{"id":1570,"date":"2018-12-11T13:34:20","date_gmt":"2018-12-11T18:34:20","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-compound-inequalities\/"},"modified":"2019-05-22T16:27:25","modified_gmt":"2019-05-22T20:27:25","slug":"solve-compound-inequalities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-compound-inequalities\/","title":{"raw":"Solve Compound Inequalities","rendered":"Solve Compound 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>Solve compound inequalities with \u201cand\u201d<\/li>\r\n \t<li>Solve compound inequalities with \u201cor\u201d<\/li>\r\n \t<li>Solve applications with compound inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835389786\" class=\"be-prepared\">\r\n<p id=\"fs-id1167835510986\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol id=\"fs-id1167835192422\" type=\"1\">\r\n \t<li>Simplify: \\(\\frac{2}{5}\\phantom{\\rule{0.2em}{0ex}}\\left(x+10\\right).\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829788421\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Simplify: \\(\\text{\u2212}\\left(x-4\\right).\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829752757\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835338613\">\r\n<h3 data-type=\"title\">Solve Compound Inequalities with \u201cand\u201d<\/h3>\r\n<p id=\"fs-id1167834195041\">Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A <span data-type=\"term\">compound inequality<\/span> is made up of two inequalities connected by the word \u201cand\u201d or the word \u201cor.\u201d For example, the following are compound inequalities.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167835360046\" class=\"unnumbered\" data-label=\"\">\\(x+3&gt;-4\\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}4x-5\\le 3\\)<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1167835300537\" class=\"unnumbered\" data-label=\"\">\\(2\\left(y+1\\right)&lt;0\\phantom{\\rule{1em}{0ex}}\\text{or}\\phantom{\\rule{1em}{0ex}}y-5\\ge -2\\)<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835357414\">\r\n<div data-type=\"title\">Compound Inequality<\/div>\r\n<p id=\"fs-id1167834061509\">A <strong data-effect=\"bold\">compound inequality<\/strong> is made up of two inequalities connected by the word \u201cand\u201d or the word \u201cor.\u201d<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167834429410\">To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.<\/p>\r\n<p id=\"fs-id1167835362358\">To solve a compound inequality with the word \u201cand,\u201d we look for all numbers that make <em data-effect=\"italics\">both<\/em> inequalities true. To solve a compound inequality with the word \u201cor,\u201d we look for all numbers that make <em data-effect=\"italics\">either<\/em> inequality true.<\/p>\r\n<p id=\"fs-id1167834196849\">Let\u2019s start with the compound inequalities with \u201cand.\u201d Our solution will be the numbers that are solutions to <em data-effect=\"italics\">both<\/em> inequalities known as the intersection of the two inequalities. Consider how the intersection of two streets\u2014the part where the streets overlap\u2014belongs to both streets.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167834308105\" data-alt=\"The figure is an illustration of two streets with their intersection shaded\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_001_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is an illustration of two streets with their intersection shaded\" \/><\/span>\r\n<p id=\"fs-id1167835489259\">To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs\u2014where the graphs overlap.<\/p>\r\n<p id=\"fs-id1167834430724\">For the compound inequality \\(x&gt;-3\\) and \\(x\\le 2,\\) we graph each inequality. We then look for where the graphs \u201coverlap\u201d. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See <a href=\"#CNX_IntAlg_Figure_02_06_002\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_06_002\"><span data-type=\"media\" id=\"fs-id1167835341616\" data-alt=\"The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_002_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.\" \/><\/span><\/div>\r\n<p id=\"fs-id1167835309111\">We can see that the numbers between \\(-3\\) and \\(2\\) are shaded on both of the first two graphs. They will then be shaded on the solution graph.<\/p>\r\n<p id=\"fs-id1167835514644\">The number \\(-3\\) is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.<\/p>\r\n<p id=\"fs-id1167834053558\">The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.<\/p>\r\n<p id=\"fs-id1167826996735\">This is how we will show our solution in the next examples.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167826864192\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835325006\">\r\n<div data-type=\"problem\" id=\"fs-id1167835420949\">\r\n<p id=\"fs-id1167835390168\">Solve \\(6x-3&lt;9\\) and \\(2x+7\\ge 3.\\) Graph the solution and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791472258\">\r\n<table class=\"unnumbered unstyled\" summary=\"6 x minus 3 is less than 9 and 2 x plus 9 is greater than or equal to 3. Step 1 is to solve each inequality. 6 x is less than 12, which simplifies to x is less than 2. 2 x is greater than or equal to negative 6, which simplifies to x is greater than or equal to negative 3. Step 2 is to graph each solution. The graph of x is less than 2 has a right parenthesis at 2 and is shaded to its left. The graph of x is greater than or equal to negative 3 has a left bracket at negative 3 and is shaded to its right. Then graph the numbers that make both inequalities true. The final graph will show all the numbers that make both inequalities true\u2014the numbers shaded on both of the first two graphs. The graph of x is less than 2 and x is greater than or equal to negative 3 has a left bracket at negative 3 and a right parenthesis at 2 and is shaded between the bracket and parenthesis. Each graph is marked at negative 3 and 2. Step 3 is to write the solution in interval notation. It is negative 3 to 2 within a bracket and a parenthesis. All the numbers that make both inequalities true are the solution to the compound inequality.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(6x-3&lt;9\\phantom{\\rule{0.55em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x+9\\ge 3\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Solve each\r\n<div data-type=\"newline\"><\/div>\r\ninequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(6x-3&lt;9\\phantom{\\rule{0.55em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x+9\\ge 3\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(6x&lt;12\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x\\ge -6\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x&lt;2\\phantom{\\rule{0.55em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\ge -3\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Graph each solution. Then graph the numbers that make both inequalities true. The final graph will show all the numbers that make both inequalities true\u2014the numbers shaded on <em data-effect=\"italics\">both<\/em> of the first two graphs.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832060115\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_003a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Write the solution in interval notation.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\">\\(\\left[-3,2\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td colspan=\"4\" data-valign=\"top\" data-align=\"left\">All the numbers that make both inequalities true are the solution to the compound inequality.<\/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-id1167835322297\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832046520\">\r\n<div data-type=\"problem\" id=\"fs-id1167835339414\">\r\n<p id=\"fs-id1167834459102\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(4x-7&lt;9\\) and \\(5x+8\\ge 3.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830703172\"><span data-type=\"media\" id=\"fs-id1167835418946\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than 4. On a number line it is shown with a closed circle at negative 1 and an open circle at 4 with shading in between the closed and open circles. Its interval notation is negative 1 to 4 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_302_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than 4. On a number line it is shown with a closed circle at negative 1 and an open circle at 4 with shading in between the closed and open circles. Its interval notation is negative 1 to 4 within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835230092\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835303505\">\r\n<div data-type=\"problem\" id=\"fs-id1167826997274\">\r\n\r\nSolve the compound inequality. Graph the solution and write the solution in interval notation: \\(3x-4&lt;5\\) and \\(4x+9\\ge 1.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167834190135\" data-alt=\"The solution is negative 2 is less than or equal to x which is less than 3. On a number line it is shown with a closed circle at negative 2 and an open circle at 3 with shading in between the closed and open circles. Its interval notation is negative 2 to 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_06_303_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than or equal to x which is less than 3. On a number line it is shown with a closed circle at negative 2 and an open circle at 3 with shading in between the closed and open circles. Its interval notation is negative 2 to 3 within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834156842\" class=\"howto\">\r\n<div data-type=\"title\">Solve a compound inequality with \u201cand.\u201d<\/div>\r\n<ol id=\"fs-id1167835234281\" class=\"stepwise\" type=\"1\">\r\n \t<li>Solve each inequality.<\/li>\r\n \t<li>Graph each solution. Then graph the numbers that make <em data-effect=\"italics\">both<\/em> inequalities true.\r\n<div data-type=\"newline\"><\/div>\r\nThis graph shows the solution to the compound inequality.<\/li>\r\n \t<li>Write the solution in interval notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835345108\">\r\n<div data-type=\"problem\" id=\"fs-id1167835332470\">\r\n<p id=\"fs-id1167835575835\">Solve \\(3\\left(2x+5\\right)\\le 18\\) and \\(2\\left(x-7\\right)&lt;-6.\\) Graph the solution and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171784049192\">\r\n<table id=\"fs-id1167835531554\" class=\"unnumbered unstyled\" summary=\"3 times the quantity 2 x plus 5 is less than or equal to 18 and 2 times the quantity x minus 7 is less than negative 6. Solve each inequality. 3 times the quantity 2 x plus 5 is less than or equal to 18 simplifies to 6 x plus 15 is less than or equal to 18 which simplifies to 6 x is less than or equal to 3, which simplifies to x is less than or equal to one-half. 2 times the quantity x minus 7 is less than negative 6 simplifies to 2 x minus 14 is less than negative 6, which simplifies to 2 x is less than 8, which simplifies to x is less than 4. Graph each solution. The graph of x is less than or equal to one-half has a right bracket at one-half and is shaded to the left. The graph of x is less than 4 has a right parenthesis and is shaded to the left. Both graphs are marked at one-half. Graph the numbers that make both inequalities true. Graph the numbers that make both inequalities true. The graph has a right bracket at one-half and is shaded to the left. It is marked at one-half. Write the solution in interval notation. It is negative infinity and one-half within a parenthesis and a bracket.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(3\\left(2x+5\\right)\\le 18\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2\\left(x-7\\right)&lt;-6\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve each\r\n<div data-type=\"newline\"><\/div>\r\ninequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(6x+15\\le 18\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x-14&lt;-6\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(6x\\le 3\\phantom{\\rule{0.5em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x&lt;8\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\le \\frac{1}{2}\\phantom{\\rule{0.4em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x&lt;4\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph each\r\n<div data-type=\"newline\"><\/div>\r\nsolution.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835180552\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_004a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph the numbers\r\n<div data-type=\"newline\"><\/div>\r\nthat make both\r\n<div data-type=\"newline\"><\/div>\r\ninequalities true.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171780895282\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_004b_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Write the solution\r\n<div data-type=\"newline\"><\/div>\r\nin interval notation.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,\\frac{1}{2}\\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\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835419122\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167832226556\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(2\\left(3x+1\\right)\\le 20\\) and \\(4\\left(x-1\\right)&lt;2.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830865391\"><span data-type=\"media\" id=\"fs-id1167835263571\" data-alt=\"The solution is x is less than three-halves. On a number line it is shown with an open circle at three-halves with shading to its left. Its interval notation is negative infinity to three-halves within a parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_304_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than three-halves. On a number line it is shown with an open circle at three-halves with shading to its left. Its interval notation is negative infinity to three-halves within a parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832056538\">\r\n<div data-type=\"problem\" id=\"fs-id1167834533400\">\r\n\r\nSolve the compound inequality. Graph the solution and write the solution in interval notation: \\(5\\left(3x-1\\right)\\le 10\\) and \\(4\\left(x+3\\right)&lt;8.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834423488\"><span data-type=\"media\" id=\"fs-id1167835346662\" data-alt=\"The solution is x is less than negative 1. On a number line it is shown with an open circle at 1 with shading to its left. Its interval notation is negative infinity to negative 1 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_305_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 1. On a number line it is shown with an open circle at 1 with shading to its left. Its interval notation is negative infinity to negative 1 within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167835254174\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835349939\">\r\n<div data-type=\"problem\" id=\"fs-id1167835218133\">\r\n<p id=\"fs-id1167835234710\">Solve \\(\\frac{1}{3}x-4\\ge -2\\) and \\(-2\\left(x-3\\right)\\ge 4.\\) Graph the solution and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791298797\">\r\n<table id=\"fs-id1167835335470\" class=\"unnumbered unstyled\" summary=\"One-third x minus 4 is greater than or equal to negative 2 and negative 2 times the quantity x minus 3 is greater than or equal to 4. Solve each inequality. One-third x minus 4 is greater than or equal to negative 2 simplifies to x is greater than or equal to 6. Negative 2 times the quantity x minus 3 is greater than or equal to 4 simplifies to negative 2 x is greater than negative 2, which simplifies to x is less than or equal to 1. Graph each solution. The graph of x is greater than or equal to 6 has a left bracket at 6 and is shaded to its right. The graph of x is less than or equal to 1 has a right bracket at 1 and is shaded to its left. Graph the numbers that make both inequalities true. Notice that there are no numbers that make the inequalities true.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\frac{1}{3}x-4\\ge -2\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(-2\\left(x-3\\right)\\ge 4\\phantom{\\rule{0.75em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve each inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\frac{1}{3}x-4\\ge -2\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(-2x+6\\ge 4\\phantom{\\rule{0.75em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\frac{1}{3}x\\ge 2\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(-2x\\ge -2\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\ge 6\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\le 1\\phantom{\\rule{0.75em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph each solution.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835322051\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_005a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph the numbers that\r\n<div data-type=\"newline\"><\/div>\r\nmake both inequalities\r\n<div data-type=\"newline\"><\/div>\r\ntrue.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835166097\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_005b_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{8em}{0ex}}\\)There are no numbers that make both inequalities true.