-id1167831837548\">Nataly is considering two job offers. The first job would pay her ?83,000 per year. The second would pay her ?66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831837555\">\r\n<p id=\"fs-id1167831837557\">?110,000<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832057715\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167832057717\">\r\n<p id=\"fs-id1167832057719\">Jake\u2019s water bill is ?24.80 per month plus ?2.20 per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than ?60?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834459074\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167834459076\">\r\n<p id=\"fs-id1167834459079\">Kiyoshi\u2019s phone plan costs ?17.50 per month plus ?0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than ?56.60?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834156998\">\r\n<p id=\"fs-id1167834157000\">260 messages<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834537854\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167834537856\">\r\n<p id=\"fs-id1167834537858\">Marlon\u2019s TV plan costs ?49.99 per month plus ?5.49 per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of ?100?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835346052\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835346054\">\r\n<p id=\"fs-id1167835346056\">Kellen wants to rent a banquet room in a restaurant for her cousin\u2019s baby shower. The restaurant charges ?350 for the banquet room plus ?32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be ?1,500?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835319247\">\r\n<p id=\"fs-id1167835319249\">35 people<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835319255\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835319257\">\r\n<p id=\"fs-id1167835319259\">Moshde runs a hairstyling business from her house. She charges ?45 for a haircut and style. Her monthly expenses are ?960. She wants to be able to put at least ?1,200 per month into her savings account order to open her own salon. How many \u201ccut &amp; styles\u201d must she do to save at least ?1,200 per month?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835367251\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835367254\">\r\n<p id=\"fs-id1167835367256\">Noe installs and configures software on home computers. He charges ?125 per job. His monthly expenses are ?1,600. How many jobs must he work in order to make a profit of at least ?2,400?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831103748\">\r\n<p id=\"fs-id1167831103750\">32 jobs<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831103755\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167831103758\">\r\n<p id=\"fs-id1167831103760\">Katherine is a personal chef. She charges ?115 per four-person meal. Her monthly expenses are ?3,150. How many four-person meals must she sell in order to make a profit of at least ?1,900?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835433645\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835433647\">\r\n<p id=\"fs-id1167835433649\">Melissa makes necklaces and sells them online. She charges ?88 per necklace. Her monthly expenses are ?3,745. How many necklaces must she sell if she wants to make a profit of at least ?1,650?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835433655\">\r\n<p id=\"fs-id1167835362346\">62 necklaces<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835362352\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835362354\">\r\n<p id=\"fs-id1167835362356\">Five student government officers want to go to the state convention. It will cost them ?110 for registration, ?375 for transportation and food, and ?42 per person for the hotel. There is ?450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge ?5 per car, how many cars must they wash in order to have enough money to pay for the trip?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835420294\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835420297\">\r\n<p id=\"fs-id1167835420299\">Cesar is planning a four-day trip to visit his friend at a college in another state. It will cost him ?198 for airfare, ?56 for local transportation, and ?45 per day for food. He has ?189 in savings and can earn ?35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835420306\">\r\n<p id=\"fs-id1167831890418\">seven lawns<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831890423\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167831890425\">\r\n<p id=\"fs-id1167831890427\">Alonzo works as a car detailer. He charges ?175 per car. He is planning to move out of his parents\u2019 house and rent his first apartment. He will need to pay ?120 for application fees, ?950 for security deposit, and first and last months\u2019 rent at ?1,140 per month. He has ?1,810 in savings. How many cars must he detail to have enough money to rent the apartment?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835355843\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167835355845\">\r\n<p id=\"fs-id1167835355847\">Eun-Kyung works as a tutor and earns ?60 per hour. She has ?792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her ?1,520 for food and drinks and ?150 for the photographer. She will also have a favor for each of the guests, and each favor will cost ?7.50. How many hours must she tutor to have enough money for the party?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834426224\">\r\n<p id=\"fs-id1167834340054\">20 hours<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"everyday\" data-depth=\"2\" id=\"fs-id1167834340060\">\r\n<h4 data-type=\"title\">Everyday Math<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167835369660\">\r\n<div data-type=\"problem\" id=\"fs-id1167835369662\">\r\n<p id=\"fs-id1167835369664\"><strong data-effect=\"bold\">Maximum load on a stage<\/strong> In 2014, a high school stage collapsed in Fullerton, California, when 250 students got on stage for the finale of a musical production. Two dozen students were injured. The stage could support a maximum of 12,750 pounds. If the average weight of a student is assumed to be 140 pounds, what is the maximum number of students who could safely be on the stage?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835327996\">\r\n<div data-type=\"problem\" id=\"fs-id1167835327998\">\r\n<p id=\"fs-id1167834156529\"><strong data-effect=\"bold\">Maximum weight on a boat<\/strong> In 2004, a water taxi sank in Baltimore harbor and five people drowned. The water taxi had a maximum capacity of 3,500 pounds (25 people with average weight 140 pounds). The average weight of the 25 people on the water taxi when it sank was 168 pounds per person. What should the maximum number of people of this weight have been?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834156540\">\r\n<p id=\"fs-id1167835301762\">20 people<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835301768\">\r\n<div data-type=\"problem\" id=\"fs-id1167835301770\">\r\n<p id=\"fs-id1167835301772\"><strong data-effect=\"bold\">Wedding budget<\/strong> Adele and Walter found the perfect venue for their wedding reception. The cost is ?9850 for up to 100 guests, plus ?38 for each additional guest. How many guests can attend if Adele and Walter want the total cost to be no more than ?12,500?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834120487\">\r\n<div data-type=\"problem\" id=\"fs-id1167834120489\">\r\n<p id=\"fs-id1167834120491\"><strong data-effect=\"bold\">Shower budget<\/strong> Penny is planning a baby shower for her daughter-in-law. The restaurant charges ?950 for up to 25 guests, plus ?31.95 for each additional guest. How many guests can attend if Penny wants the total cost to be no more than ?1,500?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835308073\">\r\n<p id=\"fs-id1167835308075\">42 guests<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835308082\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167835349227\">\r\n<div data-type=\"problem\" id=\"fs-id1167835349229\">\r\n<p id=\"fs-id1167835349232\">Explain why it is necessary to reverse the inequality when solving \\(-5x&gt;10.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835347768\">\r\n<div data-type=\"problem\" id=\"fs-id1167835347770\">\r\n<p id=\"fs-id1167835347772\">Explain why it is necessary to reverse the inequality when solving \\(\\frac{n}{-3}&lt;12.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834185411\">\r\n<p id=\"fs-id1167834185413\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834185419\">\r\n<div data-type=\"problem\" id=\"fs-id1167834423473\">\r\n<p id=\"fs-id1167834423475\">Find your last month\u2019s phone bill and the hourly salary you are paid at your job. Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it. Do you feel this is an appropriate number of hours? Is this the appropriate phone plan for you?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835319903\">\r\n<div data-type=\"problem\" id=\"fs-id1167835319905\">\r\n<p id=\"fs-id1167831191520\">Find out how many units you have left, after this term, to achieve your college goal and estimate the number of units you can take each term in college. Calculate the number of terms it will take you to achieve your college goal by writing an appropriate inequality and then solving it. Is this an acceptable number of terms until you meet your goal? What are some ways you could accelerate this process?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831191528\">\r\n<p id=\"fs-id1167831191530\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834191261\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1167834191266\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167826783877\" data-alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was graph inequalities on the number line. In row 3, the I can was solve linear inequalities. In row 4, the I can was translate words to an inequality and solve. In row 5, the I can was solve applications with linear inequalities.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was graph inequalities on the number line. In row 3, the I can was solve linear inequalities. In row 4, the I can was translate words to an inequality and solve. In row 5, the I can was solve applications with linear inequalities.\" \/><\/span>\r\n<p id=\"fs-id1167834517174\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Graph inequalities on the number line<\/li>\n<li>Solve linear inequalities<\/li>\n<li>Translate words to an inequality and solve<\/li>\n<li>Solve applications with linear inequalities<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834327442\" class=\"be-prepared\">\n<p id=\"fs-id1167835311227\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835233718\" type=\"1\">\n<li>Translate from algebra to English: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c025ab750f542b00483c2b02d028f2b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#62;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"54\" style=\"vertical-align: -1px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167829937221\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Translate to an algebraic expression: 15 is less than <em data-effect=\"italics\">x<\/em>.\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836513192\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831985734\">\n<h3 data-type=\"title\">Graph Inequalities on the Number Line<\/h3>\n<p id=\"fs-id1167832152811\">What number would make the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> true? Are you thinking, \u201c<em data-effect=\"italics\">x<\/em> could be four\u201d? That\u2019s correct, but <em data-effect=\"italics\">x<\/em> could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-694bf2d7d217172791cdee040da90e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167834472568\">We show all the solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.<\/p>\n<p id=\"fs-id1167835359606\">We can also represent inequalities using <strong data-effect=\"bold\">interval notation<\/strong>. There is no upper end to the solution to this inequality. In interval notation, we express <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa6bc95909d9315ebcae3a094b0e1e09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> The symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c44e61f4f297faa7e306b7ae24b62d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"16\" style=\"vertical-align: 0px;\" \/> is read as \u201c<strong data-effect=\"bold\">infinity<\/strong>.\u201d It is not an actual number.<\/p>\n<p id=\"fs-id1167832143991\"><a href=\"#CNX_IntAlg_Figure_02_05_001\" class=\"autogenerated-content\">(Figure)<\/a> shows both the number line and the interval notation.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_05_001\">\n<div class=\"bc-figcaption figcaption\">The inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> is graphed on this number line and written in interval notation.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835380309\" data-alt=\"The figure shows the inquality, x is greater than 3, graphed on a number line from negative 5 to 5. There is shading that starts at 3 and extends to numbers to its right. The solution for the inequality is written in interval notation. It is the interval from 3 to infinity, not including 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the inquality, x is greater than 3, graphed on a number line from negative 5 to 5. There is shading that starts at 3 and extends to numbers to its right. The solution for the inequality is written in interval notation. It is the interval from 3 to infinity, not including 3.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167835532500\">We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.<\/p>\n<p id=\"fs-id1167832116621\">The inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34c375dd6b737c15ca9401bc0e9e9e5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/> means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29163feacef7bfd88b9b5d136f8fef91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/> We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See <a href=\"#CNX_IntAlg_Figure_02_05_002\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167835369795\">There is no lower end to those numbers. We write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34c375dd6b737c15ca9401bc0e9e9e5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/> in interval notation as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5dfb26beaa049763806dc2f234cb3146_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\" \/> The symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c34c49d3b7f33e1444d7a199a677caa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"16\" style=\"vertical-align: 0px;\" \/> is read as \u201cnegative infinity.\u201d <a href=\"#CNX_IntAlg_Figure_02_05_002\" class=\"autogenerated-content\">(Figure)<\/a> shows both the number line and interval notation.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_05_002\">\n<div class=\"bc-figcaption figcaption\">The inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34c375dd6b737c15ca9401bc0e9e9e5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/> is graphed on this number line and written in interval notation.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835364648\" data-alt=\"The figure shows the inquality, x is less than or equal to l, graphed on a number line from negative 5 to 5. There is shading that starts at 1 and extends to numbers to its left. The solution for the inequality is written in interval notation. It is the interval from negative infinity to one, including 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the inquality, x is less than or equal to l, graphed on a number line from negative 5 to 5. There is shading that starts at 1 and extends to numbers to its left. The solution for the inequality is written in interval notation. It is the interval from negative infinity to one, including 1.\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832067399\">\n<div data-type=\"title\">Inequalities, Number Lines, and Interval Notation<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835304648\" data-alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1171791696312\">The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835334972\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835305891\">\n<div data-type=\"problem\" id=\"fs-id1167834120797\">\n<p id=\"fs-id1167834061620\">Graph each inequality on the number line and write in interval notation.<\/p>\n<p id=\"fs-id1167831871394\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfc84780b14328431bcc372ff904772d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ccd550142d8e855056a81215c569209_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76a0bbe2b3a9634ad0904e6d7b6c8813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831071507\">\n<p id=\"fs-id1167835419516\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835233973\" class=\"unnumbered unstyled\" summary=\"Shade to the right of negative 3 and put a left bracket at negative 3. The solution of the inequality x is greater than negative 3 is indicated on a number line with a left bracket at a and shading to the right. Write in intervak notation. The solution in interval notation is the interval from negative 3 to infinity enclosed within a left bracket and right parenthesis.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834252412\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Shade to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f54336ab5f34f79c0d075fa22f60ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> and put a bracket at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831112379\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835375599\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_004c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834306926\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167832060163\" class=\"unnumbered unstyled\" summary=\"Shade to the left of negative 2.5 and put a parenthesis at negative 2.5. The solution of the inequality x is less than 2.5 is indicated on a number line with a right bracket at 2.5 and shading to the left. Write in interval notation. The solution in interval notation is the interval from negative infinity to 2.5 enclosed within parentheses.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830962240\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Shade to the left of 2.5 and put a parenthesis at 2.5.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835322275\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_005b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830770108\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_005c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834535692\" class=\"unnumbered unstyled\" summary=\"Shade to the left of negative three-fifths and put a right bracket at negative three-fifths. The solution of the inequality x is less than or equal to negative three-fifths is indicated on a number line with a right bracket at negative three-fifths and shading to the left. Write in interval notation. The solution in interval notation is the interval from negative infinity to negative three-fifths enclosed within a left parentheses and right bracket.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832057978\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Shade to the left of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64f51c0a30eda14153a911a151d9775b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/> and put a bracket at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-646af2e58768b797e3c8f5d6143c4f0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327153\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_006b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834061478\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_006c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835283921\">\n<div data-type=\"problem\" id=\"fs-id1167835595498\">\n<p id=\"fs-id1167835268064\">Graph each inequality on the number line and write in interval notation: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0d114ca900168936b5c270433aff883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a9adac4c3934e690a9c3e3eceb7215c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#45;&#49;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a5a858319355754eec3e2f9b510b3e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835337188\">\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832054934\" data-alt=\"The graph of the inequality x is greater than 2 is indicated on a number line with a left parenthesis at 2 and shading to the right. The solution in interval notation is the interval from 2 to infinity enclosed within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_301_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is greater than 2 is indicated on a number line with a left parenthesis at 2 and shading to the right. The solution in interval notation is the interval from 2 to infinity enclosed within parentheses.\" \/><\/span><\/p>\n<p id=\"fs-id1167834156696\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831892896\" data-alt=\"The graph of the inequality x is less than or equal to negative 1.5 is indicated on a number line with a right bracket at negative 1.5 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 1.5 enclosed within a left parenthesis and right bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_302_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is less than or equal to negative 1.5 is indicated on a number line with a right bracket at negative 1.5 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 1.5 enclosed within a left parenthesis and right bracket.\" \/><\/span><\/p>\n<p id=\"fs-id1167826983806\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835310783\" data-alt=\"The graph of the inequality x is greater than or rqual to three-fourths is indicated on a number line with a left bracket at three-fourths and shading to the right. The solution in interval notation is the interval from three-fourths to infinity enclosed within a left bracket and left parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_303_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is greater than or rqual to three-fourths is indicated on a number line with a left bracket at three-fourths and shading to the right. The solution in interval notation is the interval from three-fourths to infinity enclosed within a left bracket and left parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835348452\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835419193\">\n<div data-type=\"problem\" id=\"fs-id1167832052461\">\n<p id=\"fs-id1167832053172\">Graph each inequality on the number line and write in interval notation: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b0e8bed0aa67feda1c62ad21bbb0bca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-753f0f3f55b2b2594ed16663caebfafe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d875967e2f3c9e81fdae62de8b652652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834413671\">\n<p id=\"fs-id1167834228113\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835319438\" data-alt=\"The graph of the inequality x is less than or equal to negative 4 is indicated on a number line with a right bracket at negative 4 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 4 enclosed within an left parenthesis and right bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_304_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is less than or equal to negative 4 is indicated on a number line with a right bracket at negative 4 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 4 enclosed within an left parenthesis and right bracket.\" \/><\/span><\/p>\n<p id=\"fs-id1167835368792\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835376724\" data-alt=\"The graph of the inequality x is greater than or equal to 0.5 is indicated on a number line with a left bracket at 0.5 and shading to the right. The solution in interval notation is the interval from 0.5 to infinity enclosed within a left bracket and right parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_305_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is greater than or equal to 0.5 is indicated on a number line with a left bracket at 0.5 and shading to the right. The solution in interval notation is the interval from 0.5 to infinity enclosed within a left bracket and right parenthesis.\" \/><\/span><\/p>\n<p id=\"fs-id1167835340509\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835375263\" data-alt=\"The graph of the inequality x is less than negative two-thirds is indicated on a number line with a right parenthesis at negative two-thirds and shading to the left. The solution in interval notation is the interval from negative infinity to negative two-thirds enclosed within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_306_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality x is less than negative two-thirds is indicated on a number line with a right parenthesis at negative two-thirds and shading to the left. The solution in interval notation is the interval from negative infinity to negative two-thirds enclosed within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835371142\">What numbers are greater than two but less than five? Are you thinking say, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2de25026ba847b5e2bfef64e105cc1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&#44;&#51;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#52;&#44;&#52;&#44;&#57;&#57;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"134\" style=\"vertical-align: -6px;\" \/> We can represent all the numbers between two and five with the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8fd299e831e558f7ac69aafd14e76ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#60;&#120;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: 0px;\" \/> We can show <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f22d63981ce11b1c46733cf750ac20b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#60;&#120;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: 0px;\" \/> on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See <a href=\"#CNX_IntAlg_Figure_02_05_007\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_05_007\"><span data-type=\"media\" id=\"fs-id1167835358311\" data-alt=\"The graph of the inequality 2 is less than x which is less than 5 shows open circles a 2 and 5 and shading in between.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_007_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of the inequality 2 is less than x which is less than 5 shows open circles a 2 and 5 and shading in between.\" \/><\/span><\/div>\n<div data-type=\"example\" id=\"fs-id1167835524181\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835356443\">\n<div data-type=\"problem\" id=\"fs-id1167835356057\">\n<p id=\"fs-id1167834252251\">Graph each inequality on the number line and write in interval notation.<\/p>\n<p id=\"fs-id1167830991263\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeeb9026ab1f6353b33ab75f444d512d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#60;&#120;&#60;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"88\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a45048a44eb47cc74098296db0fc3917_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#92;&#108;&#101;&#32;&#120;&#60;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b2f6cbf8d35b791c6333dce03e2f57b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835197472\">\n<p id=\"fs-id1167835321054\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835351265\" class=\"unnumbered unstyled\" summary=\"Negative 3 is less than x which is less than 4. There open circles at negative 3 and 4 and shading between negative 3 and 4 on the number line. Put parenthese at negative 3 and 4. Write in interval notation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Shade between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and 4.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put a parentheses at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and 4.<\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835200506\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834219734\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167827943690\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834201950\" class=\"unnumbered unstyled\" summary=\"Negative 6 is less than or equal to x which is less than negative 1. There is closed circle at negative 6 and an open circle at negative 1 and shading between negative 6 and negative 1 on the number line. Put a bracket at negative 6 and a parenthesis at 1. Write in interval notation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834346774\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Shade between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and \u22121.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put a bracket at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-350a72d4937e92680f68559bba291ce8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a parenthesis at \u22121.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835324999\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835333898\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835622135\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835236159\" class=\"unnumbered unstyled\" summary=\"0 is less than or equal to x which is less than or equal to 2.5. There are closed circles at 0 and 2.5 and shading between negative 0 and 2.5 on the number line. Put brackets at 0 and 2.5. Write in interval notation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831920485\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Shade between 0 and 2.5.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put a bracket at 0 and at 2.5.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835330839\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831880568\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835327661\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835370620\">\n<div data-type=\"problem\" id=\"fs-id1167835364899\">\n<p id=\"fs-id1167832060132\">Graph each inequality on the number line and write in interval notation:<\/p>\n<p id=\"fs-id1167835344258\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-885f1ec4d2cfe0ea80ea09d8ccd90ba6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#60;&#120;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-172f67fc08139f14a4b082a9327ad228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#120;&#60;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa71c2220d3e4b502cf0369eacb5ee2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#52;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831890694\">\n<p id=\"fs-id1167835381590\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831040708\" data-alt=\"Negative 2 is less x which is less than 1. There are open circles at negative 2 and 1 and shading between negative 2 and 1 on the number line. Put parentheses at negative 2 and 1. Write in interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_307_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 2 is less x which is less than 1. There are open circles at negative 2 and 1 and shading between negative 2 and 1 on the number line. Put parentheses at negative 2 and 1. Write in interval notation.\" \/><\/span><\/p>\n<p id=\"fs-id1167835321144\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835327878\" data-alt=\"Negative 5 is less than or equal to x which is less than negative 4. There is a closed circle at negative 6 and an open circle at negative 4 and shading between negative 5 and negative 4 on the number line. Put a bracket at negative 5 and a parenthesis at negative 4. Write in interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_308_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 5 is less than or equal to x which is less than negative 4. There is a closed circle at negative 6 and an open circle at negative 4 and shading between negative 5 and negative 4 on the number line. Put a bracket at negative 5 and a parenthesis at negative 4. Write in interval notation.\" \/><\/span><\/p>\n<p id=\"fs-id1167831076658\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835346063\" data-alt=\"1 is less than or equal to x which is less than 4.25. There is closed circle at 1 and a closed circle at 4.25 and shading between 1 and 4.25 on the number line. Put brackets at 1 and 4.25. Write in interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_309_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"1 is less than or equal to x which is less than 4.25. There is closed circle at 1 and a closed circle at 4.25 and shading between 1 and 4.25 on the number line. Put brackets at 1 and 4.25. Write in interval notation.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834193336\">\n<div data-type=\"problem\" id=\"fs-id1167834382409\">\n<p id=\"fs-id1167835330830\">Graph each inequality on the number line and write in interval notation:<\/p>\n<p id=\"fs-id1171789787515\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-faee54bc177e55fbc2f6b4192af5304f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#60;&#120;&#60;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"87\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b09fb4c6b7b9d784ce9b4ac3472ee20f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#32;&#120;&#60;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9eebf95f1fd2adaea17d6f013560834a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"89\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835307740\">\n<p id=\"fs-id1167831949153\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167830700636\" data-alt=\"Negative 6 is less than x which is less than 2. There is an open circle at negative 6 and an open circle at 2 and shading between negative 6 and 2 on the number line. Put parentheses at negative 6 and 2. Write in interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_310_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 6 is less than x which is less than 2. There is an open circle at negative 6 and an open circle at 2 and shading between negative 6 and 2 on the number line. Put parentheses at negative 6 and 2. Write in interval notation.\" \/><\/span><\/p>\n<p id=\"fs-id1167834138113\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835371362\" data-alt=\"Negative 3 is less than or equal to x which is less than negative 1. There is a closed circle at negative 3 and an open circle at negative 1 and shading between negative 3 and negative 1 on the number line. Put a bracket at negative 3 and a parenthesis at negative 1. Write in interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_311_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 3 is less than or equal to x which is less than negative 1. There is a closed circle at negative 3 and an open circle at negative 1 and shading between negative 3 and negative 1 on the number line. Put a bracket at negative 3 and a parenthesis at negative 1. Write in interval notation.\" \/><\/span><\/p>\n<p id=\"fs-id1167835510464\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"2.5 is less than or equal to x which is less thanor equal to 6. There is a closed circle at 2.5 and a closed circle at 6 and shading between 2.5 and 6 on the number line. Put brackets at 2.5 and 6. Write in interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_312_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"2.5 is less than or equal to x which is less thanor equal to 6. There is a closed circle at 2.5 and a closed circle at 6 and shading between 2.5 and 6 on the number line. Put brackets at 2.5 and 6. Write in interval notation.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834346402\">\n<h3 data-type=\"title\">Solve Linear Inequalities<\/h3>\n<p id=\"fs-id1167834104913\">A linear inequality is much like a linear equation\u2014but the equal sign is replaced with an inequality sign. A <strong data-effect=\"bold\">linear inequality<\/strong> is an inequality in one variable that can be written in one of the forms, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31369616de17507aa957f4e537d2b27b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#60;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0efb75c1750011fed8f628ad7d021d67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#92;&#108;&#101;&#32;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0b804c65180c3fdd571994350ec11ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#62;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aac2c38d99507c2d05b5f2afe20006b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#92;&#103;&#101;&#32;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167834138378\">\n<div data-type=\"title\">Linear Inequality<\/div>\n<p id=\"fs-id1167826978602\">A linear inequality is an inequality in one variable that can be written in one of the following forms where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df2afd8a9cb6add6da009531338a6949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/>:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835303848\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac7f61481ba02fb87b2177d58c736ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#60;&#99;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#120;&#43;&#98;&#92;&#108;&#101;&#32;&#99;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#120;&#43;&#98;&#62;&#99;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#120;&#43;&#98;&#92;&#103;&#101;&#32;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"455\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834061592\">When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.<\/p>\n<p id=\"fs-id1167832212157\">We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831895186\" data-alt=\"Negative 4 is less than 2. Negative 4 minus 5 is less than 2 minus 5. Negative 9 is less than negative 3, which is true. Negative 4 is less than 2. Negative 4 plus 7 is less than 2 plus 7. 3 is less than 9, which is true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_011_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 4 is less than 2. Negative 4 minus 5 is less than 2 minus 5. Negative 9 is less than negative 3, which is true. Negative 4 is less than 2. Negative 4 plus 7 is less than 2 plus 7. 3 is less than 9, which is true.\" \/><\/span><\/p>\n<p id=\"fs-id1167835387130\">Notice that the inequality sign stayed the same.<\/p>\n<p id=\"fs-id1167835422282\">This leads us to the Addition and Subtraction Properties of Inequality.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835324958\">\n<div data-type=\"title\">Addition and Subtraction Property of Inequality<\/div>\n<p id=\"fs-id1167835309171\">For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c,<\/em> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e888c3ab01418f0512e04ee004c9c88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167834394945\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7846bc31244e15411205f9bc8a00a32f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#97;&#43;&#99;&#60;&#98;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#45;&#99;&#60;&#98;&#45;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#43;&#99;&#62;&#98;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#45;&#99;&#62;&#98;&#45;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"248\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"fs-id1167834537743\">We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.<\/p>\n<\/div>\n<p id=\"fs-id1167835349726\">What happens to an inequality when we divide or multiply both sides by a constant?<\/p>\n<p id=\"fs-id1167835305967\">Let\u2019s first multiply and divide both sides by a positive number.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167828421190\" data-alt=\"10 is less than 15. 10 times 5 is less than 15 times 5. 50 is less than 75 is true. 10 is less than 15. 10 divided by 5 is less than 15 divided by 5. 2 is less than 3 is true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_012_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"10 is less than 15. 10 times 5 is less than 15 times 5. 50 is less than 75 is true. 10 is less than 15. 10 divided by 5 is less than 15 divided by 5. 2 is less than 3 is true.\" \/><\/span><\/p>\n<p id=\"fs-id1167835334176\">The inequality signs stayed the same.<\/p>\n<p id=\"fs-id1167835375431\">Does the inequality stay the same when we divide or multiply by a negative number?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835331131\" data-alt=\"10 is less than 15 10 times negative 5 is blank 15 times negative 5? Negative 50 is blank negative 75. Negative 50 is greater than negative 75. 10 is less than 15. 10 divided by negative 5 is blank 15 divided by negative 5. Negative 2 is blank negative 3. Negative 2 is blank negative 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_013_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"10 is less than 15 10 times negative 5 is blank 15 times negative 5? Negative 50 is blank negative 75. Negative 50 is greater than negative 75. 