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{8em}{0ex}}\\)This is a contradiction so 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-id1167835254500\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835343501\">\r\n<div data-type=\"problem\" id=\"fs-id1167830699646\">\r\n<p id=\"fs-id1167831081633\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(\\frac{1}{4}x-3\\ge -1\\) and \\(-3\\left(x-2\\right)\\ge 2.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834473807\"><span data-type=\"media\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph of the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_306_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph of the number line or interval notation.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831824593\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835341663\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167832076363\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(\\frac{1}{5}x-5\\ge -3\\) and \\(-4\\left(x-1\\right)\\ge -2.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834185418\"><span data-type=\"media\" id=\"fs-id1167835180564\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_307_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834539332\">Sometimes we have a compound inequality that can be written more concisely. For example, \\(a&lt;x\\) and \\(x&lt;b\\) can be written simply as \\(a&lt;x&lt;b\\) and then we call it a <span data-type=\"term\" class=\"no-emphasis\">double inequality<\/span>. The two forms are equivalent.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167835323809\">\r\n<div data-type=\"title\">Double Inequality<\/div>\r\n<p id=\"fs-id1167835232788\">A double inequality is a compound inequality such as \\(a&lt;x&lt;b.\\) It is equivalent to \\(a&lt;x\\) and \\(x&lt;b.\\)<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167836286959\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\text{Other forms:}\\hfill &amp; &amp; &amp; \\begin{array}{ccccccccccccc}a&lt;x&lt;b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a&lt;x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x&lt;b\\hfill \\\\ a\\le x\\le b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a\\le x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x\\le b\\hfill \\\\ a&gt;x&gt;b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a&gt;x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x&gt;b\\hfill \\\\ a\\ge x\\ge b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a\\ge x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x\\ge b\\hfill \\end{array}\\hfill \\end{array}\\)<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835180817\">To solve a double inequality we perform the same operation on all three \u201cparts\u201d of the double inequality with the goal of isolating the variable in the center.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835312277\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835198760\">\r\n<div data-type=\"problem\" id=\"fs-id1167826987980\">\r\n<p id=\"fs-id1167834196398\">Solve \\(-4\\le 3x-7&lt;8.\\) Graph the solution and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835262370\">\r\n<table class=\"unnumbered unstyled\" summary=\"Negative 4 is less than or equal to 3 x minus 7 which is less than 8. Add 7 to all three parts. Negative 4 plus 7 is less than or equal to 3 x minus 7 plus 7 which is less than 8 plus 7. Simplify. 3 is less than or equal to 3 x which is less than 15. Divided each part by 3. 3 divided by 3 is less than or equal to 3 x divided by 3 which is less than 15 divided by 3. Simplify. The result is 1 is less than or equal to x which is less than 5. Graph the solution. The solution on a number line is a left bracket at 1, a right parenthesis at 5, and shading between the bracket and parenthesis. Write the solution in interval notation. It is 1 to 5 within a bracket and a parenthesis.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832059597\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Add 7 to all three parts.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835333058\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006b_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830865371\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006c_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Divide each part by three.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834161868\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006d_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835281520\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006e_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph the solution.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835330291\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006f_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" 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-id1167835330107\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006g_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<p id=\"fs-id1167835377148\">When written as a double inequality, \\(1\\le x&lt;5,\\) it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.<\/p>\r\nAnother way to graph the solution of \\(1\\le x&lt;5\\) is to graph both the solution of \\(x\\ge 1\\) and the solution of \\(x&lt;5.\\) We would then find the numbers that make both inequalities true as we did in previous examples.\r\n<div data-type=\"note\" id=\"fs-id1167834484818\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167827957278\">\r\n<div data-type=\"problem\" id=\"fs-id1167826778624\">\r\n<p id=\"fs-id1167835198886\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(-5\\le 4x-1&lt;7.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835231901\"><span data-type=\"media\" id=\"fs-id1167835370215\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 1 to 2 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_308_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 1 to 2 within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832227152\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835232564\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167835345122\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(-3&lt;2x-5\\le 1.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831922098\"><span data-type=\"media\" id=\"fs-id1167835180911\" data-alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 with shading between the closed and open circles. Its interval notation is negative 1 to 3 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_309_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 with shading between the closed and open circles. Its interval notation is negative 1 to 3 within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835170774\">\r\n<h3 data-type=\"title\">Solve Compound Inequalities with \u201cor\u201d<\/h3>\r\n<p id=\"fs-id1167835213802\">To solve a <span data-type=\"term\" class=\"no-emphasis\">compound inequality<\/span> with \u201cor\u201d, we start out just as we did with the compound inequalities with \u201cand\u201d\u2014we solve the two inequalities. Then we find all the numbers that make <em data-effect=\"italics\">either<\/em> inequality true.<\/p>\r\n<p id=\"fs-id1167835410279\">Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.<\/p>\r\n<p id=\"fs-id1167830769733\">To write the solution in <span data-type=\"term\" class=\"no-emphasis\">interval notation<\/span>, we will often use the <span data-type=\"term\" class=\"no-emphasis\">union symbol<\/span>, \\(\\cup \\), to show the union of the solutions shown in the graphs.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167835365153\" class=\"howto\">\r\n<div data-type=\"title\">Solve a compound inequality with \u201cor.\u201d<\/div>\r\n<ol id=\"fs-id1167832015984\" class=\"stepwise\" type=\"1\">\r\n \t<li>Solve each inequality.<\/li>\r\n \t<li>Graph each solution. Then graph the numbers that make either inequality true.<\/li>\r\n \t<li>Write the solution in interval notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167832015695\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835371428\">\r\n<div data-type=\"problem\" id=\"fs-id1167835305033\">\r\n<p id=\"fs-id1167831910174\">Solve \\(5-3x\\le -1\\) or \\(8+2x\\le 5.\\) Graph the solution and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791523544\">\r\n<table id=\"fs-id1167834064709\" class=\"unnumbered unstyled\" summary=\"5 minus 3 x is less than or equal to negative 1 or 8 plus 2 x is less than or equal to 5. Solve each inequality. 5 minus 3 x is less than or equal to negative 1 simplifies to negative 3 x is less than or equal to negative 6, which simplifies to x is greater than or equal to 2. 8 plus 2 x is less than or equal to 5 simplifies to 2 x is less than or equal to negative 3, which simplifies to x is less than or equal to negative three halves. Graph each solution. The graph of x is greater than or equal to 2 has a left bracket at 2 and is shaded to its right. The graph of x is less than or equal to negative three halves has a right bracket at negative three halves and is shaded to its left. Graph numbers that make either inequality true. The graph shows a right bracket at negative three-halves with shading to the left and a left bracket at 2 with shading to the right. So, the interval notation of the solution is the union of negative infinity to negative three-halves within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(5-3x\\le -1\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(8+2x\\le 5\\phantom{\\rule{1.25em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve each inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(5-3x\\le -1\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(8+2x\\le 5\\phantom{\\rule{1.25em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(-3x\\le -6\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\phantom{\\rule{3em}{0ex}}2x\\le -3\\phantom{\\rule{0.1em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\ge 2\\phantom{\\rule{0.65em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\phantom{\\rule{3.2em}{0ex}}x\\le -\\frac{3}{2}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph each solution.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535213\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph numbers that\r\n<div data-type=\"newline\"><\/div>\r\nmake either inequality\r\n<div data-type=\"newline\"><\/div>\r\ntrue.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835302326\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,-\\frac{3}{2}\\right]\\cup \\left[2,\\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-id1167835305860\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835377510\">\r\n<div data-type=\"problem\" id=\"fs-id1167831047475\">\r\n\r\nSolve the compound inequality. Graph the solution and write the solution in interval notation: \\(1-2x\\le -3\\) or \\(7+3x\\le 4.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834537118\"><span data-type=\"media\" id=\"fs-id1167835307405\" data-alt=\"The solution is x is greater than or equal to 2 or x is less than or equal to 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 2 and 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_06_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to 2 or x is less than or equal to 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 2 and infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832053266\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831880592\">\r\n<div data-type=\"problem\" id=\"fs-id1167832053590\">\r\n<p id=\"fs-id1167832058374\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(2-5x\\le -3\\) or \\(5+2x\\le 3.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835264543\"><span data-type=\"media\" id=\"fs-id1167832060300\" data-alt=\"The solution is x is greater than or equal to 1 or x is less than or equal to negative 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 1 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 1 and 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_06_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to 1 or x is less than or equal to negative 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 1 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 1 and infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167835303828\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834184113\">\r\n<div data-type=\"problem\" id=\"fs-id1167835640387\">\r\n<p id=\"fs-id1167832056412\">Solve \\(\\frac{2}{3}x-4\\le 3\\) or \\(\\frac{1}{4}\\left(x+8\\right)\\ge -1.\\) Graph the solution and write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171791538670\">\r\n<table id=\"fs-id1167835326892\" class=\"unnumbered unstyled can-break\" summary=\"Two-third x minus 4 is less than or equal to 3 or one-fourth times the quantity x plus 8 is greater than or equal to negative 1. Solve each inequality. Two-third x minus 4 is less than or equal to 3 simplifies to two-thirds x is less than or equal to 7 which simplifies to three-halves times two-thirds x is less than or equal to three-halves times 7 ones, which simplifies to x is less than or equal to twenty-one halves. One-fourth times the quantity x plus 8 simplifies to one-fourth x plus 2 is greater than negative 1 which simplifies to one-fourth x is greater than negative 3 which simplifies to x is greater than or equal to negative 12. Graph each solution. The graph of x is less than twenty-one halves has a right bracket at twenty-one halves and is shaded to the left. The graph of x is greater than or equal to negative 12 has a left bracket at negative 12 and is shaded to the right. Graph numbers that make either inequality true. All values are shaded on the number line. The solution is all real numbers. The interval notation is negative infinity to infinity within parentheses.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\frac{2}{3}x-4\\le 3\\phantom{\\rule{1.2em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\frac{1}{4}\\left(x+8\\right)\\ge -1\\phantom{\\rule{1.8em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve each\r\n<div data-type=\"newline\"><\/div>\r\ninequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(\\phantom{\\rule{0.4em}{0ex}}3\\left(\\frac{2}{3}x-4\\right)\\le 3\\left(3\\right)\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(4\u00b7\\frac{1}{4}\\left(x+8\\right)\\ge 4\u00b7\\left(-1\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x-12\\le 9\\phantom{\\rule{1.2em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x+8\\ge -4\\phantom{\\rule{1.75em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(2x\\le 21\\phantom{\\rule{0.7em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\ge -12\\phantom{\\rule{1.25em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\le \\frac{21}{2}\\phantom{\\rule{0.55em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\le \\frac{21}{2}\\phantom{\\rule{0.55em}{0ex}}\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\r\n<td data-valign=\"top\" data-align=\"right\">\\(x\\ge -12\\phantom{\\rule{1.25em}{0ex}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph each\r\n<div data-type=\"newline\"><\/div>\r\nsolution.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832010254\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph numbers\r\n<div data-type=\"newline\"><\/div>\r\nthat make either\r\n<div data-type=\"newline\"><\/div>\r\ninequality true.<\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835264484\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\">The solution covers all real numbers.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td colspan=\"3\" 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-id1167835279876\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834246594\">\r\n<div data-type=\"problem\" id=\"fs-id1167835308584\">\r\n<p id=\"fs-id1167831923646\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(\\frac{3}{5}x-7\\le -1\\) or \\(\\frac{1}{3}\\left(x+6\\right)\\ge -2.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834234020\"><span data-type=\"media\" id=\"fs-id1167834064229\" data-alt=\"The solution 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_06_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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-id1167831883742\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834099096\">\r\n<div data-type=\"problem\" id=\"fs-id1167834099098\">\r\n<p id=\"fs-id1167832055008\">Solve the compound inequality. Graph the solution and write the solution in interval notation: \\(\\frac{3}{4}x-3\\le 3\\) or \\(\\frac{2}{5}\\left(x+10\\right)\\ge 0.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831955858\"><span data-type=\"media\" id=\"fs-id1167834534344\" data-alt=\"The solution 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_06_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826851372\">\r\n<h3 data-type=\"title\">Solve Applications with Compound Inequalities<\/h3>\r\n<p id=\"fs-id1167835421464\">Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.