10 is less than 15. 10 divided by negative 5 is blank 15 divided by negative 5. Negative 2 is blank negative 3. Negative 2 is blank negative 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167835324788\">Notice that when we filled in the inequality signs, the inequality signs reversed their direction.<\/p>\n<p id=\"fs-id1167834085015\">When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.<\/p>\n<p id=\"fs-id1167831893684\">This gives us the Multiplication and Division Property of Inequality.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834301124\">\n<div data-type=\"title\">Multiplication and Division Property of Inequality<\/div>\n<p id=\"fs-id1167834423555\">For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834446398\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64e6d87aa7a1a1b84c789e52656db08d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#111;&#114;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#32;&#98;&#121;&#32;&#97;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#60;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#62;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#60;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#62;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#62;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#62;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#111;&#114;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#32;&#98;&#121;&#32;&#97;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#60;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#60;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#62;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#62;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#60;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#60;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"285\" width=\"352\" style=\"vertical-align: -139px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834308106\">When we divide or multiply an inequality by <em data-effect=\"italics\">a<\/em>:<\/p>\n<ul id=\"fs-id1167831933992\" data-bullet-style=\"bullet\">\n<li>positive number, the inequality stays the same.<\/li>\n<li>negative number, the inequality reverses.<\/li>\n<\/ul>\n<p id=\"fs-id1167835375489\">Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fc359f677f1b6342220b3ed093677cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#109;&#101;&#97;&#110;&#105;&#110;&#103;&#32;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#60;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"287\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1167834114104\">Think about it as \u201cIf Xander is taller than Andy, then Andy is shorter than Xander.\u201d<\/p>\n<div data-type=\"example\" id=\"fs-id1167835419716\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826781531\">\n<div data-type=\"problem\" id=\"fs-id1167835237688\">\n<p id=\"fs-id1167831943976\">Solve each inequality. Graph the solution on the number line, and write the solution in interval notation.<\/p>\n<p id=\"fs-id1167835312278\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e96b587a8f3445458568bb1cf5591a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7117c8391dd4bef1623a9e98a52572cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#121;&#60;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"60\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc5d3a905b5ac16427e1cd3efc8f3119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834501314\">\n<p id=\"fs-id1167835390468\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167828434586\" class=\"unnumbered unstyled\" summary=\"sides of the inequality. x minus three-eighths plus three-eighths is less than or equal to three-fourths plus three-eighths. Simplify. x is less than nine-eighths. Graph the solution on a number line with a right bracket at nine-eighths with shading to the left. Write the solution, negative infinity to nine-eighths, within a parenthesis and bracket.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835331608\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a50f9f96c608e65c404d41e0ef6f8a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> to both sides of the inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835301636\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835381345\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835377147\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835595108\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_014e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835333643\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834324727\" class=\"unnumbered unstyled\" summary=\"9 y is less than 54. Divide both sides of the inequality by 9. Because 9 is less than 0, the inequality stays the same. 9 y divided by 9 is less than 54 divided by 9. Simplify. y is less than 6. Graph the solution on the number line with a right parenthesis at 6 with shading to the left. Write the solution, negative infinity to 6, within parentheses.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834463151\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Divide both sides of the inequality by 9; since<\/p>\n<div data-type=\"newline\"><\/div>\n<p>9 is positive, the inequality stays the same.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835366990\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831107033\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835365733\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834506066\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_015e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835332678\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835363141\" class=\"unnumbered unstyled\" summary=\"Negative 15 is less than three-fifths z . Multiply both sides of the inequality by five-thirds. Five-thirds times negative 15 is less than five-thirds times three-fifths z. Since five-thirds is greater than 0, the inequality stays the same. Simplify. Negative 25 is less than z. Rewrite with the variable on the left. Z is greater than negative 25. Graph the solution on the number line with a left parenthesis at negative 25 with shading to the right. Write the solution in interval notation. It is negative 25 to infinity within parentheses.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834191317\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply both sides of the inequality by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f7f3c900f8678a74a55aadcc29cf7be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17520c3fd9dde5f581265ed00b139631_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is positive, the inequality stays the same.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834439012\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835371351\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Rewrite with the variable on the left.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826978025\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835415285\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_016f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835342789\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834301181\">\n<div data-type=\"problem\" id=\"fs-id1167835595340\">\n<p id=\"fs-id1167832051471\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\n<p id=\"fs-id1171791609623\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e870c2527c1621990d9d871eb36853d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54d36c3af5a92feab54e07d3d5ddfc11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#99;&#62;&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-111542a5579f1fd9f62694b0f88f5cd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835181898\">\n<p id=\"fs-id1167827957280\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167830757790\" data-alt=\"p is less than eleven-twelfths. The solution on the number line has a right bracket at eleven-twelfths with shading to the right. The solution in interval notation is, eleven-twelfths to infinity within a bracket and parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_313_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"p is less than eleven-twelfths. The solution on the number line has a right bracket at eleven-twelfths with shading to the right. The solution in interval notation is, eleven-twelfths to infinity within a bracket and parenthesis.\" \/><\/span><\/p>\n<p id=\"fs-id1167831239596\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834459154\" data-alt=\"c is less than 8. The solution on the number line has a left bracket at 8 with shading to the right. The solution in interval notation is, 8 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_314_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"c is less than 8. The solution on the number line has a left bracket at 8 with shading to the right. The solution in interval notation is, 8 to infinity within parentheses.\" \/><\/span><\/p>\n<p id=\"fs-id1167832153827\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835262243\" data-alt=\"m is greater than or equal to 8. The solution on the number line has a right bracket at 64 with shading to the right. The solution in interval notation is, 64 to infinity within a bracket and parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_315_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"m is greater than or equal to 8. The solution on the number line has a right bracket at 64 with shading to the right. The solution in interval notation is, 64 to infinity within a bracket and parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831112374\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834156710\">\n<div data-type=\"problem\" id=\"fs-id1167835236457\">\n<p id=\"fs-id1167832134181\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\n<p id=\"fs-id1171791492048\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-033bcb06b5b9e57def04f158fd3a8c03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"81\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18acd14fa93f2545bffd0d47d0fa28de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#100;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"68\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f857c9864ccd75e7438ddc7a086c159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#52;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"76\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826857110\">\n<p id=\"fs-id1167830836919\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835318449\" data-alt=\"r is less than or equal to eleven-twelfths. The solution on the number line has a left bracket at eleven-twelfths with shading to the left. The solution in interval notation is negative infinity to eleven-twelfths within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_316_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"r is less than or equal to eleven-twelfths. The solution on the number line has a left bracket at eleven-twelfths with shading to the left. The solution in interval notation is negative infinity to eleven-twelfths within a parenthesis and a bracket.\" \/><\/span><\/p>\n<p id=\"fs-id1167832060027\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832053973\" data-alt=\"c is less than or equal to 5. The solution on the number line has a right bracket at 5 with shading to the left. The solution in interval notation is negative infinity to 5 within a parentheses and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_317_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"c is less than or equal to 5. The solution on the number line has a right bracket at 5 with shading to the left. The solution in interval notation is negative infinity to 5 within a parentheses and a bracket.\" \/><\/span><\/p>\n<p id=\"fs-id1167835595823\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831887927\" data-alt=\"n is greater than negative 18. The solution on the number line has a left parenthesis at negative 18 with shading to the right. The solution in interval notation is negative 18 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_318_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"n is greater than negative 18. The solution on the number line has a left parenthesis at negative 18 with shading to the right. The solution in interval notation is negative 18 to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835421968\">Be careful when you multiply or divide by a negative number\u2014remember to reverse the inequality sign.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835356688\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835356690\">\n<div data-type=\"problem\" id=\"fs-id1167835376561\">\n<p id=\"fs-id1167835376563\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<p id=\"fs-id1167834190410\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86a6d18006d704cad268122aaa9d35cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#109;&#92;&#103;&#101;&#32;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b77273c69b7e23058eef6c530d1fbeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#45;&#50;&#125;&#92;&#103;&#101;&#32;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835519089\">\n<p id=\"fs-id1167828433884\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835377381\" class=\"unnumbered unstyled\" summary=\"Negative 13 m is greater than or equal to 65. Divided both sides of the inequality by negative 13. Negative 13 m divided y negative 13 is less than or equal to 65 divided by negative 13. Since negative 13 is less than 0, the inequality reverses. Simplify. m is less than or equal to negative 5. Graph the solution on the number line with a right bracket at negative 5 with shading to the left. Write the solution in interval notation. It is negative infinity to negative 5 within a parenthesis and bracket.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834523712\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Divide both sides of the inequality by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94b30ac2beac0a045057118f6cb38f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee7bea24371f97fb5673002404153494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> is a negative, the inequality reverses.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832075837\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835326056\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835347022\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835340006\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_017e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834423509\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835339996\" class=\"unnumbered unstyled\" summary=\"n divided by negative 2 is greater than or equal to 8. Multiply both sides of the inequality by negative 2. Negative times the quantity n divided by negative 2 is less thann or equal to negative 2 times 8 Since negative 2 is less than 0, the inequality reverses. Simplify. n is less than or equal to negative 16. Graph the solution on the number line with a right bracket at negative 16 with shading to the left. Write the solution in interval notation. It is negative infinity to negative 16 within a parenthesis and bracket.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835253796\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply both sides of the inequality by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d36c434a1919e0aaa1a4125fdaa40853_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is a negative, the inequality reverses.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835309044\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830694037\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832060542\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835238691\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_018e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835366231\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835366234\">\n<div data-type=\"problem\" id=\"fs-id1167826857188\">\n<p id=\"fs-id1167826857190\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\n<p id=\"fs-id1171791461128\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23fc4a4ec83d742bf4acdf1177ff16ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#113;&#60;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2eb6a2db687f862a5f9d92d861d0f62c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#45;&#49;&#50;&#125;&#92;&#108;&#101;&#32;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835349959\">\n<p id=\"fs-id1167835349961\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832043282\" data-alt=\"q is greater than or equal to negative 4. The solution on the number line has a left parenthesis at negative 4 with shading to the right. The solution in interval notation is negative 4 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_319_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"q is greater than or equal to negative 4. The solution on the number line has a left parenthesis at negative 4 with shading to the right. The solution in interval notation is negative 4 to infinity within parentheses.\" \/><\/span><\/p>\n<p id=\"fs-id1167834190452\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"k is greater than or equal to negative 180. The solution on the number line has a left bracket at negative 180 with shading to the right. The solution in interval notation is negative 180 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_320_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"k is greater than or equal to negative 180. The solution on the number line has a left bracket at negative 180 with shading to the right. The solution in interval notation is negative 180 to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835510891\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167828447049\">\n<div data-type=\"problem\" id=\"fs-id1167828447052\">\n<p id=\"fs-id1167835327276\">Solve each inequality, graph the solution on the number line, and write the solution in interval notation:<\/p>\n<p id=\"fs-id1171789604716\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48adaf9cb6dbc6c26784b61f2095d3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#114;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#45;&#55;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f7e90f4d368bffc2ec3fdcb0bb8db95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#117;&#125;&#123;&#45;&#52;&#125;&#92;&#103;&#101;&#32;&#45;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479773\">\n<p id=\"fs-id1167835167078\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835239491\" data-alt=\"r is greater than or equal to 10. The solution on the number line has a left bracket at 10 with shading to the right. The solution in interval notation is 10 to infinity within a bracket and parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_321_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"r is greater than or equal to 10. The solution on the number line has a left bracket at 10 with shading to the right. The solution in interval notation is 10 to infinity within a bracket and parenthesis.\" \/><\/span><\/p>\n<p id=\"fs-id1167832153412\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834494894\" data-alt=\"u is less than or equal to 64. The solution on the number line has a right bracket at 64 with shading to the left. The solution in interval notation is negative infinity to 64 within parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_322_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"u is less than or equal to 64. The solution on the number line has a right bracket at 64 with shading to the left. The solution in interval notation is negative infinity to 64 within parenthesis and a bracket.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834058969\">Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but make sure to pay close attention when we multiply or divide to isolate the variable.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835324646\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835324648\">\n<div data-type=\"problem\" id=\"fs-id1167834505676\">\n<p id=\"fs-id1167834505679\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d86474e556f802a53d5f64e5421931d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#121;&#92;&#108;&#101;&#32;&#49;&#49;&#121;&#43;&#49;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -4px;\" \/> graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835363956\">\n<table id=\"fs-id1167835338034\" class=\"unnumbered unstyled\" summary=\"6 y is less than or equal to 11 y plus 17. Subtract 11 y from both sides to collect the variables on the left. 6 y minus 11 y is less than or equal to 11 y minus 11 y plus 17. Simplify. Negative 5 y is less than or equal to 17. Divided both sides of the inequality by negative 5 and reverse the inequality. Negative 5 y is less than negative 5 is greater than or equal to 17 divided by negative 5. Simplify. y is greater than or equal to 17 divided by negative 5. Graph the solution on the number line with a left bracket at negative 17 divided by 5 with shading to the right. Write the solution in interval notation. It is negative 17 divided by 5 to infinity within brackets.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830924155\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7289a1d79b223845474f11bd313b44d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> from both sides to collect<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the variables on the left.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834123782\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835334058\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Divide both sides of the inequality by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df698f042bebf2b342ef200505b06a72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"26\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>and reverse the inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826804444\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167828411035\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831871398\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834505663\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_019g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835321328\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832153759\">\n<div data-type=\"problem\" id=\"fs-id1167832153761\">\n<p id=\"fs-id1167832153764\">Solve the inequality, graph the solution on the number line, and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-822f0a329d6b0b8648a6d0d26453c02b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#113;&#92;&#103;&#101;&#32;&#55;&#113;&#45;&#50;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834346527\"><span data-type=\"media\" id=\"fs-id1167834346529\" data-alt=\"q is less than or equal to 23 divided by 4. The solution on the number line has a right bracket at 23 divided by 4 with shading to the left. The solution in interval notation is negative infinity to 23 divided by 4 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_323_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"q is less than or equal to 23 divided by 4. The solution on the number line has a right bracket at 23 divided by 4 with shading to the left. The solution in interval notation is negative infinity to 23 divided by 4 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835362248\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835416783\">\n<div data-type=\"problem\" id=\"fs-id1167835416786\">\n<p id=\"fs-id1167835416788\">Solve the inequality, graph the solution on the number line, and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1871c7d941c1957477ecf84d5d1226d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#60;&#49;&#48;&#120;&#43;&#49;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834531312\"><span data-type=\"media\" id=\"fs-id1167834531314\" data-alt=\"x is greater than negative 19 divided by 4. The solution on the number line has a left parenthesis at negative 19 divided by 4 with shading to the right. The solution in interval notation is negative 19 divided by 4 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_324_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is greater than negative 19 divided by 4. The solution on the number line has a left parenthesis at negative 19 divided by 4 with shading to the right. The solution in interval notation is negative 19 divided by 4 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834396533\">When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don\u2019t have to multiply or divide by a negative\u2014which means we don\u2019t have to remember to reverse the inequality sign.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834184909\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834382372\">\n<div data-type=\"problem\" id=\"fs-id1167834382374\">\n<p id=\"fs-id1167834382376\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-602ef228edc0fd6db8c539c4f0f2c0cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#112;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#55;&#112;&#45;&#50;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"198\" style=\"vertical-align: -4px;\" \/> graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791694265\">\n<table id=\"fs-id1167835356650\" class=\"unnumbered unstyled can-break\" style=\"height: 446px\" summary=\"8 p plus 3 times the quantity p minus 12 is greater tgab 7 p minus 28. Simplify each side as much as possible. Distribute. The result is 8 p plus 3 p minus 36 is greater than 7 p minus 28. Combine like terms. The result is 11 p minus 36 is greater than 7 p minus 28. Subtract 7 p from both sides to collect the variables on the left, since 11 is greater than 7. Simplify. 4 p minus 36 is greater than negative 28. Add 36 to both sides to collect the constants on the right. 4 p minus 36 plus 36 is greater than negative 28 plus 36. Simplify. 4 p is greater than 8. Divided both sides of the inequality by 4. The inequality stays the same. 4 p divided by 4 is greater than 8 divided by 4. Simplify. p is greater than 2. Graph the solution on the number line with a left paarenthesis at 2 with shading to the right. Write the solution in interval notation. It 2 to infinity within parentheses.\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\"><\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b7b35db1283dca254cb75d83752d449_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#112;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#55;&#112;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"194\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify each side as much as possible.<\/td>\n<td style=\"height: 15px;width: 367px\"><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Distribute.<\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16d1d454ec4415caa65c2f934b2e0568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#112;&#43;&#51;&#112;&#45;&#51;&#54;&#62;&#55;&#112;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"178\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Combine like terms.<\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96909cd186f9b6c24aa98fe825436638_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#49;&#112;&#45;&#51;&#54;&#62;&#55;&#112;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 106px\">\n<td style=\"height: 106px;width: 237.533px\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58bfcd4057f1d18a954d8b481b113d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\" \/> from both sides to collect the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>variables on the left, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33d00308601093d8c4875d887ff69aee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#62;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"height: 106px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74c91c41c97d5781c75408c67ed08510_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#49;&#112;&#45;&#51;&#54;&#45;&#55;&#112;&#62;&#55;&#112;&#45;&#50;&#56;&#45;&#55;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"225\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify.<\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-974c0e4642dbd8b00d1ee79526734827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#57;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#112;&#45;&#51;&#54;&#62;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 90px\">\n<td style=\"height: 90px;width: 237.533px\" data-align=\"left\">Add 36 to both sides to collect the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>constants on the right.<\/td>\n<td style=\"height: 90px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e1fa79ec6207711ef277ae111c8d2c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#112;&#45;&#51;&#54;&#43;&#51;&#54;&#62;&#45;&#50;&#56;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"192\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify.<\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-baa378a280d9db824e3bfd912768274d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#112;&#62;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 90px\">\n<td style=\"height: 90px;width: 237.533px\" data-align=\"left\">Divide both sides of the inequality by<\/p>\n<div data-type=\"newline\"><\/div>\n<p>4; the inequality stays the same.<\/td>\n<td style=\"height: 90px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07ee2ed5bda19d45170a3a6ec50b340f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#112;&#125;&#123;&#52;&#125;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Simplify.<\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b06d4d854b2a6af43706d3b52a864ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#112;&#62;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 40px\">\n<td style=\"height: 40px;width: 237.533px\" data-align=\"left\">Graph the solution on the number line.<\/td>\n<td style=\"height: 40px;width: 367px\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345850\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_020a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 237.533px\" data-align=\"left\">Write the solution in interval notation.<\/td>\n<td style=\"height: 15px;width: 367px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c5d2c4afdd4281523928709724c400_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835172025\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834422978\">\n<div data-type=\"problem\" id=\"fs-id1167834422980\">\n<p id=\"fs-id1167834422982\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef7400d745221546765a7c80635325fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#121;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#53;&#121;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"187\" style=\"vertical-align: -4px;\" \/>, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830700828\"><span data-type=\"media\" id=\"fs-id1167830700832\" data-alt=\"y is greater than negative 6. The solution on the number line has a left parenthesis at negative 6 with shading to the right. The solution in interval notation is negative 6 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_325_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"y is greater than negative 6. The solution on the number line has a left parenthesis at negative 6 with shading to the right. The solution in interval notation is negative 6 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831920268\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831920272\">\n<div data-type=\"problem\" id=\"fs-id1167831920274\">\n<p id=\"fs-id1167831920276\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b7f43bad381a881d55ed9fa4f1c2d8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#117;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#49;&#48;&#117;&#43;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"197\" style=\"vertical-align: -4px;\" \/>, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835367653\"><span data-type=\"media\" id=\"fs-id1167835367656\" data-alt=\"u is greater than negative 10. The solution on the number line has a left parenthesis at 10 with shading to the right. The solution in interval notation is 10 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_326_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"u is greater than negative 10. The solution on the number line has a left parenthesis at 10 with shading to the right. The solution in interval notation is 10 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831894185\">Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831872446\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831872448\">\n<div data-type=\"problem\" id=\"fs-id1167831872451\">\n<p id=\"fs-id1167834060914\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36253b8e1c5e599f6d95ff07e1d8f903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"242\" style=\"vertical-align: -4px;\" \/> graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791366521\">\n<table id=\"fs-id1167835595614\" class=\"unnumbered unstyled can-break\" summary=\"Simplify each side of 8 x minus 2 times the quantity 5 minus x is less than the product of 4 and the quantity x plus 9 plus 6 x. Distribute. 8 x minus 10 plus 2 x is less than 4 x plus 36 plus 6 x. Combine like terms. 10 x minus 10 is less than 10 x plus 36. Subtract 10 x from both sides to collect the variables on the left. 10 x minus 10 minus 10 x is less than 10 x plus 36 minus 10 x. Simplify. The result is negative 10 is less than 36. The x\u2019s are gone, and we havea true inequality. The inequality is an identity. Graph the solution all real numbers on the number line. The number line is shaded for all values. Write the solution in interval notation. It is negative infinity to infinity within parentheses.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Simplify each side as much as possible.<\/td>\n<td style=\"width: 183px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9ab22ad991fe7c719067fb4633bd8e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f42d8252d5bcd87189ebca10e92f3e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"238\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Distribute.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b45e6ccae08d42c507a7dad07fd4a39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#45;&#49;&#48;&#43;&#50;&#120;&#60;&#52;&#120;&#43;&#51;&#54;&#43;&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"222\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Combine like terms.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41243737d2fe02b7314482b67169f574_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#45;&#49;&#48;&#60;&#49;&#48;&#120;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"157\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Subtract 10<em data-effect=\"italics\">x<\/em> from both sides to collect<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the variables on the left.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-785c40a441d45f7a45010b200b807f12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#45;&#49;&#48;&#45;&#49;&#48;&#120;&#60;&#49;&#48;&#120;&#43;&#51;&#54;&#45;&#49;&#48;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"256\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Simplify.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ed559e857936eafb5e6da8e23b9f9a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#60;&#51;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>\u2019s are gone, and we have a true<\/p>\n<div data-type=\"newline\"><\/div>\n<p>statement.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\">The inequality is an identity.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is all real numbers.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Graph the solution on the number line.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835639989\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_021a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184px\" data-align=\"left\">Write the solution in interval notation.<\/td>\n<td style=\"width: 183px\"><\/td>\n<td style=\"width: 223px\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832076309\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832076312\">\n<div data-type=\"problem\" id=\"fs-id1167835347614\">\n<p id=\"fs-id1167835347616\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78ff7598a07b2d11561cbed2b5ac0869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#98;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"228\" style=\"vertical-align: -4px;\" \/>, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835343911\"><span data-type=\"media\" id=\"fs-id1167835343914\" data-alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_327_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835343967\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831025106\">\n<div data-type=\"problem\" id=\"fs-id1167831025108\">\n<p id=\"fs-id1167831025111\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-126a61cbd15a9e3d44dc7be7acec5618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#104;&#45;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"247\" style=\"vertical-align: -4px;\" \/>, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834059504\"><span data-type=\"media\" id=\"fs-id1167834059508\" data-alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_328_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1166401420698\">We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835609996\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835609999\">\n<div data-type=\"problem\" id=\"fs-id1167835610001\">\n<p id=\"fs-id1167835610003\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3e116edd7eee94140e6c95d4c416b8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#97;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#52;&#125;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"147\" style=\"vertical-align: -6px;\" \/> graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835345971\">\n<table id=\"fs-id1167835345973\" class=\"unnumbered unstyled can-break\" summary=\"One-third a minus one-eighth a is greater than five twenty-fourths a plus three-fourths. Multiply both sides by the L C D, 24, to clear the fractions. 24 times the quantity One-third a minus one-eighth a is greater than 24 times the quantity five twenty-fourths a plus three-fourth. Simplify. 8 a minus 3 a is greater than 5 a plus 18. Combine like terms. Subtract 5 a from both sides to collect the variables on the left. 5 a minus 5 a is greater than 5 a minus 5 a plus 18. Simplify. 0 is less than 18. The statement is false. The inequality is a contradiction. There is no solution. As a result, there is no graph on the number line or interval notation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834213893\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply both sides by the LCD, 24,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to clear the fractions.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835423366\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830963400\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834429828\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9360ef9fb862d7bf0e669eb7b4606fea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> from both sides to collect the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>variables on the left.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835368315\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830704524\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The statement is false.<\/td>\n<td data-align=\"left\">The inequality is a contradiction.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>There is no solution.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph the solution on the number line.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832041553\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_022g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-align=\"left\">There is no solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835345876\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835345879\">\n<div data-type=\"problem\" id=\"fs-id1167835345881\">\n<p id=\"fs-id1167834473392\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc53865fc3ff512683f4f417c73388ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#120;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"143\" style=\"vertical-align: -7px;\" \/>, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834097799\"><span data-type=\"media\" id=\"fs-id1167834097803\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_329_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834186297\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834186300\">\n<div data-type=\"problem\" id=\"fs-id1167835422250\">\n<p id=\"fs-id1167835422252\">Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac4bc51f5c73a376563a789f2f0dbdf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#122;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#122;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#53;&#125;&#122;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -7px;\" \/>, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834394525\"><span data-type=\"media\" id=\"fs-id1167834196537\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_330_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831896625\">\n<h3 data-type=\"title\">Translate to an Inequality and Solve<\/h3>\n<p id=\"fs-id1167831896630\">To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like \u201cmore than\u201d and \u201cless than.