<\/p>\r\n<p id=\"fs-id1167834428991\">Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a <span data-type=\"term\" class=\"no-emphasis\">compound inequality<\/span>. Last, we will solve the compound inequality.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167834185809\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834185811\">\r\n<div data-type=\"problem\" id=\"fs-id1167831846923\">\r\n<p id=\"fs-id1167835317903\">Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.<\/p>\r\n<p id=\"fs-id1167835317906\">During the summer, a property owner will pay ?24.72 plus ?1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to ?57.06 and ?171.02. How many hcf can the owner use if he wants his usage to stay in the normal range?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831036915\">\r\n<table id=\"fs-id1167835416828\" class=\"unnumbered unstyled\" summary=\"Identify what we are looking for. We are looking for the number of h c f he can use and stay in the normal usage billing range. Name what we are looking for. Let x e equal to the number of h c f he can use. Translate to an inequality. The bill is 24 dollars and 72 cents plus 1 dollar and 54 cents times the number of h c f he uses. That is 24.72 plus 1.54 x. His bill will be between or equal to 57 dollars and 6 cents and 171 dollars and 2 cents. That is 57.06 is less than or equal to 24.72 plus 1.54 x which is less than 171.02. Solve the inequality. 57.06 minus 24.72 is less than or equal to 24.72 minus 24.72 plus 1.54 x which is less than 171.02 minus 24.72. 32.34 is less than or equal to 1.54 x which is less than or equal to 146.3. 32.34 divided by 1.54 is less than or equal to 1.54 x divided by 1.54 which is less than or equal to 146.3 divided by 1.54. The result is 21 is less than or equal to x which is less than or equal to 95. Answer the question. The property owner can use 21 to 95 h c f and still fall within the normal usage billing range.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Identify what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The number of hcf he can use and stay in the \u201cnormal usage\u201d billing range.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Name what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(x=\\) the number of hcf he can use.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Translate to an inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Bill is ?24.72 plus ?1.54 times the number of hcf he uses or \\(24.72+1.54x.\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835614910\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve the inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834151973\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Answer the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The property owner can use 21\u201395 hcf and still fall within the \u201cnormal usage\u201d billing range.<\/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-id1167835322185\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835269568\">\r\n<div data-type=\"problem\" id=\"fs-id1167834214035\">\r\n<p id=\"fs-id1167834214037\">Due to the drought in California, many communities now have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.<\/p>\r\n<p id=\"fs-id1167835325768\">During the summer, a property owner will pay ?24.72 plus ?1.32 per hcf for Conservation Usage. The bill for Conservation Usage would be between or equal to ?31.32 and ?52.12. How many hcf can the owner use if she wants her usage to stay in the conservation range?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835325771\">\r\n<p id=\"fs-id1167835307820\">The homeowner can use 5\u201320 hcf and still fall within the \u201cconservation usage\u201d billing range.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832010482\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831911604\">\r\n<div data-type=\"problem\" id=\"fs-id1167831911606\">\r\n<p id=\"fs-id1167832153621\">Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.<\/p>\r\n<p id=\"fs-id1167832153624\">During the winter, a property owner will pay ?24.72 plus ?1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to ?49.36 and ?86.32. How many hcf will he be allowed to use if he wants his usage to stay in the normal range?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826880107\">\r\n<p id=\"fs-id1167826880110\">The homeowner can use 16\u201340 hcf and still fall within the \u201cnormal usage\u201d billing range.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835419051\" class=\"media-2\">\r\n<p id=\"fs-id1167835237414\">Access this online resource for additional instruction and practice with solving compound inequalities.<\/p>\r\n\r\n<ul id=\"fs-id1167835350269\" data-bullet-style=\"bullet\">\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37compinequalit\">Compound inequalities<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835609421\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1167834430991\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">How to solve a compound inequality with \u201cand\u201d<\/strong>\r\n<ol id=\"fs-id1167834161616\" class=\"stepwise\" type=\"1\">\r\n \t<li>Solve each inequality.<\/li>\r\n \t<li>Graph each solution. Then graph the numbers that make <em data-effect=\"italics\">both<\/em> inequalities true. This graph shows the solution to the compound inequality.<\/li>\r\n \t<li>Write the solution in interval notation.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Double Inequality<\/strong>\r\n<ul id=\"fs-id1167834192375\" data-bullet-style=\"bullet\">\r\n \t<li>A <strong data-effect=\"bold\">double inequality<\/strong> is a compound inequality such as \\(a&lt;x&lt;b\\). It is equivalent to \\(a&lt;x\\) and \\(x&lt;b.\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccc}\\text{Other forms:}\\hfill &amp; &amp; &amp; \\begin{array}{ccccccccccccc}a&lt;x&lt;b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a&lt;x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x&lt;b\\hfill \\\\ a\\le x\\le b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a\\le x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x\\le b\\hfill \\\\ a&gt;x&gt;b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a&gt;x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x&gt;b\\hfill \\\\ a\\ge x\\ge b\\hfill &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; a\\ge x\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; x\\ge b\\hfill \\end{array}\\hfill \\end{array}\\)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to solve a compound inequality with \u201cor\u201d<\/strong>\r\n<ol id=\"fs-id1167835343531\" class=\"stepwise\" type=\"1\">\r\n \t<li>Solve each inequality.<\/li>\r\n \t<li>Graph each solution. Then graph the numbers that make either inequality true.<\/li>\r\n \t<li>Write the solution in interval notation.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835305325\">\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834085098\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<strong data-effect=\"bold\">Solve Compound Inequalities with \u201cand\u201d<\/strong>\r\n<p id=\"fs-id1167835353641\">In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167835410923\">\r\n<div data-type=\"problem\" id=\"fs-id1167835410925\">\r\n<p id=\"fs-id1167835389977\">\\(x&lt;3\\) and \\(x\\ge 1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835365420\">\r\n<div data-type=\"problem\" id=\"fs-id1167835365422\">\r\n<p id=\"fs-id1167831891680\">\\(x\\le 4\\) and \\(x&gt;-2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835236771\"><span data-type=\"media\" id=\"fs-id1167831239004\" data-alt=\"The solution is negative 2 is less than x which is less than or equal to 4. Its graph has an open circle at 1negative 2 and a closed circle at 4 with shading between the open and closed circles. Its interval notation is negative 2 to 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_06_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than x which is less than or equal to 4. Its graph has an open circle at 1negative 2 and a closed circle at 4 with shading between the open and closed circles. Its interval notation is negative 2 to 4 within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835172044\">\r\n<div data-type=\"problem\">\r\n\r\n\\(x\\ge -4\\) and \\(x\\le -1\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832226738\">\r\n<div data-type=\"problem\" id=\"fs-id1167832226740\">\r\n<p id=\"fs-id1167835345642\">\\(x&gt;-6\\) and \\(x&lt;-3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832058417\"><span data-type=\"media\" id=\"fs-id1167832036194\" data-alt=\"The solution is negative 6 is less than x which is less than negative 3. Its graph has an open circle at negative 6 and an open circle at negative 3 with shading between open circles. Its interval notation is negative 6 to negative 3 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 6 is less than x which is less than negative 3. Its graph has an open circle at negative 6 and an open circle at negative 3 with shading between open circles. Its interval notation is negative 6 to negative 3 within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835479842\">\r\n<div data-type=\"problem\" id=\"fs-id1167835479844\">\r\n<p id=\"fs-id1167835334128\">\\(5x-2&lt;8\\) and \\(6x+9\\ge 3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831920497\">\r\n<div data-type=\"problem\" id=\"fs-id1167831920499\">\r\n<p id=\"fs-id1167834431736\">\\(4x-1&lt;7\\) and \\(2x+8\\ge 4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835361932\"><span data-type=\"media\" id=\"fs-id1167834061494\" data-alt=\"The solution is negative 2 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 2 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 2 to 2 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 2 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 2 to 2 within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834228215\">\r\n<div data-type=\"problem\" id=\"fs-id1167834228218\">\r\n<p id=\"fs-id1167835640049\">\\(4x+6\\le 2\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(2x+1\\ge -5\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835519126\">\r\n<div data-type=\"problem\" id=\"fs-id1167831896617\">\r\n<p id=\"fs-id1167831896619\">\\(4x-2\\le 4\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(7x-1&gt;-8\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835332277\"><span data-type=\"media\" id=\"fs-id1167835331595\" data-alt=\"The solution is negative 1 is less than x which is less than or equal to three-halves. Its graph has an open circle at negative 1 and a closed circle at three-halves with shading between the open and closed circles. Its interval notation is negative 1 to three-halves within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_321_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than or equal to three-halves. Its graph has an open circle at negative 1 and a closed circle at three-halves with shading between the open and closed circles. Its interval notation is negative 1 to three-halves within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834061295\">\r\n<div data-type=\"problem\" id=\"fs-id1167835416534\">\r\n<p id=\"fs-id1167835416536\">\\(2x-11&lt;5\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3x-8&gt;-5\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834376303\">\r\n<div data-type=\"problem\" id=\"fs-id1167835330479\">\r\n<p id=\"fs-id1167835330481\">\\(7x-8&lt;6\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(5x+7&gt;-3\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835310981\"><span data-type=\"media\" id=\"fs-id1167834194399\" data-alt=\"The solution is negative 2 is less than x which is less than 2. Its graph has an open circle at negative 2 and an open circle at 2 with shading between the open circles. Its interval notation is negative 2 to 2 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_323_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than x which is less than 2. Its graph has an open circle at negative 2 and an open circle at 2 with shading between the open circles. Its interval notation is negative 2 to 2 within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835280396\">\r\n<div data-type=\"problem\" id=\"fs-id1167835280398\">\r\n\r\n\\(4\\left(2x-1\\right)\\le 12\\) and\r\n<div data-type=\"newline\"><\/div>\r\n\\(2\\left(x+1\\right)&lt;4\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835395541\">\r\n<div data-type=\"problem\" id=\"fs-id1167835395544\">\r\n<p id=\"fs-id1167835395546\">\\(5\\left(3x-2\\right)\\le 5\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3\\left(x+3\\right)&lt;3\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167827943653\"><span data-type=\"media\" id=\"fs-id1167827943656\" data-alt=\"The solution is x is less than negative 2. Its graph has an open circle at negative 2 and is shaded to the left. Its interval notation is negative infinity to negative 2 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_325_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2. Its graph has an open circle at negative 2 and is shaded to the left. Its interval notation is negative infinity to negative 2 within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832054794\">\r\n<div data-type=\"problem\" id=\"fs-id1167832054796\">\r\n<p id=\"fs-id1167830700823\">\\(3\\left(2x-3\\right)&gt;3\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(4\\left(x+5\\right)\\ge 4\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167828401864\">\r\n<div data-type=\"problem\" id=\"fs-id1167834382460\">\r\n<p id=\"fs-id1167834382462\">\\(-3\\left(x+4\\right)&lt;0\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(-1\\left(3x-1\\right)\\le 7\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832044173\"><span data-type=\"media\" id=\"fs-id1167832044176\" data-alt=\"The solution is x is greater than or equal to negative 2. Its graph has a closed circle at negative 2 and is shaded to the right. Its interval notation is negative 2 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_06_327_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to negative 2. Its graph has a closed circle at negative 2 and is shaded to the right. Its interval notation is negative 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835305263\">\r\n<div data-type=\"problem\" id=\"fs-id1167835305265\">\r\n<p id=\"fs-id1167835305267\">\\(\\frac{1}{2}\\left(3x-4\\right)\\le 1\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\frac{1}{3}\\left(x+6\\right)\\le 4\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835336477\">\r\n<div data-type=\"problem\" id=\"fs-id1167835301441\">\r\n<p id=\"fs-id1167835301443\">\\(\\frac{3}{4}\\left(x-8\\right)\\le 3\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\frac{1}{5}\\left(x-5\\right)\\le 3\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834132852\"><span data-type=\"media\" id=\"fs-id1167832051713\" data-alt=\"The solution is x is less than or equal to 12. Its graph has a closed circle at 12 and is shaded to the left. Its interval notation is negative infinity to 12 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_329_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to 12. Its graph has a closed circle at 12 and is shaded to the left. Its interval notation is negative infinity to 12 within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835622016\">\r\n<div data-type=\"problem\" id=\"fs-id1167830924493\">\r\n<p id=\"fs-id1167830924495\">\\(5x-2\\le 3x+4\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3x-4\\ge 2x+1\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835374454\">\r\n<div data-type=\"problem\" id=\"fs-id1167835374456\">\r\n<p id=\"fs-id1167835374458\">\\(\\frac{3}{4}x-5\\ge -2\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(-3\\left(x+1\\right)\\ge 6\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834063755\"><span data-type=\"media\" id=\"fs-id1167834191412\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_331_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835358541\">\r\n<div data-type=\"problem\" id=\"fs-id1167835346454\">\r\n<p id=\"fs-id1167835346457\">\\(\\frac{2}{3}x-6\\ge -4\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(-4\\left(x+2\\right)\\ge 0\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832056107\">\r\n<div data-type=\"problem\" id=\"fs-id1167832056109\">\r\n<p id=\"fs-id1167835468473\">\\(\\frac{1}{2}\\left(x-6\\right)+2&lt;-5\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(4-\\frac{2}{3}x&lt;6\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835416928\"><span data-type=\"media\" id=\"fs-id1167834189874\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_333_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834053651\">\r\n<div data-type=\"problem\" id=\"fs-id1167834053654\">\r\n<p id=\"fs-id1167834376167\">\\(-5\\le 4x-1&lt;7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835216015\">\r\n<div data-type=\"problem\" id=\"fs-id1167835216017\">\r\n<p id=\"fs-id1167835216019\">\\(-3&lt;2x-5\\le 1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826814007\"><span data-type=\"media\" id=\"fs-id1167826814012\" data-alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 and is shaded between the open and closed circles. Its interval notation is 1 to 3 