\u201d But others are not as obvious. <a href=\"#fs-id1167826983841\" class=\"autogenerated-content\">(Figure)<\/a> shows some common phrases that indicate inequalities.<\/p>\n<table id=\"fs-id1167826983841\" summary=\"This table has four columns and two rows. The first row is a header row and it labels each column, \u201cgreater than,\u201d \u201cgreater than or equal to,\u201d and \u201cless than,\u201d and \u201cless than or equal to.\u201d In column 1, the phrases are \u201cis greater than,\u201d \u201cis more than,\u201d \u201cis larget than,\u201d and \u201cexceeds.\u201d. In column two, the phrases are \u201cis greater than or equal to,\u201d \u201cis at least,\u201d \u201cis no less than,\u201d and \u201cis the minimum.\u201d In column 3, the phrases are \u201cis less than,\u201d \u201cis smaller than,\u201d \u201chas fewer than,\u201d and \u201cis lower than.\u201d In column 4 the phrases are \u201cis less than or equal to,\u201d \u201cis at most,\u201d \u201cis no more than,\u201d and \u201cis the maximum.\u201d\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bed2a7b3a4cb8554a7ceb3c94e90a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5fae947beab7b30f45d5b6603772f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e2b165150917469474a6d203f27e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">is greater than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is more than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is larger than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>exceeds<\/td>\n<td data-valign=\"middle\" data-align=\"left\">is greater than or equal to<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is at least<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is no less than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is the minimum<\/td>\n<td data-valign=\"middle\" data-align=\"left\">is less than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is smaller than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>has fewer than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is lower than<\/td>\n<td data-valign=\"middle\" data-align=\"left\">is less than or equal to<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is at most<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is no more than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is the maximum<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1167835330450\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835330452\">\n<div data-type=\"problem\" id=\"fs-id1167835330454\">\n<p id=\"fs-id1167835328637\">Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834539042\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c0e2c7947ea2c5185d4ac75bdd4e733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#119;&#101;&#110;&#116;&#121;&#45;&#115;&#101;&#118;&#101;&#110;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#52;&#56;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"300\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835326571\">\n<table id=\"fs-id1167835326573\" class=\"unnumbered unstyled\" summary=\"or equal to 48. Solve by adding 27 to both sides. x minus 27 plus 27 is greater than or equal to 48 plus 27. Simplify. The result is x is greater than or equal to 75. Graph the solution on the number line with a left bracket at 75 with shading to the right. Write the solution in interval notation. It is 75 to infinity within a bracket and parenthesis.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835489337\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Translate.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830757362\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve\u2014add 27 to both sides.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835333039\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834061828\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Graph on the number line.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835345148\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in interval notation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832065810\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_023f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835229537\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835229541\">\n<div data-type=\"problem\" id=\"fs-id1167835229543\">\n<p id=\"fs-id1167835229545\">Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.<\/p>\n<p id=\"fs-id1167831887724\">Nineteen less than <em data-effect=\"italics\">p<\/em> is no less than 47.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830703809\"><span data-type=\"media\" id=\"fs-id1167830703813\" data-alt=\"p minus 19 is greater than or equal to 47. Its solution is p is greater than or equal to 66. The solution on the number line has a left bracket at 66 with shading to the right. The solution in interval notation is 66 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_331_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"p minus 19 is greater than or equal to 47. Its solution is p is greater than or equal to 66. The solution on the number line has a left bracket at 66 with shading to the right. The solution in interval notation is 66 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826869988\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834132735\">\n<div data-type=\"problem\" id=\"fs-id1167834132737\">\n<p id=\"fs-id1167834132739\">Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.<\/p>\n<p id=\"fs-id1167830699113\">Four more than <em data-effect=\"italics\">a<\/em> is at most 15.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830699122\"><span data-type=\"media\" id=\"fs-id1167832058199\" data-alt=\"a plus 4 is less than or equal to 15. Its solution is a is less than or equal to 11. The solution on the number line has a right bracket at 11with shading to the left. The solution in interval notation is negative infinity to 11 within a parenthesis and bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_332_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"a plus 4 is less than or equal to 15. Its solution is a is less than or equal to 11. The solution on the number line has a right bracket at 11with shading to the left. The solution in interval notation is negative infinity to 11 within a parenthesis and bracket.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831893517\">\n<h3 data-type=\"title\">Solve Applications with Linear Inequalities<\/h3>\n<p id=\"fs-id1167834213968\">Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.<\/p>\n<p id=\"fs-id1167834213973\">We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.<\/p>\n<p id=\"fs-id1167835532881\">Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835532887\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835532889\">\n<div data-type=\"problem\" id=\"fs-id1167835623451\">\n<p id=\"fs-id1167835623453\">Dawn won a mini-grant of ?4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost ?254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171784074417\">\n<p id=\"fs-id1167834535668\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f0db650c00aeb0825018932a61bbbd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#32;&#82;&#101;&#97;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#32;&#73;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#109;&#97;&#120;&#105;&#109;&#117;&#109;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#32;&#68;&#97;&#119;&#110;&#32;&#99;&#97;&#110;&#32;&#98;&#117;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#32;&#78;&#97;&#109;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#111;&#111;&#115;&#101;&#32;&#97;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#32;&#116;&#111;&#32;&#114;&#101;&#112;&#114;&#101;&#115;&#101;&#110;&#116;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#117;&#97;&#110;&#116;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#32;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#32;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#103;&#105;&#118;&#101;&#115;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#102;&#111;&#114;&#109;&#97;&#116;&#105;&#111;&#110;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#105;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#105;&#110;&#116;&#111;&#32;&#97;&#110;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#50;&#53;&#52;&#46;&#49;&#50;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#63;&#52;&#44;&#48;&#48;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#52;&#46;&#49;&#50;&#110;&#92;&#108;&#101;&#32;&#52;&#48;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#53;&#46;&#32;&#83;&#111;&#108;&#118;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#117;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#97;&#32;&#119;&#104;&#111;&#108;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#44;&#32;&#115;&#111;&#32;&#114;&#111;&#117;&#110;&#100;&#32;&#116;&#111;&#32;&#49;&#53;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#110;&#92;&#108;&#101;&#32;&#49;&#53;&#46;&#55;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#110;&#92;&#108;&#101;&#32;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#54;&#46;&#32;&#67;&#104;&#101;&#99;&#107;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#109;&#97;&#107;&#101;&#32;&#115;&#117;&#114;&#101;&#32;&#105;&#116;&#32;&#109;&#97;&#107;&#101;&#115;&#32;&#115;&#101;&#110;&#115;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#111;&#117;&#110;&#100;&#105;&#110;&#103;&#32;&#100;&#111;&#119;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#105;&#99;&#101;&#32;&#116;&#111;&#32;&#63;&#50;&#53;&#48;&#44;&#32;&#49;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#32;&#119;&#111;&#117;&#108;&#100;&#32;&#99;&#111;&#115;&#116;&#32;&#63;&#51;&#44;&#55;&#53;&#48;&#44;&#32;&#119;&#104;&#105;&#108;&#101;&#32;&#49;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#32;&#119;&#111;&#117;&#108;&#100;&#32;&#98;&#101;&#32;&#63;&#52;&#44;&#48;&#48;&#48;&#46;&#32;&#83;&#111;&#32;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#120;&#105;&#109;&#117;&#109;&#32;&#111;&#102;&#32;&#49;&#53;&#32;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#32;&#97;&#116;&#32;&#63;&#50;&#53;&#52;&#46;&#49;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#101;&#109;&#115;&#32;&#114;&#101;&#97;&#115;&#111;&#110;&#97;&#98;&#108;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#55;&#46;&#32;&#65;&#110;&#115;&#119;&#101;&#114;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#32;&#119;&#105;&#116;&#104;&#32;&#97;&#32;&#99;&#111;&#109;&#112;&#108;&#101;&#116;&#101;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#97;&#119;&#110;&#32;&#99;&#97;&#110;&#32;&#98;&#117;&#121;&#32;&#97;&#32;&#109;&#97;&#120;&#105;&#109;&#117;&#109;&#32;&#111;&#102;&#32;&#49;&#53;&#32;&#116;&#97;&#98;&#108;&#101;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"588\" width=\"840\" style=\"vertical-align: -289px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831887544\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831887548\">\n<div data-type=\"problem\" id=\"fs-id1167831887551\">\n<p id=\"fs-id1167831887553\">Angie has ?20 to spend on juice boxes for her son\u2019s preschool picnic. Each pack of juice boxes costs ?2.63. What is the maximum number of packs she can buy?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835341023\">\n<p id=\"fs-id1167835341026\">Angie can buy 7 packs of juice.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826814163\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826814167\">\n<div data-type=\"problem\" id=\"fs-id1167826814169\">\n<p id=\"fs-id1167826814171\">Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost ?42.75 per person for dinner, including tip and tax. His budget for the party is ?500. What is the maximum number of people Daniel can have at the party?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831955653\">\n<p id=\"fs-id1167831955656\">Daniel can have 11 people at the party.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834065806\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834065808\">\n<div data-type=\"problem\" id=\"fs-id1167834065810\">\n<p id=\"fs-id1167834065812\">Taleisha\u2019s phone plan costs her ?28.80 a month plus ?0.20 per text message. How many text messages can she send\/receive and keep her monthly phone bill no more than ?50?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791453866\">\n<p id=\"fs-id1167835173750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bde8543b6b9a2b6da2e40e264d2ac6bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#32;&#82;&#101;&#97;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#32;&#73;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#116;&#101;&#120;&#116;&#32;&#109;&#101;&#115;&#115;&#97;&#103;&#101;&#115;&#32;&#84;&#97;&#108;&#101;&#105;&#115;&#104;&#97;&#32;&#99;&#97;&#110;&#32;&#109;&#97;&#107;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#32;&#78;&#97;&#109;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#111;&#111;&#115;&#101;&#32;&#97;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#32;&#116;&#111;&#32;&#114;&#101;&#112;&#114;&#101;&#115;&#101;&#110;&#116;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#117;&#97;&#110;&#116;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#116;&#101;&#120;&#116;&#32;&#109;&#101;&#115;&#115;&#97;&#103;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#32;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#105;&#118;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#102;&#111;&#114;&#109;&#97;&#116;&#105;&#111;&#110;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#105;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#105;&#110;&#116;&#111;&#32;&#97;&#110;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#50;&#56;&#46;&#56;&#48;&#32;&#112;&#108;&#117;&#115;&#32;&#63;&#48;&#46;&#50;&#48;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#101;&#120;&#116;&#32;&#109;&#101;&#115;&#115;&#97;&#103;&#101;&#115;&#32;&#105;&#115;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#111;&#114;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#63;&#53;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#56;&#46;&#56;&#48;&#43;&#48;&#46;&#50;&#48;&#116;&#92;&#108;&#101;&#32;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#53;&#46;&#32;&#83;&#111;&#108;&#118;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#50;&#116;&#92;&#108;&#101;&#32;&#50;&#49;&#46;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#92;&#108;&#101;&#32;&#49;&#48;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#101;&#120;&#116;&#32;&#109;&#101;&#115;&#115;&#97;&#103;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#54;&#46;&#32;&#67;&#104;&#101;&#99;&#107;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#109;&#97;&#107;&#101;&#32;&#115;&#117;&#114;&#101;&#32;&#105;&#116;&#32;&#109;&#97;&#107;&#101;&#115;&#32;&#115;&#101;&#110;&#115;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#101;&#115;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#56;&#46;&#56;&#48;&#43;&#48;&#46;&#50;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#55;&#46;&#32;&#87;&#114;&#105;&#116;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#115;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#97;&#108;&#101;&#105;&#115;&#104;&#97;&#32;&#99;&#97;&#110;&#32;&#115;&#101;&#110;&#100;&#47;&#114;&#101;&#99;&#101;&#105;&#118;&#101;&#32;&#110;&#111;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#54;&#32;&#116;&#101;&#120;&#116;&#32;&#109;&#101;&#115;&#115;&#97;&#103;&#101;&#115;&#32;&#116;&#111;&#32;&#107;&#101;&#101;&#112;&#32;&#104;&#101;&#114;&#32;&#98;&#105;&#108;&#108;&#32;&#110;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#63;&#53;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"498\" width=\"833\" style=\"vertical-align: -243px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167828420745\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167828420749\">\n<div data-type=\"problem\" id=\"fs-id1167828420751\">\n<p id=\"fs-id1167828420753\">Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges ?75 a week plus ?0.25 a mile. How many miles can they travel during the week and still keep within their ?200 budget?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835194797\">\n<p>Sergio and Lizeth can travel no more than 500 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834300216\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834300220\">\n<div data-type=\"problem\" id=\"fs-id1167834300222\">\n<p id=\"fs-id1167834300224\">Rameen\u2019s heating bill is ?5.42 per month plus ?1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of ?87.50.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831191514\">\n<p id=\"fs-id1167831191516\">Rameen can use no more than 76 therms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826814017\">Profit is the money that remains when the costs have been subtracted from the revenue. In the next example, we will find the number of jobs a small businesswoman needs to do every month in order to make a certain amount of profit.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826814022\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826814024\">\n<div data-type=\"problem\" id=\"fs-id1167835352861\">\n<p id=\"fs-id1167835352864\">Felicity has a calligraphy business. She charges ?2.50 per wedding invitation. Her monthly expenses are ?650. How many invitations must she write to earn a profit of at least ?2,800 per month?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791497893\">\n<p id=\"fs-id1167826940798\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6597ad4b8d20d49eda612e9db762eebf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#32;&#82;&#101;&#97;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#32;&#73;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#105;&#110;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#32;&#70;&#101;&#108;&#105;&#99;&#105;&#116;&#121;&#32;&#110;&#101;&#101;&#100;&#115;&#32;&#116;&#111;&#32;&#119;&#114;&#105;&#116;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#32;&#78;&#97;&#109;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#106;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#105;&#110;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#111;&#111;&#115;&#101;&#32;&#97;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#32;&#116;&#111;&#32;&#114;&#101;&#112;&#114;&#101;&#115;&#101;&#110;&#116;&#32;&#105;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#32;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#46;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#105;&#118;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#102;&#111;&#114;&#109;&#97;&#116;&#105;&#111;&#110;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#105;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#105;&#110;&#116;&#111;&#32;&#97;&#110;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#50;&#46;&#53;&#48;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#105;&#110;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#117;&#115;&#32;&#63;&#54;&#53;&#48;&#32;&#105;&#115;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#63;&#50;&#44;&#56;&#48;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#46;&#53;&#48;&#106;&#45;&#54;&#53;&#48;&#92;&#103;&#101;&#32;&#50;&#44;&#56;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#53;&#46;&#32;&#83;&#111;&#108;&#118;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#46;&#53;&#106;&#92;&#103;&#101;&#32;&#51;&#44;&#52;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#106;&#92;&#103;&#101;&#32;&#49;&#44;&#51;&#56;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#54;&#46;&#32;&#67;&#104;&#101;&#99;&#107;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#109;&#97;&#107;&#101;&#32;&#115;&#117;&#114;&#101;&#32;&#105;&#116;&#32;&#109;&#97;&#107;&#101;&#115;&#32;&#115;&#101;&#110;&#115;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#32;&#70;&#101;&#108;&#105;&#99;&#105;&#116;&#121;&#32;&#119;&#114;&#111;&#116;&#101;&#32;&#49;&#52;&#48;&#48;&#32;&#105;&#110;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#44;&#32;&#104;&#101;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#114;&#111;&#102;&#105;&#116;&#32;&#119;&#111;&#117;&#108;&#100;&#32;&#98;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#46;&#53;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;&#53;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#50;&#44;&#56;&#53;&#48;&#46;&#32;&#84;&#104;&#105;&#115;&#32;&#105;&#115;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#63;&#50;&#56;&#48;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#55;&#46;&#32;&#87;&#114;&#105;&#116;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#115;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#101;&#108;&#105;&#99;&#105;&#116;&#121;&#32;&#109;&#117;&#115;&#116;&#32;&#119;&#114;&#105;&#116;&#101;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#49;&#44;&#51;&#56;&#48;&#32;&#105;&#110;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"434\" width=\"822\" style=\"vertical-align: -212px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826997011\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826997015\">\n<div data-type=\"problem\" id=\"fs-id1167826997017\">\n<p id=\"fs-id1167835370722\">Caleb has a pet sitting business. He charges ?32 per hour. His monthly expenses are ?2,272. How many hours must he work in order to earn a profit of at least ?800 per month?