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_335_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 and is shaded between the open and closed circles. Its interval notation is 1 to 3 within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834193415\">\r\n<div data-type=\"problem\" id=\"fs-id1167834193417\">\r\n<p id=\"fs-id1167830693732\">\\(5&lt;4x+1&lt;9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835166910\">\r\n<div data-type=\"problem\" id=\"fs-id1167835166912\">\r\n<p id=\"fs-id1167835166914\">\\(-1&lt;3x+2&lt;8\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835352827\"><span data-type=\"media\" id=\"fs-id1167835352832\" data-alt=\"The solution is negative 1 is less than x which is less than 2. Its graph has an open circle at negative 1 an open circle at 2 and is shaded between. Its interval notation is negative 1 to 2 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_337_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than 2. Its graph has an open circle at negative 1 an open circle at 2 and is shaded between. Its interval notation is negative 1 to 2 within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834372491\">\r\n<div data-type=\"problem\" id=\"fs-id1167834372493\">\r\n<p id=\"fs-id1167835395570\">\\(-8&lt;5x+2\\le -3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826869972\">\r\n<div data-type=\"problem\" id=\"fs-id1167826869974\">\r\n<p id=\"fs-id1167826864186\">\\(-6\\le 4x-2&lt;-2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834079422\"><span data-type=\"media\" id=\"fs-id1167834079426\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than or 0. Its graph has a closed circle at negative 1 and an open circle at 0 and is shaded between the closed and open circles. Its interval notation is negative 1 to 0 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_339_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than or 0. Its graph has a closed circle at negative 1 and an open circle at 0 and is shaded between the closed and open circles. Its interval notation is negative 1 to 0 within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835233619\"><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cor\u201d<\/strong><\/p>\r\n<p id=\"fs-id1167834196649\">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-id1167835350578\">\r\n<div data-type=\"problem\" id=\"fs-id1167835350580\">\r\n<p id=\"fs-id1167835350582\">\\(x\\le -2\\) or \\(x&gt;3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834120546\">\r\n<div data-type=\"problem\" id=\"fs-id1167831881001\">\r\n<p id=\"fs-id1167831881003\">\\(x\\le -4\\) or \\(x&gt;-3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835192948\"><span data-type=\"media\" id=\"fs-id1167832043596\" data-alt=\"The solution is x is less than or equal to negative 4 or x is greater than negative 3. The graph of the solutions on a number line has a closed circle at negative 4 and shading to the left and an open circle at negative 3 with shading to the right. The interval notation is the union of negative infinity to negative 4 within a parenthesis and a bracket and negative 3 and infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_341_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative 4 or x is greater than negative 3. The graph of the solutions on a number line has a closed circle at negative 4 and shading to the left and an open circle at negative 3 with shading to the right. The interval notation is the union of negative infinity to negative 4 within a parenthesis and a bracket and negative 3 and infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834146970\">\r\n<div data-type=\"problem\" id=\"fs-id1167834146972\">\r\n<p id=\"fs-id1167834146974\">\\(x&lt;2\\) or \\(x\\ge 5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832026058\">\r\n<div data-type=\"problem\" id=\"fs-id1167832026060\">\r\n<p id=\"fs-id1167834061807\">\\(x&lt;0\\) or \\(x\\ge 4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834183567\"><span data-type=\"media\" id=\"fs-id1167832075536\" data-alt=\"The solution is x is less than 0 or x is greater than or equal to 2. The graph of the solutions on a number line has an open circle at 0 and shading to the left and a closed circle at 4 with shading to the right. The interval notation is the union of negative infinity to 0 within parentheses and 4 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_06_343_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 0 or x is greater than or equal to 2. The graph of the solutions on a number line has an open circle at 0 and shading to the left and a closed circle at 4 with shading to the right. The interval notation is the union of negative infinity to 0 within parentheses and 4 to infinity within a bracket and parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826997912\">\r\n<div data-type=\"problem\" id=\"fs-id1167826997915\">\r\n<p id=\"fs-id1167826997917\">\\(2+3x\\le 4\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(5-2x\\le -1\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835234301\">\r\n<div data-type=\"problem\" id=\"fs-id1167835511148\">\r\n<p id=\"fs-id1167835511150\">\\(4-3x\\le -2\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(2x-1\\le -5\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835368318\"><span data-type=\"media\" id=\"fs-id1167830704458\" data-alt=\"The solution is x is less than or equal to negative 2 or x is greater than or equal to 2. The graph of the solutions on a number line has a closed circle at negative 2 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 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_06_345_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative 2 or x is greater than or equal to 2. The graph of the solutions on a number line has a closed circle at negative 2 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831106365\">\r\n<div data-type=\"problem\" id=\"fs-id1167835329690\">\r\n<p id=\"fs-id1167835329693\">\\(2\\left(3x-1\\right)&lt;4\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3x-5&gt;1\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830868590\">\r\n<div data-type=\"problem\" id=\"fs-id1167830868592\">\r\n<p id=\"fs-id1167832056301\">\\(3\\left(2x-3\\right)&lt;-5\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(4x-1&gt;3\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835514235\"><span data-type=\"media\" id=\"fs-id1167835514239\" data-alt=\"The solution is x is less than two-thirds or x is greater than 1. The graph of the solutions on a number line has an open circle at two-thirds and shading to the left and an open circle at 1 with shading to the right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 and infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_347_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than two-thirds or x is greater than 1. The graph of the solutions on a number line has an open circle at two-thirds and shading to the left and an open circle at 1 with shading to the right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 and infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834252380\">\r\n<div data-type=\"problem\" id=\"fs-id1167834464423\">\r\n<p id=\"fs-id1167834464426\">\\(\\frac{3}{4}x-2&gt;4\\) or \\(4\\left(2-x\\right)&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830699114\">\r\n<div data-type=\"problem\" id=\"fs-id1167830699116\">\r\n<p id=\"fs-id1167830699118\">\\(\\frac{2}{3}x-3&gt;5\\) or \\(3\\left(5-x\\right)&gt;6\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830865772\"><span data-type=\"media\" id=\"fs-id1167834193250\" data-alt=\"The solution is x is less than 3 or x is greater than 12. The graph of the solutions on a number line has an open circle at 3 and shading to the left and an open circle at 12 with shading to the right. The interval notation is the union of negative infinity to 3 within parentheses and 12 and infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_349_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 3 or x is greater than 12. The graph of the solutions on a number line has an open circle at 3 and shading to the left and an open circle at 12 with shading to the right. The interval notation is the union of negative infinity to 3 within parentheses and 12 and infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167827958539\">\r\n<div data-type=\"problem\" id=\"fs-id1167827958541\">\r\n<p id=\"fs-id1167827958543\">\\(3x-2&gt;4\\) or \\(5x-3\\le 7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831107016\">\r\n<div data-type=\"problem\" id=\"fs-id1167835509690\">\r\n<p id=\"fs-id1167835509692\">\\(2\\left(x+3\\right)\\ge 0\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3\\left(x+4\\right)\\le 6\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830704314\"><span data-type=\"media\" id=\"fs-id1167835489053\" data-alt=\"The solution 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_06_351_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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 data-type=\"exercise\" id=\"fs-id1167834372539\">\r\n<div data-type=\"problem\" id=\"fs-id1167834372541\">\r\n<p id=\"fs-id1167834372543\">\\(\\frac{1}{2}x-3\\le 4\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\frac{1}{3}\\left(x-6\\right)\\ge -2\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834429124\">\r\n<div data-type=\"problem\" id=\"fs-id1167834429126\">\r\n<p id=\"fs-id1167835423006\">\\(\\frac{3}{4}x+2\\le -1\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\frac{1}{2}\\left(x+8\\right)\\ge -3\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832151191\"><span data-type=\"media\" id=\"fs-id1167830836979\" data-alt=\"The solution 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_06_353_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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<p id=\"fs-id1167834227956\"><strong data-effect=\"bold\">Mixed practice<\/strong><\/p>\r\n<p id=\"fs-id1167834227961\">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-id1167831923362\">\r\n<div data-type=\"problem\" id=\"fs-id1167831923364\">\r\n<p id=\"fs-id1167831923366\">\\(3x+7\\le 1\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(2x+3\\ge -5\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832015546\">\r\n<div data-type=\"problem\" id=\"fs-id1167832015548\">\r\n<p id=\"fs-id1167832015550\">\\(6\\left(2x-1\\right)&gt;6\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(5\\left(x+2\\right)\\ge 0\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835415130\"><span data-type=\"media\" id=\"fs-id1167835415134\" data-alt=\"The solution is x is less than 1. Its graph has an open circle at negative 1 is shaded to the right. Its interval notation is 1 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_355_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 1. Its graph has an open circle at negative 1 is shaded to the right. Its interval notation is 1 to infinity within parentheses.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831832694\">\r\n<div data-type=\"problem\" id=\"fs-id1167831832696\">\r\n<p id=\"fs-id1167835343859\">\\(4-7x\\ge -3\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(5\\left(x-3\\right)+8&gt;3\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834537127\">\r\n<div data-type=\"problem\" id=\"fs-id1167834563618\">\r\n<p id=\"fs-id1167834563621\">\\(\\frac{1}{2}x-5\\le 3\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\frac{1}{4}\\left(x-8\\right)\\ge -3\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834185831\"><span data-type=\"media\" id=\"fs-id1167834448967\" data-alt=\"The solution 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_06_357_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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 data-type=\"exercise\" id=\"fs-id1167826880124\">\r\n<div data-type=\"problem\" id=\"fs-id1167826880126\">\r\n<p id=\"fs-id1167826880129\">\\(-5\\le 2x-1&lt;7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834535529\">\r\n<div data-type=\"problem\" id=\"fs-id1167835377724\">\r\n<p id=\"fs-id1167835377726\">\\(\\frac{1}{5}\\left(x-5\\right)+6&lt;4\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3-\\frac{2}{3}x&lt;5\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834274010\"><span data-type=\"media\" id=\"fs-id1167834274014\" 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_06_359_img_new.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 data-type=\"exercise\" id=\"fs-id1167834184189\">\r\n<div data-type=\"problem\" id=\"fs-id1167834515957\">\r\n<p id=\"fs-id1167834515959\">\\(4x-2&gt;6\\) or<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(3x-1\\le -2\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167835356561\">\r\n<p id=\"fs-id1167832054573\">\\(6x-3\\le 1\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(5x-1&gt;-6\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167826978569\" data-alt=\"The solution is negative 1 is less than x which is less than or equal to two-thirds. Its graph has an open circle at negative 1 and a closed circle at two-thirds and is shaded between the open and closed circles. Its interval notation is negative 1 to two-thirds within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_361_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than or equal to two-thirds. Its graph has an open circle at negative 1 and a closed circle at two-thirds and is shaded between the open and closed circles. Its interval notation is negative 1 to two-thirds within a parenthesis and a bracket.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831880376\">\r\n<div data-type=\"problem\" id=\"fs-id1167831880378\">\r\n<p id=\"fs-id1167831880381\">\\(-2\\left(3x-4\\right)\\le 2\\) and<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(-4\\left(x-1\\right)&lt;2\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835173744\">\r\n<div data-type=\"problem\" id=\"fs-id1167835173746\">\r\n<p id=\"fs-id1167835173748\">\\(-5\\le 3x-2\\le 4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826870092\"><span data-type=\"media\" id=\"fs-id1167834563744\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and a closed circle at 2 and is shaded between the closed circles. Its interval notation is negative 1 to 4 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_363_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and a closed circle at 2 and is shaded between the closed circles. Its interval notation is negative 1 to 4 within brackets.\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835350832\"><strong data-effect=\"bold\">Solve Applications with Compound Inequalities<\/strong><\/p>\r\n<p id=\"fs-id1167835350838\">In the following exercises, solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167831911300\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167831911302\">\r\n<p id=\"fs-id1167831911304\">Penelope is playing a number game with her sister June. Penelope is thinking of a number and wants June to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Penelope might be thinking of.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826802327\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167826802329\">\r\n<p id=\"fs-id1167831103348\">Gregory is thinking of a number and he wants his sister Lauren to guess the number. His first clue is that six less than twice his number is between four and forty-two. Write a compound inequality that shows the range of numbers that Gregory might be thinking of.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831103354\">\r\n<p id=\"fs-id1167831103356\">\\(5\\le n\\le 24\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834190496\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167834190498\">\r\n<p id=\"fs-id1167834190501\">Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 18 feet. The perimeter of the dog run must be at least 42 feet and no more than 72 feet. Use a compound inequality to find the range of values for the width of the dog run.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835360423\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167834229072\">\r\n<p id=\"fs-id1167834229074\">Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834229081\">\r\n<p id=\"fs-id1167834581467\">\\(6\\le w\\le 12\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"everyday\" data-depth=\"2\" id=\"fs-id1167835355931\">\r\n<h4 data-type=\"title\">Everyday Math<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167827943008\">\r\n<div data-type=\"problem\" id=\"fs-id1167827943010\">\r\n<p id=\"fs-id1167827943013\"><strong data-effect=\"bold\">Blood Pressure<\/strong> A person\u2019s blood pressure is measured with two numbers. The systolic blood pressure measures the pressure of the blood on the arteries as the heart beats. The diastolic blood pressure measures the pressure while the heart is resting.<\/p>\r\n<p id=\"fs-id1167835342667\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be your systolic blood pressure. Research and then write the compound inequality that shows you what a normal systolic blood pressure should be for someone your age.