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370728\">\n<p id=\"fs-id1167835370730\">Caleb must work at least 96 hours.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832076580\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832076584\">\n<div data-type=\"problem\" id=\"fs-id1167832076586\">\n<p id=\"fs-id1167832015429\">Elliot has a landscape maintenance business. His monthly expenses are ?1,100. If he charges ?60 per job, how many jobs must he do to earn a profit of at least ?4,000 a month?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832015435\">\n<p id=\"fs-id1167832015437\">Elliot must work at least 85 jobs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167827988113\">There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.<\/p>\n<div data-type=\"example\" id=\"fs-id1167827988118\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834099288\">\n<div data-type=\"problem\" id=\"fs-id1167834099290\">\n<p id=\"fs-id1167834099292\">Malik is planning a six-day summer vacation trip. He has ?840 in savings, and he earns ?45 per hour for tutoring. The trip will cost him ?525 for airfare, ?780 for food and sightseeing, and ?95 per night for the hotel. How many hours must he tutor to have enough money to pay for the trip?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791551409\">\n<p id=\"fs-id1167834194392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a15f69bd32cffda0209588f3e697f949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#32;&#82;&#101;&#97;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#32;&#73;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#104;&#111;&#117;&#114;&#115;&#32;&#77;&#97;&#108;&#105;&#107;&#32;&#109;&#117;&#115;&#116;&#32;&#116;&#117;&#116;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#32;&#78;&#97;&#109;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#121;&#111;&#117;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#111;&#111;&#115;&#101;&#32;&#97;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#32;&#116;&#111;&#32;&#114;&#101;&#112;&#114;&#101;&#115;&#101;&#110;&#116;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#117;&#97;&#110;&#116;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#104;&#111;&#117;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#32;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#46;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#105;&#118;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#102;&#111;&#114;&#109;&#97;&#116;&#105;&#111;&#110;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#105;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#101;&#120;&#112;&#101;&#110;&#115;&#101;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#111;&#114;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#105;&#110;&#99;&#111;&#109;&#101;&#46;&#32;&#84;&#104;&#101;&#32;&#99;&#111;&#115;&#116;&#32;&#111;&#102;&#32;&#97;&#105;&#114;&#102;&#97;&#114;&#101;&#32;&#112;&#108;&#117;&#115;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#116;&#32;&#111;&#102;&#32;&#102;&#111;&#111;&#100;&#32;&#97;&#110;&#100;&#32;&#115;&#105;&#103;&#104;&#116;&#115;&#101;&#101;&#105;&#110;&#103;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#104;&#111;&#116;&#101;&#108;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#105;&#108;&#108;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#110;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#118;&#105;&#110;&#103;&#115;&#32;&#112;&#108;&#117;&#115;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#109;&#111;&#117;&#110;&#116;&#32;&#101;&#97;&#114;&#110;&#101;&#100;&#32;&#116;&#117;&#116;&#111;&#114;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#32;&#105;&#110;&#116;&#111;&#32;&#97;&#110;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#50;&#53;&#43;&#55;&#56;&#48;&#43;&#57;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#56;&#52;&#48;&#43;&#52;&#53;&#104;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#53;&#46;&#32;&#83;&#111;&#108;&#118;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#49;&#44;&#56;&#55;&#53;&#92;&#108;&#101;&#32;&#56;&#52;&#48;&#43;&#52;&#53;&#104;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#44;&#48;&#51;&#53;&#92;&#108;&#101;&#32;&#52;&#53;&#104;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#92;&#108;&#101;&#32;&#104;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#103;&#101;&#32;&#50;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#54;&#46;&#32;&#67;&#104;&#101;&#99;&#107;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#109;&#97;&#107;&#101;&#32;&#115;&#117;&#114;&#101;&#32;&#105;&#116;&#32;&#109;&#97;&#107;&#101;&#115;&#32;&#115;&#101;&#110;&#115;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#50;&#51;&#32;&#105;&#110;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#56;&#55;&#53;&#92;&#108;&#101;&#32;&#56;&#52;&#48;&#43;&#52;&#53;&#104;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#56;&#55;&#53;&#92;&#108;&#101;&#32;&#56;&#52;&#48;&#43;&#52;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#56;&#55;&#53;&#92;&#108;&#101;&#32;&#49;&#56;&#55;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#55;&#46;&#32;&#87;&#114;&#105;&#116;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#115;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#97;&#108;&#105;&#107;&#32;&#109;&#117;&#115;&#116;&#32;&#116;&#117;&#116;&#111;&#114;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#50;&#51;&#32;&#104;&#111;&#117;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"742\" width=\"791\" style=\"vertical-align: -366px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835370593\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835370597\">\n<div data-type=\"problem\" id=\"fs-id1167835370599\">\n<p id=\"fs-id1167835370602\">Brenda\u2019s best friend is having a destination wedding and the event will last three days. Brenda has ?500 in savings and can earn ?15 an hour babysitting. She expects to pay ?350 airfare, ?375 for food and entertainment and ?60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835327683\">\n<p id=\"fs-id1167831921533\">Brenda must babysit at least 27 hours.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831921539\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831921544\">\n<div data-type=\"problem\" id=\"fs-id1167831239022\">\n<p id=\"fs-id1167831239025\">Josue wants to go on a 10-night road trip with friends next spring. It will cost him ?180 for gas, ?450 for food, and ?49 per night to share a motel room. He has ?520 in savings and can earn ?30 per driveway shoveling snow. How many driveways must he shovel to have enough money to pay for the trip?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831239032\">\n<p id=\"fs-id1167835377790\">Josue must shovel at least 20 driveways.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835377798\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167834060088\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Inequalities, Number Lines, and Interval Notation<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb6d9041221754fcaf5c6b21dab13732_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#103;&#101;&#32;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#60;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#108;&#101;&#32;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"332\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835342769\" data-alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_024_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows that the solution of the inequality x is greater than a is indicated on a number line with a left parenthesis at a and shading to the right, and that the solution in interval notation is the interval from a to infinity enclosed in parentheses. It shows the solution of the inequality x is greater than or equal to a is indicated on a number line with an left bracket at a and shading to the right, and that the solution in interval notation is the interval a to infinity within a left bracket and right parenthesis. It shows that the solution of the inequality x is less than a is indicated on a number line with a right parenthesis at a and shading to the left, and that the solution in interval notation is the the interval negative infinity to a within parentheses. It shows that the solution of the inequality x is less than or equal to a is indicated on anumber line with a right bracket at a and shading to the left, and that the solution in interval notation is negative infinity to a within a left parenthesis and right bracket.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Linear Inequality<\/strong>\n<ul id=\"fs-id1167831920234\" data-bullet-style=\"open-circle\">\n<li>A <strong data-effect=\"bold\">linear inequality<\/strong> is an inequality in one variable that can be written in one of the following forms where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48a45175b1a7fe48e696f192d336269a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"51\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167831884342\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccabe36c9c2cd0276cbf4214a5103cff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#60;&#99;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#120;&#43;&#98;&#92;&#108;&#101;&#32;&#99;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#120;&#43;&#98;&#62;&#99;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#120;&#43;&#98;&#92;&#103;&#101;&#32;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"508\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Addition and Subtraction Property of Inequality<\/strong>\n<ul id=\"fs-id1167831922537\" data-bullet-style=\"open-circle\">\n<li>For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e888c3ab01418f0512e04ee004c9c88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167834194116\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7846bc31244e15411205f9bc8a00a32f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#97;&#43;&#99;&#60;&#98;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#45;&#99;&#60;&#98;&#45;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#43;&#99;&#62;&#98;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#45;&#99;&#62;&#98;&#45;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"248\" style=\"vertical-align: -13px;\" \/><\/div>\n<\/li>\n<li>We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Multiplication and Division Property of Inequality<\/strong>\n<ul id=\"fs-id1167826997478\" data-bullet-style=\"open-circle\">\n<li>For any numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>,\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78049857fd7924c9545f9f103facd46c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#111;&#114;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#32;&#98;&#121;&#32;&#97;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#60;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#62;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#60;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#62;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#62;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#62;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#111;&#114;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#32;&#98;&#121;&#32;&#97;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#60;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#60;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#62;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#62;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#60;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#60;&#98;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"219\" width=\"356\" style=\"vertical-align: -106px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Phrases that indicate inequalities<\/strong>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834516117\" class=\"unnumbered\" summary=\"This table has four columns and two rows. The first row is a header row and it labels each column, \u201cgreater than,\u201d \u201cgreater than or equal to,\u201d and \u201cless than,\u201d and \u201cless than or equal to.\u201d In column 1, the phrases are \u201cis greater than,\u201d \u201cis more than,\u201d \u201cis larget than,\u201d and \u201cexceeds.\u201d. In column two, the phrases are \u201cis greater than or equal to,\u201d \u201cis at least,\u201d \u201cis no less than,\u201d and \u201cis the minimum.\u201d In column 3, the phrases are \u201cis less than,\u201d \u201cis smaller than,\u201d \u201chas fewer than,\u201d and \u201cis lower than.\u201d In column 4 the phrases are \u201cis less than or equal to,\u201d \u201cis at most,\u201d \u201cis no more than,\u201d and \u201cis the maximum.\u201d\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bed2a7b3a4cb8554a7ceb3c94e90a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5fae947beab7b30f45d5b6603772f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e2b165150917469474a6d203f27e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">is greater than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is more than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is larger than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>exceeds<\/td>\n<td data-valign=\"middle\" data-align=\"left\">is greater than or equal to<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is at least<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is no less than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is the minimum<\/td>\n<td data-valign=\"middle\" data-align=\"left\">is less than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is smaller than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>has fewer than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is lower than<\/td>\n<td data-valign=\"middle\" data-align=\"left\">is less than or equal to<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is at most<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is no more than<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>is the maximum<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832055404\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167832055407\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167832055414\"><strong data-effect=\"bold\">Graph Inequalities on the Number Line<\/strong><\/p>\n<p id=\"fs-id1167832059729\">In the following exercises, graph each inequality on the number line and write in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832059733\">\n<div data-type=\"problem\" id=\"fs-id1167832059735\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b270a7e2bb18b56387023c4670dcc5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c029d017a90a170d23da2c5619934e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#51;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca7e34ab9ea8ffc2f3ab5cf62e8b0f1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832058612\">\n<div data-type=\"problem\" id=\"fs-id1167832058614\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c49d9cc64b220a8227bfc035ca02696a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#45;&#48;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4096541a73965dde20098d399384a65b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832152484\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831239691\" data-alt=\"The solution for x is greater than 3 on a number line has a left bracket 3 with shading to the right. The solution in interval notation is 3 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_336_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is greater than 3 on a number line has a left bracket 3 with shading to the right. The solution in interval notation is 3 to infinity within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834130205\" data-alt=\"The solution for x is less than or equal to negative 0.5 on a number line has a right bracket at negative 0.5 with shading to the left. The solution in interval notation is negative infinity to negative 0.5 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_337_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is less than or equal to negative 0.5 on a number line has a right bracket at negative 0.5 with shading to the left. The solution in interval notation is negative infinity to negative 0.5 within a parenthesis and a bracket.\" \/><\/span><\/p>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835306019\" data-alt=\"The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_338_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835318324\">\n<div data-type=\"problem\" id=\"fs-id1167835318326\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d3d8f7668f2defed69c14bdb24574d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ccd550142d8e855056a81215c569209_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-550ce1f282434fbe027fd256bf4ac6ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835415275\">\n<div data-type=\"problem\" id=\"fs-id1167835415277\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b157a9b63bb99ace04045920c37207c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2c179d195826a7e574cb8027015b67c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#49;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dda6edd7aa9c59ff8d8baec22f7b6865_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834501202\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835366746\" data-alt=\"The solution for x is less than or equal to 5 on a number line has a right bracket with shading to the left. The solution in interval notation is negative infinity to 5 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_342_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is less than or equal to 5 on a number line has a right bracket with shading to the left. The solution in interval notation is negative infinity to 5 within a parenthesis and a bracket.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831106373\" data-alt=\"The solution for x is greater than or equal to negative 1.5 on a number line has a left bracket with shading to the right. The solution in interval notation is negative 1.5 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_343_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is greater than or equal to negative 1.5 on a number line has a left bracket with shading to the right. The solution in interval notation is negative 1.5 to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834133022\" data-alt=\"The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_344_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834505878\">\n<div data-type=\"problem\" id=\"fs-id1167834505880\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29975ed14e5e39c77318da2f68ab3d0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#60;&#120;&#60;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b09fb4c6b7b9d784ce9b4ac3472ee20f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#32;&#120;&#60;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6352da57bd0465113d9a4c91ea82a594_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#49;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834097969\">\n<div data-type=\"problem\" id=\"fs-id1167834097971\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-049599758a2288a6dd98a2ce2cf3d840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#60;&#120;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"88\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46b710422e5dd0980deb8046fb9b70f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#120;&#60;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77a24705eec65a8fb0938ebfe1fa705c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#51;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834523866\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834064750\" data-alt=\"Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_348_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834238997\" data-alt=\"Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_349_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832053454\" data-alt=\"0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_350_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834279403\">\n<div data-type=\"problem\" id=\"fs-id1167834279405\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3554b6b6bae5d0a863c763b9688be84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#60;&#120;&#60;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"88\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45b869fada011a13e862a6f70c55a9de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#60;&#120;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bed2b3a3a2bf1eaeb8d49dfbc5c0e06c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#50;&#53;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"111\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835387177\">\n<div data-type=\"problem\" id=\"fs-id1167835387179\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6d2de4be3fb8dfb4842091a9a51135d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#60;&#120;&#60;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9091edc55c0093512fc99aaa7cc292bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#60;&#120;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3adea2c75ac7253e197c7c0a6b8881cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;&#55;&#53;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"111\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834583310\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834408545\" data-alt=\"Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_354_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835376533\" data-alt=\"Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_355_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834064673\" data-alt=\"Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_356_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835311117\"><strong data-effect=\"bold\">Solve Linear Inequalities<\/strong><\/p>\n<p id=\"fs-id1167835311123\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831922822\">\n<div data-type=\"problem\" id=\"fs-id1167831922824\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c21b7658731355a4c3e22cf7d4ad2dce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"81\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ad33f13b5f868069725f59af3bcfd84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#62;&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13c5a5770867b8380f5d52bc453aeb78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835282992\">\n<div data-type=\"problem\" id=\"fs-id1167835229616\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1a80bd17ee1668ec1d0504d758ae876_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb13d78a746f1eb41af9589e256e9840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#121;&#60;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-311d2ff403bd5acd36826d0403baabe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#48;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835363560\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831239561\" data-alt=\"The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_360_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834120655\" data-alt=\"The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_361_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831882629\" data-alt=\"The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_362_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835308740\">\n<div data-type=\"problem\" id=\"fs-id1167835308742\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fbc52d9f5429686da6b349191fcd4eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#50;&#48;&#125;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"103\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc768b066cb59f74edf7efb481cef857_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#116;&#92;&#103;&#101;&#32;&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f660bd94df16eb59eb6435def3c4391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#106;&#92;&#103;&#101;&#32;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834185458\">\n<div data-type=\"problem\" id=\"fs-id1167834372403\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44be974eded68bae94c846486949420c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#50;&#125;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"102\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acc603c62f9050ebdd7543ffdafdf4e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#115;&#60;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"73\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71352fea45d0777c2b59362ca717a6d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;&#103;&#92;&#108;&#101;&#32;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834533585\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832015591\" data-alt=\"The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_366_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834134703\" data-alt=\"The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_367_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835306108\" data-alt=\"The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_368_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835234273\">\n<div data-type=\"problem\" id=\"fs-id1167835234276\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-179e9045fc789824f8f5e8b42fb3075b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#117;&#92;&#103;&#101;&#32;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-469c542b32bf106e3f4e7365648cb04e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#45;&#51;&#125;&#92;&#108;&#101;&#32;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831846739\">\n<div data-type=\"problem\" id=\"fs-id1167831846741\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65a1e7421c636dfb0f2e58c383418613_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#118;&#92;&#108;&#101;&#32;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"73\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77820316a7065484ff09a0d776a5f7bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#45;&#49;&#48;&#125;&#92;&#103;&#101;&#32;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"68\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064605\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831890647\" data-alt=\"The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_371_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835330571\" data-alt=\"The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_372_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832226771\">\n<div data-type=\"problem\" id=\"fs-id1167832226773\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be08d9ab4d3bc2976e743af893d307ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#99;&#60;&#49;&#50;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"80\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-500ad553f328fdcda437aee89480e503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835517775\">\n<div data-type=\"problem\" id=\"fs-id1167835517777\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cc5452d8e8554f5eda47c85d99c7463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#100;&#62;&#49;&#48;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"81\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17fe85721301a6692ce0d88700d7ec92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835305580\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167830699874\" data-alt=\"The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_375_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834395948\" data-alt=\"The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_376_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835357194\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835357198\">\n<div data-type=\"problem\" id=\"fs-id1167835357200\">\n<p id=\"fs-id1167835433698\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a21fa12b54d371e8fbfdbab233ef26f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#118;&#92;&#103;&#101;&#32;&#57;&#118;&#45;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"99\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835512207\">\n<div data-type=\"problem\" id=\"fs-id1167835512210\">\n<p id=\"fs-id1167835512212\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0d1bad3f631d9b57fbec824eb1554ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#117;&#92;&#108;&#101;&#32;&#56;&#117;&#45;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835329406\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835329410\" data-alt=\"The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_378_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835575754\">\n<p id=\"fs-id1167835575756\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e8be6cc969f88fc74f2eb2acb8bf0d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#113;&#60;&#55;&#113;&#45;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835512806\">\n<div data-type=\"problem\" id=\"fs-id1167835512808\">\n<p id=\"fs-id1167835512810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-baf23edd4b941f524af86536cfc6d087_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#112;&#62;&#49;&#52;&#112;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835302412\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834132718\" data-alt=\"The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_380_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115257\">\n<div data-type=\"problem\" id=\"fs-id1167831115259\">\n<p id=\"fs-id1167831115261\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89f8c47beeb9a6a33e192ed2a600ff3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#120;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#49;&#48;&#120;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"206\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826849467\">\n<div data-type=\"problem\" id=\"fs-id1167826849469\">\n<p id=\"fs-id1167826849471\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bbe3855ba5ceafd5577facc36599e9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#121;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#52;&#121;&#45;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826781047\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167826781051\" data-alt=\"The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_382_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479515\">\n<div data-type=\"problem\" id=\"fs-id1167835479517\">\n<p id=\"fs-id1167835479519\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af05303e4830a47ed9a149c7ed22c09c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#104;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#55;&#104;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835361714\">\n<div data-type=\"problem\" id=\"fs-id1167835361716\">\n<p id=\"fs-id1167835362972\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae40c4e35f79baedd4905b58f9d323d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#107;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#55;&#107;&#45;&#50;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830837665\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834428919\" data-alt=\"The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_384_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832151015\">\n<div data-type=\"problem\" id=\"fs-id1167832151017\">\n<p id=\"fs-id1167832151020\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b98082b5ddf89c397fe4fd8826033ba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#109;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#45;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"268\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831832843\">\n<div data-type=\"problem\" id=\"fs-id1167831832845\">\n<p id=\"fs-id1167831832848\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-692dbe98f08bd9d5211ef044706d7bab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#110;&#45;&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#57;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"249\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826997719\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167826997723\" data-alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_386_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834133774\">\n<div data-type=\"problem\" id=\"fs-id1167834094539\">\n<p id=\"fs-id1167834094542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5bae97ee6932e259949f4359f3ac199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#98;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#98;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&#98;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835262304\">\n<div data-type=\"problem\" id=\"fs-id1167835262306\">\n<p id=\"fs-id1167831883461\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc5ba67c1e04ea1f4bed26e4686af0aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#117;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#117;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#55;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"255\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835379518\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835379522\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_388_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835530439\">\n<div data-type=\"problem\" id=\"fs-id1167835530442\">\n<p id=\"fs-id1167835530444\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cc6cee21ac4032661f11fa89a592ed6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#103;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#45;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#43;&#52;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"215\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831117186\">\n<div data-type=\"problem\" id=\"fs-id1167831117188\">\n<p id=\"fs-id1167831117190\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ecbe752d904ec1cfaa78249cefc7a16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#104;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#104;&#43;&#57;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"225\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228156\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834228160\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_390_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184410\">\n<div data-type=\"problem\" id=\"fs-id1167834184412\">\n<p id=\"fs-id1167834279140\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53136775052ba8f41095241c7d93e594_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#97;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"141\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834228204\">\n<div data-type=\"problem\" id=\"fs-id1167834228206\">\n<p id=\"fs-id1167834228208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c93077631ecd029c6eabb1ae74b87e15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#118;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#118;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#49;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#118;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"259\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835514619\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835514623\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_392_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835519277\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835519281\">\n<div data-type=\"problem\" id=\"fs-id1167835519283\">\n<p id=\"fs-id1167831887178\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50e60c81373fa087b0b492e11eb8bfed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#107;&#92;&#108;&#101;&#32;&#45;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832214484\">\n<div data-type=\"problem\" id=\"fs-id1167835215265\">\n<p id=\"fs-id1167835215267\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-236410a06bde0a55b514e954b2e164e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#107;&#92;&#103;&#101;&#32;&#45;&#55;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835186339\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835186343\" data-alt=\"The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_394_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184690\">\n<div data-type=\"problem\" id=\"fs-id1167834184692\">\n<p id=\"fs-id1167835303474\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91b7e8a4050e137383dff671a171a20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#112;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#53;&#112;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"228\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835622040\">\n<div data-type=\"problem\" id=\"fs-id1167835622042\">\n<p id=\"fs-id1167835622044\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d14373e0d9e217361cc8f5ce479a6ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#113;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#45;&#51;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#113;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"226\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834346562\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834346566\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_396_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826996576\">\n<div data-type=\"problem\" id=\"fs-id1167826996578\">\n<p id=\"fs-id1167834228852\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cc1d0c4c409d0dcb31e541769698953_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;&#120;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"87\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835381056\">\n<div data-type=\"problem\" id=\"fs-id1167835381058\">\n<p id=\"fs-id1167835381060\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c2427179cd472b6cba3d4fafdf3f727_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#56;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828434538\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167828434543\" data-alt=\"The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_398_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835288340\">\n<div data-type=\"problem\" id=\"fs-id1167835288342\">\n<p id=\"fs-id1167835288344\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-325e800a1fc8edee50897f0c9205aec7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#43;&#51;&#52;&#60;&#45;&#57;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830836892\">\n<div data-type=\"problem\" id=\"fs-id1167832031145\">\n<p id=\"fs-id1167832031147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7ea95db70330c0d1df926a8b8a0ae89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#43;&#50;&#57;&#62;&#45;&#54;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832043391\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832043395\" data-alt=\"The solution is g is greater than negative 90. The solution on a number line has a left parenthesis at negative 90 with shading to the right. The solution in interval notation is negative 90 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_400_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is g is greater than negative 90. The solution on a number line has a left parenthesis at negative 90 with shading to the right. The solution in interval notation is negative 90 to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834252306\">\n<div data-type=\"problem\" id=\"fs-id1167834252309\">\n<p id=\"fs-id1167834252311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa4178aa3072d4cfb4cf30131d0e7f7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#49;&#56;&#125;&#92;&#103;&#101;&#32;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835349143\">\n<div data-type=\"problem\" id=\"fs-id1167835349145\">\n<p id=\"fs-id1167835349147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99f68e42394e41343240f1282a41f3b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#49;&#51;&#125;&#92;&#108;&#101;&#32;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835517862\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835421713\" data-alt=\"The solution is n is less than or equal to negative 78. The solution on a number line has a right bracket at negative 78 with shading to the left. The solution in interval notation is negative infinity to negative 78 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_402_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is n is less than or equal to negative 78. The solution on a number line has a right bracket at negative 78 with shading to the left. The solution in interval notation is negative infinity to negative 78 within a parenthesis and a bracket.