<\/p>\r\n<p id=\"fs-id1167826788433\"><span class=\"token\">\u24d1<\/span> Let <em data-effect=\"italics\">y<\/em> be your diastolic blood pressure. Research and then write the compound inequality that shows you what a normal diastolic blood pressure should be for someone your age.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834214012\">\r\n<div data-type=\"problem\">\r\n\r\n<strong data-effect=\"bold\">Body Mass Index<\/strong> (BMI) is a measure of body fat is determined using your height and weight.\r\n<p id=\"fs-id1167834319818\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be your BMI. Research and then write the compound inequality to show the BMI range for you to be considered normal weight.<\/p>\r\n<p id=\"fs-id1167826941158\"><span class=\"token\">\u24d1<\/span> Research a BMI calculator and determine your BMI. Is it a solution to the inequality in part (a)?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834193729\">\r\n<p id=\"fs-id1167834193731\"><span class=\"token\">\u24d0<\/span> answers vary <span class=\"token\">\u24d1<\/span> answers vary<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835336642\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167835283002\">\r\n<div data-type=\"problem\" id=\"fs-id1167835283005\">\r\n<p id=\"fs-id1167835283007\">In your own words, explain the difference between the properties of equality and the properties of inequality.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835341545\">\r\n<div data-type=\"problem\" id=\"fs-id1167835341547\">\r\n<p id=\"fs-id1167835341549\">Explain the steps for solving the compound inequality \\(2-7x\\ge -5\\) or \\(4\\left(x-3\\right)+7&gt;3.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834228394\">\r\n<p id=\"fs-id1167834228396\">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-id1167830770224\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1167830770229\"><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-id1167835420960\" data-alt=\"This table has four columns and four 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 solve compound inequalities with \u201cand.\u201d In row 3, the I can was solve compound inequalities with \u201cor.\u201d In row 4, the I can was solve applications with compound inequalities.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four 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 solve compound inequalities with \u201cand.\u201d In row 3, the I can was solve compound inequalities with \u201cor.\u201d In row 4, the I can was solve applications with compound inequalities.\" \/><\/span>\r\n<p id=\"fs-id1167835365234\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\" class=\"textbox shaded\">\r\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\r\n<dl id=\"fs-id1167834377510\">\r\n \t<dt>compound inequality<\/dt>\r\n \t<dd id=\"fs-id1167834377515\">A compound inequality is made up of two inequalities connected by the word \u201cand\u201d or the word \u201cor.\u201d<\/dd>\r\n<\/dl>\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>Solve compound inequalities with \u201cand\u201d<\/li>\n<li>Solve compound inequalities with \u201cor\u201d<\/li>\n<li>Solve applications with compound inequalities<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835389786\" class=\"be-prepared\">\n<p id=\"fs-id1167835510986\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835192422\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f1d05561ed6b807289386fa61bfd3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#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;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"86\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829788421\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e8536235d20b43cd0166d88bdab5064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829752757\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835338613\">\n<h3 data-type=\"title\">Solve Compound Inequalities with \u201cand\u201d<\/h3>\n<p id=\"fs-id1167834195041\">Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A <span data-type=\"term\">compound inequality<\/span> is made up of two inequalities connected by the word \u201cand\u201d or the word \u201cor.\u201d For example, the following are compound inequalities.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835360046\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3532aef379d2847dcb3f4a810b319d89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;&#62;&#45;&#52;&#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;&#97;&#110;&#100;&#125;&#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;&#52;&#120;&#45;&#53;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"233\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" id=\"fs-id1167835300537\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01bf0012b4bd37b6cb77bef13737bcc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#48;&#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;&#111;&#114;&#125;&#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;&#121;&#45;&#53;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"235\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1167835357414\">\n<div data-type=\"title\">Compound Inequality<\/div>\n<p id=\"fs-id1167834061509\">A <strong data-effect=\"bold\">compound inequality<\/strong> is made up of two inequalities connected by the word \u201cand\u201d or the word \u201cor.\u201d<\/p>\n<\/div>\n<p id=\"fs-id1167834429410\">To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.<\/p>\n<p id=\"fs-id1167835362358\">To solve a compound inequality with the word \u201cand,\u201d we look for all numbers that make <em data-effect=\"italics\">both<\/em> inequalities true. To solve a compound inequality with the word \u201cor,\u201d we look for all numbers that make <em data-effect=\"italics\">either<\/em> inequality true.<\/p>\n<p id=\"fs-id1167834196849\">Let\u2019s start with the compound inequalities with \u201cand.\u201d Our solution will be the numbers that are solutions to <em data-effect=\"italics\">both<\/em> inequalities known as the intersection of the two inequalities. Consider how the intersection of two streets\u2014the part where the streets overlap\u2014belongs to both streets.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834308105\" data-alt=\"The figure is an illustration of two streets with their intersection shaded\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_001_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is an illustration of two streets with their intersection shaded\" \/><\/span><\/p>\n<p id=\"fs-id1167835489259\">To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs\u2014where the graphs overlap.<\/p>\n<p id=\"fs-id1167834430724\">For the compound inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b2daf07fe69e53ef183aac104448cff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45494479160b17d971554b7a46312c1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> we graph each inequality. We then look for where the graphs \u201coverlap\u201d. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See <a href=\"#CNX_IntAlg_Figure_02_06_002\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_06_002\"><span data-type=\"media\" id=\"fs-id1167835341616\" data-alt=\"The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_002_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.\" \/><\/span><\/div>\n<p id=\"fs-id1167835309111\">We can see that the numbers 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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> are shaded on both of the first two graphs. They will then be shaded on the solution graph.<\/p>\n<p id=\"fs-id1167835514644\">The number <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;\" \/> is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.<\/p>\n<p id=\"fs-id1167834053558\">The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.<\/p>\n<p id=\"fs-id1167826996735\">This is how we will show our solution in the next examples.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826864192\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835325006\">\n<div data-type=\"problem\" id=\"fs-id1167835420949\">\n<p id=\"fs-id1167835390168\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-037e324c4a9e27f4e91529cef55bac77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#45;&#51;&#60;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c15e442b95c20a832abdbd31229dbae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#55;&#92;&#103;&#101;&#32;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -3px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791472258\">\n<table class=\"unnumbered unstyled\" summary=\"6 x minus 3 is less than 9 and 2 x plus 9 is greater than or equal to 3. Step 1 is to solve each inequality. 6 x is less than 12, which simplifies to x is less than 2. 2 x is greater than or equal to negative 6, which simplifies to x is greater than or equal to negative 3. Step 2 is to graph each solution. The graph of x is less than 2 has a right parenthesis at 2 and is shaded to its left. The graph of x is greater than or equal to negative 3 has a left bracket at negative 3 and is shaded to its right. Then graph the numbers that make both inequalities true. The final graph will show all the numbers that make both inequalities true\u2014the numbers shaded on both of the first two graphs. The graph of x is less than 2 and x is greater than or equal to negative 3 has a left bracket at negative 3 and a right parenthesis at 2 and is shaded between the bracket and parenthesis. Each graph is marked at negative 3 and 2. Step 3 is to write the solution in interval notation. It is negative 3 to 2 within a bracket and a parenthesis. All the numbers that make both inequalities true are the solution to the compound inequality.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53cdc406cb082c8378762e6773e2391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#45;&#51;&#60;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3417b190f46762f0effab323a89eb8a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#57;&#92;&#103;&#101;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Solve each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inequality.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53cdc406cb082c8378762e6773e2391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#45;&#51;&#60;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3417b190f46762f0effab323a89eb8a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#57;&#92;&#103;&#101;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e10f6467b965119b11a755fab2ae7d74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#60;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7739f2163d650971be01af6f5a93ae25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#92;&#103;&#101;&#32;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0fdca2fa1d3ff3cdb139ca5441f43f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\n<td data-valign=\"top\" data-align=\"right\"><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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Graph each solution. Then graph the numbers that make both inequalities true. The final graph will show all the numbers that make both inequalities true\u2014the numbers shaded on <em data-effect=\"italics\">both<\/em> of the first two graphs.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832060115\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_003a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Write the solution in interval notation.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba6525fc28865aeda14321f5889c1b44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"4\" data-valign=\"top\" data-align=\"left\">All the numbers that make both inequalities true are the solution to the compound inequality.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835322297\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832046520\">\n<div data-type=\"problem\" id=\"fs-id1167835339414\">\n<p id=\"fs-id1167834459102\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f327c348e4df9f387f6496af25c7f5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#55;&#60;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83b6a52a19ba378b08673a9302e69294_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#43;&#56;&#92;&#103;&#101;&#32;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830703172\"><span data-type=\"media\" id=\"fs-id1167835418946\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than 4. On a number line it is shown with a closed circle at negative 1 and an open circle at 4 with shading in between the closed and open circles. Its interval notation is negative 1 to 4 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_06_302_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than 4. On a number line it is shown with a closed circle at negative 1 and an open circle at 4 with shading in between the closed and open circles. Its interval notation is negative 1 to 4 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835230092\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835303505\">\n<div data-type=\"problem\" id=\"fs-id1167826997274\">\n<p>Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20f077ed930883af45b8598ddcc5fd57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#52;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"81\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6e71ff97cd0f1331de181695abc3e6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#57;&#92;&#103;&#101;&#32;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167834190135\" data-alt=\"The solution is negative 2 is less than or equal to x which is less than 3. On a number line it is shown with a closed circle at negative 2 and an open circle at 3 with shading in between the closed and open circles. Its interval notation is negative 2 to 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_06_303_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than or equal to x which is less than 3. On a number line it is shown with a closed circle at negative 2 and an open circle at 3 with shading in between the closed and open circles. Its interval notation is negative 2 to 3 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834156842\" class=\"howto\">\n<div data-type=\"title\">Solve a compound inequality with \u201cand.\u201d<\/div>\n<ol id=\"fs-id1167835234281\" class=\"stepwise\" type=\"1\">\n<li>Solve each inequality.<\/li>\n<li>Graph each solution. Then graph the numbers that make <em data-effect=\"italics\">both<\/em> inequalities true.\n<div data-type=\"newline\"><\/div>\n<p>This graph shows the solution to the compound inequality.<\/li>\n<li>Write the solution in interval notation.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835345108\">\n<div data-type=\"problem\" id=\"fs-id1167835332470\">\n<p id=\"fs-id1167835575835\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1495ad9dfd736f9cf1856dab4e35fb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7299302d6e7c45e6fb07b36f76c0eb81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171784049192\">\n<table id=\"fs-id1167835531554\" class=\"unnumbered unstyled\" summary=\"3 times the quantity 2 x plus 5 is less than or equal to 18 and 2 times the quantity x minus 7 is less than negative 6. Solve each inequality. 3 times the quantity 2 x plus 5 is less than or equal to 18 simplifies to 6 x plus 15 is less than or equal to 18 which simplifies to 6 x is less than or equal to 3, which simplifies to x is less than or equal to one-half. 2 times the quantity x minus 7 is less than negative 6 simplifies to 2 x minus 14 is less than negative 6, which simplifies to 2 x is less than 8, which simplifies to x is less than 4. Graph each solution. The graph of x is less than or equal to one-half has a right bracket at one-half and is shaded to the left. The graph of x is less than 4 has a right parenthesis and is shaded to the left. Both graphs are marked at one-half. Graph the numbers that make both inequalities true. Graph the numbers that make both inequalities true. The graph has a right bracket at one-half and is shaded to the left. It is marked at one-half. Write the solution in interval notation. It is negative infinity and one-half within a parenthesis and a bracket.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1495ad9dfd736f9cf1856dab4e35fb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c123b4cd21e9101bc15d77ad0a6be35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inequality.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cec97eea7309a5347da498143715a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#43;&#49;&#53;&#92;&#108;&#101;&#32;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c4cfd49b2287921b087ed15f0028a9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#49;&#52;&#60;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"105\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08b494c1f63f9ff61c2b27047bb15ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#92;&#108;&#101;&#32;&#51;&#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=\"15\" width=\"52\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6d7f5bc883f8674241383ff70f045d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#60;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a169ce9ba074ba42e132e210dc48853_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9df5fedb1994a9e620747376a9dbe8ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>solution.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835180552\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_004a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the numbers<\/p>\n<div data-type=\"newline\"><\/div>\n<p>that make both<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inequalities true.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171780895282\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_004b_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in interval notation.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ec2598ce15e2581a8a8855649091066_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;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/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-id1167835419122\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832226556\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4ff47e7870fc01613cbfe6e15c3dbbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be35d5a47c981f45683544be3f4d402a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830865391\"><span data-type=\"media\" id=\"fs-id1167835263571\" data-alt=\"The solution is x is less than three-halves. On a number line it is shown with an open circle at three-halves with shading to its left. Its interval notation is negative infinity to three-halves within a parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_304_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than three-halves. On a number line it is shown with an open circle at three-halves with shading to its left. Its interval notation is negative infinity to three-halves within a parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832056538\">\n<div data-type=\"problem\" id=\"fs-id1167834533400\">\n<p>Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbaccf51a9939acbb71f5bb74a6ae5de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4edbbc1edbfe38dcd3381b878244bf37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834423488\"><span data-type=\"media\" id=\"fs-id1167835346662\" data-alt=\"The solution is x is less than negative 1. On a number line it is shown with an open circle at 1 with shading to its left. Its interval notation is negative infinity to negative 1 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_305_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 1. On a number line it is shown with an open circle at 1 with shading to its left. Its interval notation is negative infinity to negative 1 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835254174\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835349939\">\n<div data-type=\"problem\" id=\"fs-id1167835218133\">\n<p id=\"fs-id1167835234710\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b6f21ca9bfd0b38bfe6ee8750f51bc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5d93ee9a1b768188066cbc1ff3d90c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791298797\">\n<table id=\"fs-id1167835335470\" class=\"unnumbered unstyled\" summary=\"One-third x minus 4 is greater than or equal to negative 2 and negative 2 times the quantity x minus 3 is greater than or equal to 4. Solve each inequality. One-third x minus 4 is greater than or equal to negative 2 simplifies to x is greater than or equal to 6. Negative 2 times the quantity x minus 3 is greater than or equal to 4 simplifies to negative 2 x is greater than negative 2, which simplifies to x is less than or equal to 1. Graph each solution. The graph of x is greater than or equal to 6 has a left bracket at 6 and is shaded to its right. The graph of x is less than or equal to 1 has a right bracket at 1 and is shaded to its left. Graph the numbers that make both inequalities true. Notice that there are no numbers that make the inequalities true.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b6f21ca9bfd0b38bfe6ee8750f51bc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdccd14d73cbf5437b2c75836180de42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each inequality.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b6f21ca9bfd0b38bfe6ee8750f51bc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83b6f38b964abe26832e015961f3d681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#120;&#43;&#54;&#92;&#103;&#101;&#32;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02f5250fc66e01b1355851084d36eed3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#92;&#103;&#101;&#32;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4f43816e2e21f983be1fff2b78c2f81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#120;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"77\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5b93744211fd80292d7166d84d45660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">and<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f4a2b53d73f3a0d63039dcfced43d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph each solution.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835322051\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_005a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the numbers that<\/p>\n<div data-type=\"newline\"><\/div>\n<p>make both inequalities<\/p>\n<div data-type=\"newline\"><\/div>\n<p>true.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835166097\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_005b_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c539148683ba11410ba46f819d41cc7_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;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>There are no numbers that make both inequalities true.<\/p>\n<div data-type=\"newline\"><\/div>\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-2c539148683ba11410ba46f819d41cc7_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;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>This is a contradiction so there is no solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835254500\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835343501\">\n<div data-type=\"problem\" id=\"fs-id1167830699646\">\n<p id=\"fs-id1167831081633\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7589f0d7bd4da1013860407a466a935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#45;&#51;&#92;&#103;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45714b0ff39884949f3f65a56fb9f71a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834473807\"><span data-type=\"media\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph of 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_06_306_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph of the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831824593\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835341663\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832076363\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d22c289910cb5b2956969740c60f48f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#53;&#92;&#103;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-308274afcda4c5e034a00455077c159b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834185418\"><span data-type=\"media\" id=\"fs-id1167835180564\" data-alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph or 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_06_307_img_new-2.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834539332\">Sometimes we have a compound inequality that can be written more concisely. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-286c3aad4070ab968d94b568825cdebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"43\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9bd77a12c27f6f773bdb04281f0960a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> can be written simply as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-561338b2996b9d3a29893526e510e6b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#120;&#60;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"75\" style=\"vertical-align: 0px;\" \/> and then we call it a <span data-type=\"term\" class=\"no-emphasis\">double inequality<\/span>. The two forms are equivalent.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835323809\">\n<div data-type=\"title\">Double Inequality<\/div>\n<p id=\"fs-id1167835232788\">A double inequality is a compound inequality such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f4992a39f400e01bd075615f5380c53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#120;&#60;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: 0px;\" \/> It is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-286c3aad4070ab968d94b568825cdebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"43\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca232f294a1ab8addba9d9715f39205_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167836286959\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb937d33e9499284c1b28727de1c946a_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;&#116;&#101;&#120;&#116;&#123;&#79;&#116;&#104;&#101;&#114;&#32;&#102;&#111;&#114;&#109;&#115;&#58;&#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;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#97;&#60;&#120;&#60;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#60;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#60;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#92;&#108;&#101;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#92;&#108;&#101;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#62;&#120;&#62;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#62;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#62;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#92;&#103;&#101;&#32;&#120;&#92;&#103;&#101;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#92;&#103;&#101;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#92;&#103;&#101;&#32;&#98;&#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=\"82\" width=\"656\" style=\"vertical-align: -36px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835180817\">To solve a double inequality we perform the same operation on all three \u201cparts\u201d of the double inequality with the goal of isolating the variable in the center.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835312277\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835198760\">\n<div data-type=\"problem\" id=\"fs-id1167826987980\">\n<p id=\"fs-id1167834196398\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eab6a679eaa49e5687a0255720c1b7f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#32;&#51;&#120;&#45;&#55;&#60;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" style=\"vertical-align: -3px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835262370\">\n<table class=\"unnumbered unstyled\" summary=\"Negative 4 is less than or equal to 3 x minus 7 which is less than 8. Add 7 to all three parts. Negative 4 plus 7 is less than or equal to 3 x minus 7 plus 7 which is less than 8 plus 7. Simplify. 3 is less than or equal to 3 x which is less than 15. Divided each part by 3. 3 divided by 3 is less than or equal to 3 x divided by 3 which is less than 15 divided by 3. Simplify. The result is 1 is less than or equal to x which is less than 5. Graph the solution. The solution on a number line is a left bracket at 1, a right parenthesis at 5, and shading between the bracket and parenthesis. Write the solution in interval notation. It is 1 to 5 within a bracket and a parenthesis.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832059597\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006a_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add 7 to all three parts.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835333058\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006b_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830865371\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006c_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide each part by three.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834161868\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006d_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835281520\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006e_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the solution.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835330291\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006f_img-2.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835330107\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_006g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835377148\">When written as a double inequality, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f881021dd117a5622033dd07f4d266c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#108;&#101;&#32;&#120;&#60;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -4px;\" \/> it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.<\/p>\n<p>Another way to graph the solution of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f12389d1166c7c8266dfdd163ee3acb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#108;&#101;&#32;&#120;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -3px;\" \/> is to graph both the solution of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f077fdba5fb39bebecba8de762e4880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/> and the solution of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05efdffb2c0686a1cbcd46596ec2490e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\" \/> We would then find the numbers that make both inequalities true as we did in previous examples.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834484818\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167827957278\">\n<div data-type=\"problem\" id=\"fs-id1167826778624\">\n<p id=\"fs-id1167835198886\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5f5dfd30b1f7e9668a2a7912796c1ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#52;&#120;&#45;&#49;&#60;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835231901\"><span data-type=\"media\" id=\"fs-id1167835370215\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 1 to 2 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_06_308_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 1 to 2 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832227152\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835232564\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835345122\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c21eac95f239d79fe45b2763309e022_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#60;&#50;&#120;&#45;&#53;&#92;&#108;&#101;&#32;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831922098\"><span data-type=\"media\" id=\"fs-id1167835180911\" data-alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 with shading between the closed and open circles. Its interval notation is negative 1 to 3 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_06_309_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 with shading between the closed and open circles. Its interval notation is negative 1 to 3 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835170774\">\n<h3 data-type=\"title\">Solve Compound Inequalities with \u201cor\u201d<\/h3>\n<p id=\"fs-id1167835213802\">To solve a <span data-type=\"term\" class=\"no-emphasis\">compound inequality<\/span> with \u201cor\u201d, we start out just as we did with the compound inequalities with \u201cand\u201d\u2014we solve the two inequalities. Then we find all the numbers that make <em data-effect=\"italics\">either<\/em> inequality true.<\/p>\n<p id=\"fs-id1167835410279\">Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.<\/p>\n<p id=\"fs-id1167830769733\">To write the solution in <span data-type=\"term\" class=\"no-emphasis\">interval notation<\/span>, we will often use the <span data-type=\"term\" class=\"no-emphasis\">union symbol<\/span>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f8eeee53de74b0c680cf3bb5952f893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#117;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -1px;\" \/>, to show the union of the solutions shown in the graphs.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835365153\" class=\"howto\">\n<div data-type=\"title\">Solve a compound inequality with \u201cor.\u201d<\/div>\n<ol id=\"fs-id1167832015984\" class=\"stepwise\" type=\"1\">\n<li>Solve each inequality.<\/li>\n<li>Graph each solution. Then graph the numbers that make either inequality true.<\/li>\n<li>Write the solution in interval notation.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167832015695\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835371428\">\n<div data-type=\"problem\" id=\"fs-id1167835305033\">\n<p id=\"fs-id1167831910174\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a61c661c9db848ca7fc09e1cd5e64d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#51;&#120;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94db7a2c534af1db8b2e06a5471fae91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#50;&#120;&#92;&#108;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -3px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791523544\">\n<table id=\"fs-id1167834064709\" class=\"unnumbered unstyled\" summary=\"5 minus 3 x is less than or equal to negative 1 or 8 plus 2 x is less than or equal to 5. Solve each inequality. 