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832053963\"><strong data-effect=\"bold\">Translate to an Inequality and Solve<\/strong><\/p>\n<p id=\"fs-id1167832053969\">In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830961003\">\n<div data-type=\"problem\" id=\"fs-id1167830961005\">\n<p id=\"fs-id1167830961008\">Three more than <em data-effect=\"italics\">h<\/em> is no less than 25.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834431153\">\n<div data-type=\"problem\" id=\"fs-id1167831116560\">\n<p id=\"fs-id1167831116562\">Six more than <em data-effect=\"italics\">k<\/em> exceeds 25.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831116571\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835417487\" data-alt=\"The inequality is k plus 6 is greater than 25. Its solution is k is greater than 19. The solution on a number line has a left parenthesis at 19 with shading to the right. The solution in interval notation is 19 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_404_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is k plus 6 is greater than 25. Its solution is k is greater than 19. The solution on a number line has a left parenthesis at 19 with shading to the right. The solution in interval notation is 19 to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831970324\">\n<div data-type=\"problem\" id=\"fs-id1167831970326\">\n<p id=\"fs-id1167831970329\">Ten less than <em data-effect=\"italics\">w<\/em> is at least 39.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826996665\">\n<div data-type=\"problem\" id=\"fs-id1167826996667\">\n<p id=\"fs-id1167826996669\">Twelve less than <em data-effect=\"italics\">x<\/em> is no less than 21.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835356207\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835356212\" data-alt=\"The inequality is x minus 12 is greater than or equal to 21. Its solution is x is greater than or equal to 33. The solution on a number line has a left bracket at 33 with shading to the right. The solution in interval notation is 33 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_406_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is x minus 12 is greater than or equal to 21. Its solution is x is greater than or equal to 33. The solution on a number line has a left bracket at 33 with shading to the right. The solution in interval notation is 33 to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835327956\">\n<div data-type=\"problem\" id=\"fs-id1167835327958\">\n<p id=\"fs-id1167835327960\">Negative five times <em data-effect=\"italics\">r<\/em> is no more than 95.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865920\">\n<div data-type=\"problem\" id=\"fs-id1167830865922\">\n<p id=\"fs-id1167830865924\">Negative two times <em data-effect=\"italics\">s<\/em> is lower than 56.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835323758\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835323763\" data-alt=\"The inequality is negative 2 s is less than 56. Its solution is s is greater than negative 28. The solution on a number line has a left parenthesis at negative 28 with shading to the right. The solution in interval notation is negative 28 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_408_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is negative 2 s is less than 56. Its solution is s is greater than negative 28. The solution on a number line has a left parenthesis at negative 28 with shading to the right. The solution in interval notation is negative 28 to infinity within parentheses.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835483755\">\n<div data-type=\"problem\" id=\"fs-id1167835483757\">\n<p id=\"fs-id1167834279498\">Nineteen less than <em data-effect=\"italics\">b<\/em> is at most <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-feffbc35629ce777232642973e88e58a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835305054\">\n<div data-type=\"problem\" id=\"fs-id1167835254296\">\n<p id=\"fs-id1167835254298\">Fifteen less than <em data-effect=\"italics\">a<\/em> is at least <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-095e1dfdc93a80eb2c8554aaa37b4f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352382\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835352386\" data-alt=\"The inequality is a minus 15 is greater than or equal to negative 7. Its solution is a is greater than or equal to 8. The solution on a number line has a left bracket at 8 with shading to the right. The solution in interval notation is 8 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_410_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a minus 15 is greater than or equal to negative 7. Its solution is a is greater than or equal to 8. The solution on a number line has a left bracket at 8 with shading to the right. The solution in interval notation is 8 to infinity within a bracket and a parenthesis.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834063416\"><strong data-effect=\"bold\">Solve Applications with Linear Inequalities<\/strong><\/p>\n<p id=\"fs-id1167834346719\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834346722\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834346724\">\n<p id=\"fs-id1167834120757\">Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830704639\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830704641\">\n<p id=\"fs-id1167830704643\">The elevator in Yehire\u2019s apartment building has a sign that says the maximum weight is 2100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835331980\">\n<p id=\"fs-id1167835331982\">A maximum of 14 people can safely ride in the elevator.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835331987\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835331990\">\n<p id=\"fs-id1167834179717\">Andre is looking at apartments with three of his friends. They want the monthly rent to be no more than ?2,360. If the roommates split the rent evenly among the four of them, what is the maximum rent each will pay?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831894139\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831894141\">\n<p id=\"fs-id1167831894143\">Arleen got a ?20 gift card for the coffee shop. Her favorite iced drink costs ?3.79. What is the maximum number of drinks she can buy with the gift card?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834246891\">\n<p id=\"fs-id1167834246893\">five drinks<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835375582\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835375584\">\n<p id=\"fs-id1167835375586\">Teegan likes to play golf. He has budgeted ?60 next month for the driving range. It costs him ?10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831894612\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831894614\">\n<p id=\"fs-id1167831894616\">Ryan charges his neighbors ?17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least ?1,500?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835512300\">\n<p id=\"fs-id1167835512303\">86 cars<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834422606\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834422608\">\n<p id=\"fs-id1167834422610\">Keshad gets paid ?2,400 per month plus 6% of his sales. His brother earns ?3,300 per month. For what amount of total sales will Keshad\u2019s monthly pay be higher than his brother\u2019s monthly pay?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835345550\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835345552\">\n<p id=\"fs-id1167835345555\">Kimuyen needs to earn ?4,150 per month in order to pay all her expenses. Her job pays her ?3,475 per month plus 4% of her total sales. What is the minimum Kimuyen\u2019s total sales must be in order for her to pay all her expenses?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826997350\">\n<p id=\"fs-id1167826997352\">?16,875<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997357\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826997360\">\n<p id=\"fs-id1167826997362\">Andre has been offered an entry-level job. The company offered him ?48,000 per year plus 3.5% of his total sales. Andre knows that the average pay for this job is ?62,000. What would Andre\u2019s total sales need to be for his pay to be at least as high as the average pay for this job?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835367892\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835367894\">\n<p id=\"fs-id1167831837548\">Nataly is considering two job offers. The first job would pay her ?83,000 per year. The second would pay her ?66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831837555\">\n<p id=\"fs-id1167831837557\">?110,000<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832057715\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832057717\">\n<p id=\"fs-id1167832057719\">Jake\u2019s water bill is ?24.80 per month plus ?2.20 per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than ?60?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834459074\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834459076\">\n<p id=\"fs-id1167834459079\">Kiyoshi\u2019s phone plan costs ?17.50 per month plus ?0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than ?56.60?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834156998\">\n<p id=\"fs-id1167834157000\">260 messages<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537854\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834537856\">\n<p id=\"fs-id1167834537858\">Marlon\u2019s TV plan costs ?49.99 per month plus ?5.49 per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of ?100?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835346052\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835346054\">\n<p id=\"fs-id1167835346056\">Kellen wants to rent a banquet room in a restaurant for her cousin\u2019s baby shower. The restaurant charges ?350 for the banquet room plus ?32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be ?1,500?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319247\">\n<p id=\"fs-id1167835319249\">35 people<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319255\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835319257\">\n<p id=\"fs-id1167835319259\">Moshde runs a hairstyling business from her house. She charges ?45 for a haircut and style. Her monthly expenses are ?960. She wants to be able to put at least ?1,200 per month into her savings account order to open her own salon. How many \u201ccut &amp; styles\u201d must she do to save at least ?1,200 per month?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835367251\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835367254\">\n<p id=\"fs-id1167835367256\">Noe installs and configures software on home computers. He charges ?125 per job. His monthly expenses are ?1,600. How many jobs must he work in order to make a profit of at least ?2,400?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831103748\">\n<p id=\"fs-id1167831103750\">32 jobs<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831103755\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831103758\">\n<p id=\"fs-id1167831103760\">Katherine is a personal chef. She charges ?115 per four-person meal. Her monthly expenses are ?3,150. How many four-person meals must she sell in order to make a profit of at least ?1,900?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835433645\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835433647\">\n<p id=\"fs-id1167835433649\">Melissa makes necklaces and sells them online. She charges ?88 per necklace. Her monthly expenses are ?3,745. How many necklaces must she sell if she wants to make a profit of at least ?1,650?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835433655\">\n<p id=\"fs-id1167835362346\">62 necklaces<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835362352\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835362354\">\n<p id=\"fs-id1167835362356\">Five student government officers want to go to the state convention. It will cost them ?110 for registration, ?375 for transportation and food, and ?42 per person for the hotel. There is ?450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge ?5 per car, how many cars must they wash in order to have enough money to pay for the trip?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835420294\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835420297\">\n<p id=\"fs-id1167835420299\">Cesar is planning a four-day trip to visit his friend at a college in another state. It will cost him ?198 for airfare, ?56 for local transportation, and ?45 per day for food. He has ?189 in savings and can earn ?35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420306\">\n<p id=\"fs-id1167831890418\">seven lawns<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831890423\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831890425\">\n<p id=\"fs-id1167831890427\">Alonzo works as a car detailer. He charges ?175 per car. He is planning to move out of his parents\u2019 house and rent his first apartment. He will need to pay ?120 for application fees, ?950 for security deposit, and first and last months\u2019 rent at ?1,140 per month. He has ?1,810 in savings. How many cars must he detail to have enough money to rent the apartment?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835355843\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835355845\">\n<p id=\"fs-id1167835355847\">Eun-Kyung works as a tutor and earns ?60 per hour. She has ?792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her ?1,520 for food and drinks and ?150 for the photographer. She will also have a favor for each of the guests, and each favor will cost ?7.50. How many hours must she tutor to have enough money for the party?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834426224\">\n<p id=\"fs-id1167834340054\">20 hours<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"everyday\" data-depth=\"2\" id=\"fs-id1167834340060\">\n<h4 data-type=\"title\">Everyday Math<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835369660\">\n<div data-type=\"problem\" id=\"fs-id1167835369662\">\n<p id=\"fs-id1167835369664\"><strong data-effect=\"bold\">Maximum load on a stage<\/strong> In 2014, a high school stage collapsed in Fullerton, California, when 250 students got on stage for the finale of a musical production. Two dozen students were injured. The stage could support a maximum of 12,750 pounds. If the average weight of a student is assumed to be 140 pounds, what is the maximum number of students who could safely be on the stage?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835327996\">\n<div data-type=\"problem\" id=\"fs-id1167835327998\">\n<p id=\"fs-id1167834156529\"><strong data-effect=\"bold\">Maximum weight on a boat<\/strong> In 2004, a water taxi sank in Baltimore harbor and five people drowned. The water taxi had a maximum capacity of 3,500 pounds (25 people with average weight 140 pounds). The average weight of the 25 people on the water taxi when it sank was 168 pounds per person. What should the maximum number of people of this weight have been?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834156540\">\n<p id=\"fs-id1167835301762\">20 people<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301768\">\n<div data-type=\"problem\" id=\"fs-id1167835301770\">\n<p id=\"fs-id1167835301772\"><strong data-effect=\"bold\">Wedding budget<\/strong> Adele and Walter found the perfect venue for their wedding reception. The cost is ?9850 for up to 100 guests, plus ?38 for each additional guest. How many guests can attend if Adele and Walter want the total cost to be no more than ?12,500?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120487\">\n<div data-type=\"problem\" id=\"fs-id1167834120489\">\n<p id=\"fs-id1167834120491\"><strong data-effect=\"bold\">Shower budget<\/strong> Penny is planning a baby shower for her daughter-in-law. The restaurant charges ?950 for up to 25 guests, plus ?31.95 for each additional guest. How many guests can attend if Penny wants the total cost to be no more than ?1,500?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835308073\">\n<p id=\"fs-id1167835308075\">42 guests<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835308082\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835349227\">\n<div data-type=\"problem\" id=\"fs-id1167835349229\">\n<p id=\"fs-id1167835349232\">Explain why it is necessary to reverse the inequality when solving <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bba762c0c95f987772e98f0f7aace63c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#120;&#62;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"77\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835347768\">\n<div data-type=\"problem\" id=\"fs-id1167835347770\">\n<p id=\"fs-id1167835347772\">Explain why it is necessary to reverse the inequality when solving <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5a2be1885be0393535746726c13a0f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#45;&#51;&#125;&#60;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834185411\">\n<p id=\"fs-id1167834185413\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834185419\">\n<div data-type=\"problem\" id=\"fs-id1167834423473\">\n<p id=\"fs-id1167834423475\">Find your last month\u2019s phone bill and the hourly salary you are paid at your job. Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it. Do you feel this is an appropriate number of hours? Is this the appropriate phone plan for you?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319903\">\n<div data-type=\"problem\" id=\"fs-id1167835319905\">\n<p id=\"fs-id1167831191520\">Find out how many units you have left, after this term, to achieve your college goal and estimate the number of units you can take each term in college. Calculate the number of terms it will take you to achieve your college goal by writing an appropriate inequality and then solving it. Is this an acceptable number of terms until you meet your goal? What are some ways you could accelerate this process?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831191528\">\n<p id=\"fs-id1167831191530\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834191261\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834191266\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826783877\" data-alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was graph inequalities on the number line. In row 3, the I can was solve linear inequalities. In row 4, the I can was translate words to an inequality and solve. In row 5, the I can was solve applications with linear inequalities.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_05_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was graph inequalities on the number line. In row 3, the I can was solve linear inequalities. In row 4, the I can was translate words to an inequality and solve. In row 5, the I can was solve applications with linear inequalities.\" \/><\/span><\/p>\n<p id=\"fs-id1167834517174\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1469","chapter","type-chapter","status-publish","hentry"],"part":991,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1469","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1469\/revisions"}],"predecessor-version":[{"id":15157,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1469\/revisions\/15157"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/991"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1469\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=1469"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=1469"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=1469"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=1469"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}