5 minus 3 x is less than or equal to negative 1 simplifies to negative 3 x is less than or equal to negative 6, which simplifies to x is greater than or equal to 2. 8 plus 2 x is less than or equal to 5 simplifies to 2 x is less than or equal to negative 3, which simplifies to x is less than or equal to negative three halves. Graph each solution. The graph of x is greater than or equal to 2 has a left bracket at 2 and is shaded to its right. The graph of x is less than or equal to negative three halves has a right bracket at negative three halves and is shaded to its left. Graph numbers that make either inequality true. The graph shows a right bracket at negative three-halves with shading to the left and a left bracket at 2 with shading to the right. So, the interval notation of the solution is the union of negative infinity to negative three-halves within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a61c661c9db848ca7fc09e1cd5e64d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#51;&#120;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c58ff9b84ec0405d76942633cc0e3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#50;&#120;&#92;&#108;&#101;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each inequality.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a61c661c9db848ca7fc09e1cd5e64d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#51;&#120;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c58ff9b84ec0405d76942633cc0e3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#50;&#120;&#92;&#108;&#101;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5ccc99e1427a0df71b32caa61cfc6ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#120;&#92;&#108;&#101;&#32;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"78\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-798142dff67679a52aa23d2b4da0cd71_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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#120;&#92;&#108;&#101;&#32;&#45;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9751c7914c7338e1dd24f889be6d30dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f73e5979f72269996a99f27e73d2a645_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;&#51;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#108;&#101;&#32;&#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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph each solution.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535213\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph numbers that<\/p>\n<div data-type=\"newline\"><\/div>\n<p>make either inequality<\/p>\n<div data-type=\"newline\"><\/div>\n<p>true.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835302326\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e25eed7023293f12aa45d7e7d653ac5c_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;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"129\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835305860\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835377510\">\n<div data-type=\"problem\" id=\"fs-id1167831047475\">\n<p>Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fc66cbdd6c5f25bd62e08c205d1fdbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#50;&#120;&#92;&#108;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da558635963bf528599537405577566b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#43;&#51;&#120;&#92;&#108;&#101;&#32;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537118\"><span data-type=\"media\" id=\"fs-id1167835307405\" data-alt=\"The solution is x is greater than or equal to 2 or x is less than or equal to 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 2 and 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_06_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to 2 or x is less than or equal to 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 2 and infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832053266\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831880592\">\n<div data-type=\"problem\" id=\"fs-id1167832053590\">\n<p id=\"fs-id1167832058374\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d43a71032c74d6c8b9dedf1c8189a1e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#45;&#53;&#120;&#92;&#108;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e44cc5935f2791e92ac3547f29458da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#50;&#120;&#92;&#108;&#101;&#32;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835264543\"><span data-type=\"media\" id=\"fs-id1167832060300\" data-alt=\"The solution is x is greater than or equal to 1 or x is less than or equal to negative 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 1 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 1 and 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_06_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to 1 or x is less than or equal to negative 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 1 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 1 and infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835303828\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834184113\">\n<div data-type=\"problem\" id=\"fs-id1167835640387\">\n<p id=\"fs-id1167832056412\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a159e7583c2f1172078966017fe1dde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0becd08fe86465d5b39f65adf0e6da81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -6px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791538670\">\n<table id=\"fs-id1167835326892\" class=\"unnumbered unstyled can-break\" summary=\"Two-third x minus 4 is less than or equal to 3 or one-fourth times the quantity x plus 8 is greater than or equal to negative 1. Solve each inequality. Two-third x minus 4 is less than or equal to 3 simplifies to two-thirds x is less than or equal to 7 which simplifies to three-halves times two-thirds x is less than or equal to three-halves times 7 ones, which simplifies to x is less than or equal to twenty-one halves. One-fourth times the quantity x plus 8 simplifies to one-fourth x plus 2 is greater than negative 1 which simplifies to one-fourth x is greater than negative 3 which simplifies to x is greater than or equal to negative 12. Graph each solution. The graph of x is less than twenty-one halves has a right bracket at twenty-one halves and is shaded to the left. The graph of x is greater than or equal to negative 12 has a left bracket at negative 12 and is shaded to the right. Graph numbers that make either inequality true. All values are shaded on the number line. The solution is all real numbers. The interval notation is negative infinity to infinity within parentheses.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f40529c9b9f0fe6b1e23bc87d6d33fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#92;&#108;&#101;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b8f6419eb78d33610436edc727aa9ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#49;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inequality.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97bcfa6f619d14667a4b993f8920403c_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;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"137\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04cdcaca885de71bbdda071ac6c38391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#52;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"148\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac80371620b7d4764dd801815e84fce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#49;&#50;&#92;&#108;&#101;&#32;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0993125d3b7c93ecef42cb79f3dacfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#56;&#92;&#103;&#101;&#32;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"87\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8871fc69687b548f7ae273d8725903e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#92;&#108;&#101;&#32;&#50;&#49;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-500bebf1e19d48b1a26244088f933a2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"65\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54d975075bef9b69ccf636019ff6f975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54d975075bef9b69ccf636019ff6f975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\">or<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-500bebf1e19d48b1a26244088f933a2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"65\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>solution.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832010254\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph numbers<\/p>\n<div data-type=\"newline\"><\/div>\n<p>that make either<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inequality true.<\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835264484\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\">The solution covers all real numbers.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"3\" 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-id1167835279876\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834246594\">\n<div data-type=\"problem\" id=\"fs-id1167835308584\">\n<p id=\"fs-id1167831923646\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5000ae806e0f247350cf7be93350f982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#120;&#45;&#55;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-994aa4cdeb4bfd7819aedc3f26d4990a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834234020\"><span data-type=\"media\" id=\"fs-id1167834064229\" data-alt=\"The solution 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_06_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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-id1167831883742\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834099096\">\n<div data-type=\"problem\" id=\"fs-id1167834099098\">\n<p id=\"fs-id1167832055008\">Solve the compound inequality. Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76ef4ec73cb435b7ccd114da60d49f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c83a5f02bd739a25671bea0f63b7b8c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831955858\"><span data-type=\"media\" id=\"fs-id1167834534344\" data-alt=\"The solution 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_06_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826851372\">\n<h3 data-type=\"title\">Solve Applications with Compound Inequalities<\/h3>\n<p id=\"fs-id1167835421464\">Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.<\/p>\n<p id=\"fs-id1167834428991\">Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a <span data-type=\"term\" class=\"no-emphasis\">compound inequality<\/span>. Last, we will solve the compound inequality.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834185809\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834185811\">\n<div data-type=\"problem\" id=\"fs-id1167831846923\">\n<p id=\"fs-id1167835317903\">Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.<\/p>\n<p id=\"fs-id1167835317906\">During the summer, a property owner will pay ?24.72 plus ?1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to ?57.06 and ?171.02. How many hcf can the owner use if he wants his usage to stay in the normal range?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831036915\">\n<table id=\"fs-id1167835416828\" class=\"unnumbered unstyled\" summary=\"Identify what we are looking for. We are looking for the number of h c f he can use and stay in the normal usage billing range. Name what we are looking for. Let x e equal to the number of h c f he can use. Translate to an inequality. The bill is 24 dollars and 72 cents plus 1 dollar and 54 cents times the number of h c f he uses. That is 24.72 plus 1.54 x. His bill will be between or equal to 57 dollars and 6 cents and 171 dollars and 2 cents. That is 57.06 is less than or equal to 24.72 plus 1.54 x which is less than 171.02. Solve the inequality. 57.06 minus 24.72 is less than or equal to 24.72 minus 24.72 plus 1.54 x which is less than 171.02 minus 24.72. 32.34 is less than or equal to 1.54 x which is less than or equal to 146.3. 32.34 divided by 1.54 is less than or equal to 1.54 x divided by 1.54 which is less than or equal to 146.3 divided by 1.54. The result is 21 is less than or equal to x which is less than or equal to 95. Answer the question. The property owner can use 21 to 95 h c f and still fall within the normal usage billing range.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The number of hcf he can use and stay in the \u201cnormal usage\u201d billing range.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Name what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7bbcde7229c9d7d6f7f2b6793961e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> the number of hcf he can use.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Translate to an inequality.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Bill is ?24.72 plus ?1.54 times the number of hcf he uses or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3c66b4d843bb3985c9f2a8ca575620e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#46;&#55;&#50;&#43;&#49;&#46;&#53;&#52;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"108\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835614910\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834151973\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Answer the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The property owner can use 21\u201395 hcf and still fall within the \u201cnormal usage\u201d billing range.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835322185\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835269568\">\n<div data-type=\"problem\" id=\"fs-id1167834214035\">\n<p id=\"fs-id1167834214037\">Due to the drought in California, many communities now have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.<\/p>\n<p id=\"fs-id1167835325768\">During the summer, a property owner will pay ?24.72 plus ?1.32 per hcf for Conservation Usage. The bill for Conservation Usage would be between or equal to ?31.32 and ?52.12. How many hcf can the owner use if she wants her usage to stay in the conservation range?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835325771\">\n<p id=\"fs-id1167835307820\">The homeowner can use 5\u201320 hcf and still fall within the \u201cconservation usage\u201d billing range.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832010482\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831911604\">\n<div data-type=\"problem\" id=\"fs-id1167831911606\">\n<p id=\"fs-id1167832153621\">Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.<\/p>\n<p id=\"fs-id1167832153624\">During the winter, a property owner will pay ?24.72 plus ?1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to ?49.36 and ?86.32. How many hcf will he be allowed to use if he wants his usage to stay in the normal range?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826880107\">\n<p id=\"fs-id1167826880110\">The homeowner can use 16\u201340 hcf and still fall within the \u201cnormal usage\u201d billing range.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835419051\" class=\"media-2\">\n<p id=\"fs-id1167835237414\">Access this online resource for additional instruction and practice with solving compound inequalities.<\/p>\n<ul id=\"fs-id1167835350269\" data-bullet-style=\"bullet\">\n<li><a href=\"https:\/\/openstax.org\/l\/37compinequalit\">Compound inequalities<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835609421\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167834430991\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to solve a compound inequality with \u201cand\u201d<\/strong>\n<ol id=\"fs-id1167834161616\" class=\"stepwise\" type=\"1\">\n<li>Solve each inequality.<\/li>\n<li>Graph each solution. Then graph the numbers that make <em data-effect=\"italics\">both<\/em> inequalities true. This graph shows the solution to the compound inequality.<\/li>\n<li>Write the solution in interval notation.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Double Inequality<\/strong>\n<ul id=\"fs-id1167834192375\" data-bullet-style=\"bullet\">\n<li>A <strong data-effect=\"bold\">double inequality<\/strong> is a compound inequality such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-561338b2996b9d3a29893526e510e6b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#120;&#60;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"75\" style=\"vertical-align: 0px;\" \/>. It is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-286c3aad4070ab968d94b568825cdebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"43\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca232f294a1ab8addba9d9715f39205_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/>\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-cb937d33e9499284c1b28727de1c946a_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;&#116;&#101;&#120;&#116;&#123;&#79;&#116;&#104;&#101;&#114;&#32;&#102;&#111;&#114;&#109;&#115;&#58;&#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;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#97;&#60;&#120;&#60;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#60;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#60;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#92;&#108;&#101;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#92;&#108;&#101;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#62;&#120;&#62;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#62;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#62;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#92;&#103;&#101;&#32;&#120;&#92;&#103;&#101;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#92;&#103;&#101;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#120;&#92;&#103;&#101;&#32;&#98;&#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=\"82\" width=\"656\" style=\"vertical-align: -36px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">How to solve a compound inequality with \u201cor\u201d<\/strong>\n<ol id=\"fs-id1167835343531\" class=\"stepwise\" type=\"1\">\n<li>Solve each inequality.<\/li>\n<li>Graph each solution. Then graph the numbers that make either inequality true.<\/li>\n<li>Write the solution in interval notation.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835305325\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834085098\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cand\u201d<\/strong><\/p>\n<p id=\"fs-id1167835353641\">In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835410923\">\n<div data-type=\"problem\" id=\"fs-id1167835410925\">\n<p id=\"fs-id1167835389977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdfb122a09ac596aa47b022ad9a5cc2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f077fdba5fb39bebecba8de762e4880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835365420\">\n<div data-type=\"problem\" id=\"fs-id1167835365422\">\n<p id=\"fs-id1167831891680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbed3567e67df3c2fbacde26ad1d4141_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f43b6a5facdba102b847a9695e3b12fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835236771\"><span data-type=\"media\" id=\"fs-id1167831239004\" data-alt=\"The solution is negative 2 is less than x which is less than or equal to 4. Its graph has an open circle at 1negative 2 and a closed circle at 4 with shading between the open and closed circles. Its interval notation is negative 2 to 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_06_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than x which is less than or equal to 4. Its graph has an open circle at 1negative 2 and a closed circle at 4 with shading between the open and closed circles. Its interval notation is negative 2 to 4 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835172044\">\n<div data-type=\"problem\">\n<p><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;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71173cb3520cc7f6bc259c3176f8a66a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832226738\">\n<div data-type=\"problem\" id=\"fs-id1167832226740\">\n<p id=\"fs-id1167835345642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53cb56bc5c351da540a86eddc5f2f18f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2683702dda8a54e9f9a54ae639396ac5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832058417\"><span data-type=\"media\" id=\"fs-id1167832036194\" data-alt=\"The solution is negative 6 is less than x which is less than negative 3. Its graph has an open circle at negative 6 and an open circle at negative 3 with shading between open circles. Its interval notation is negative 6 to negative 3 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 6 is less than x which is less than negative 3. Its graph has an open circle at negative 6 and an open circle at negative 3 with shading between open circles. Its interval notation is negative 6 to negative 3 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479842\">\n<div data-type=\"problem\" id=\"fs-id1167835479844\">\n<p id=\"fs-id1167835334128\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae08d31c0b0b238d0b8e4f696d7a3ac3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#50;&#60;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c6266894621d856067148c33d84a564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#43;&#57;&#92;&#103;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920497\">\n<div data-type=\"problem\" id=\"fs-id1167831920499\">\n<p id=\"fs-id1167834431736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c156bae8d51b5aab1c0bbc515b9e684e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#49;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3da4f9a41372cf2838063e58336a6af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#56;&#92;&#103;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835361932\"><span data-type=\"media\" id=\"fs-id1167834061494\" data-alt=\"The solution is negative 2 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 2 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 2 to 2 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_06_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 2 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 2 to 2 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834228215\">\n<div data-type=\"problem\" id=\"fs-id1167834228218\">\n<p id=\"fs-id1167835640049\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7f4c2e9795cdbbc43a3c0fc164f76da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#54;&#92;&#108;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: -3px;\" \/> and<\/p>\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-51602221b7ece198736066382c6bcab2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#49;&#92;&#103;&#101;&#32;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835519126\">\n<div data-type=\"problem\" id=\"fs-id1167831896617\">\n<p id=\"fs-id1167831896619\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fdb4fde6d628fa524399a16c10fff04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#50;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/> and<\/p>\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-247207ae467cac176fbcdffb7b9a7780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#45;&#49;&#62;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"96\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835332277\"><span data-type=\"media\" id=\"fs-id1167835331595\" data-alt=\"The solution is negative 1 is less than x which is less than or equal to three-halves. Its graph has an open circle at negative 1 and a closed circle at three-halves with shading between the open and closed circles. Its interval notation is negative 1 to three-halves 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_06_321_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than or equal to three-halves. Its graph has an open circle at negative 1 and a closed circle at three-halves with shading between the open and closed circles. Its interval notation is negative 1 to three-halves within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834061295\">\n<div data-type=\"problem\" id=\"fs-id1167835416534\">\n<p id=\"fs-id1167835416536\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e353e0dd015f43818caf39e03917e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#49;&#49;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -1px;\" \/> and<\/p>\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-a0e07739f190b181fb95e6b40eabd7a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#56;&#62;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"95\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834376303\">\n<div data-type=\"problem\" id=\"fs-id1167835330479\">\n<p id=\"fs-id1167835330481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ce27e7e49ca2042360ddde7c874e23d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#45;&#56;&#60;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: 0px;\" \/> and<\/p>\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-5d07f615a6f1039b610b2c36ce29b0d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#43;&#55;&#62;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835310981\"><span data-type=\"media\" id=\"fs-id1167834194399\" data-alt=\"The solution is negative 2 is less than x which is less than 2. Its graph has an open circle at negative 2 and an open circle at 2 with shading between the open circles. Its interval notation is negative 2 to 2 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_323_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than x which is less than 2. Its graph has an open circle at negative 2 and an open circle at 2 with shading between the open circles. Its interval notation is negative 2 to 2 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835280396\">\n<div data-type=\"problem\" id=\"fs-id1167835280398\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a301b54cea5e7d93be0218cead222908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/> and<\/p>\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-3ecfe30b635357411c5f1b8402d66cb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835395541\">\n<div data-type=\"problem\" id=\"fs-id1167835395544\">\n<p id=\"fs-id1167835395546\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-594549f6ba861961afca3b6bf3603539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/> and<\/p>\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-c9bc007cc801fce57f7a9fe31809e6ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827943653\"><span data-type=\"media\" id=\"fs-id1167827943656\" data-alt=\"The solution is x is less than negative 2. Its graph has an open circle at negative 2 and is shaded to the left. Its interval notation is negative infinity to negative 2 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_325_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2. Its graph has an open circle at negative 2 and is shaded to the left. Its interval notation is negative infinity to negative 2 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832054794\">\n<div data-type=\"problem\" id=\"fs-id1167832054796\">\n<p id=\"fs-id1167830700823\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-004227314f3337ad785e4b3b39dc841b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/> and<\/p>\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-e05a7d17e27276f59b9a176cfa7093b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828401864\">\n<div data-type=\"problem\" id=\"fs-id1167834382460\">\n<p id=\"fs-id1167834382462\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fdfb878b562473c1b9d2aceafc0202a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/> and<\/p>\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-5d3a8bacd5e6e58a921e6833875760f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832044173\"><span data-type=\"media\" id=\"fs-id1167832044176\" data-alt=\"The solution is x is greater than or equal to negative 2. Its graph has a closed circle at negative 2 and is shaded to the right. Its interval notation is negative 2 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_06_327_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to negative 2. Its graph has a closed circle at negative 2 and is shaded to the right. Its interval notation is negative 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835305263\">\n<div data-type=\"problem\" id=\"fs-id1167835305265\">\n<p id=\"fs-id1167835305267\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27328d5bc78f7242ea7e88ebfb7a59fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/> and<\/p>\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-6c09dd7912bbe90454340ef88b1124e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835336477\">\n<div data-type=\"problem\" id=\"fs-id1167835301441\">\n<p id=\"fs-id1167835301443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42d2770607f772d81512c58df285621e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/> and<\/p>\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-8183d31820cacf2f601a97d132635335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834132852\"><span data-type=\"media\" id=\"fs-id1167832051713\" data-alt=\"The solution is x is less than or equal to 12. Its graph has a closed circle at 12 and is shaded to the left. Its interval notation is negative infinity to 12 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_06_329_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to 12. Its graph has a closed circle at 12 and is shaded to the left. Its interval notation is negative infinity to 12 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835622016\">\n<div data-type=\"problem\" id=\"fs-id1167830924493\">\n<p id=\"fs-id1167830924495\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-578b5b449dc0a105c8aea2276035ab15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#50;&#92;&#108;&#101;&#32;&#51;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"123\" style=\"vertical-align: -3px;\" \/> and<\/p>\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-12cef22a5b5e496cb0027c223b1378a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#52;&#92;&#103;&#101;&#32;&#50;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835374454\">\n<div data-type=\"problem\" id=\"fs-id1167835374456\">\n<p id=\"fs-id1167835374458\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac7a99a4906a6d3eeb67b3331ea24f53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#53;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> and<\/p>\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-1419ad0c339488c86f099a4d6c2f2053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834063755\"><span data-type=\"media\" id=\"fs-id1167834191412\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or 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_06_331_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835358541\">\n<div data-type=\"problem\" id=\"fs-id1167835346454\">\n<p id=\"fs-id1167835346457\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83b2a0f09ec32e0d3d5522489605cdbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#54;&#92;&#103;&#101;&#32;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -6px;\" \/> and<\/p>\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-60f78d4878afff0dc5dc77cd665c6127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832056107\">\n<div data-type=\"problem\" id=\"fs-id1167832056109\">\n<p id=\"fs-id1167835468473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85b43d0d8d4ac5cda2bfd776920d4cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#60;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"142\" style=\"vertical-align: -6px;\" \/> and<\/p>\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-9171a3d7c8f069d68099a29064720191_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#60;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835416928\"><span data-type=\"media\" id=\"fs-id1167834189874\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or 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_06_333_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834053651\">\n<div data-type=\"problem\" id=\"fs-id1167834053654\">\n<p id=\"fs-id1167834376167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f36910e5b6a6fe7fd7aa55ac249489bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#52;&#120;&#45;&#49;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835216015\">\n<div data-type=\"problem\" id=\"fs-id1167835216017\">\n<p id=\"fs-id1167835216019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb5b45dc181ab946221533c9ba3205b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#60;&#50;&#120;&#45;&#53;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826814007\"><span data-type=\"media\" id=\"fs-id1167826814012\" data-alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 and is shaded between the open and closed circles. Its interval notation is 1 to 3 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_06_335_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 and is shaded between the open and closed circles. Its interval notation is 1 to 3 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834193415\">\n<div data-type=\"problem\" id=\"fs-id1167834193417\">\n<p id=\"fs-id1167830693732\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0123af502bce24e0922b22a6995c6c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#60;&#52;&#120;&#43;&#49;&#60;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"115\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835166910\">\n<div data-type=\"problem\" id=\"fs-id1167835166912\">\n<p id=\"fs-id1167835166914\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fab107da6f81836348c66846158b834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#60;&#51;&#120;&#43;&#50;&#60;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"127\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352827\"><span data-type=\"media\" id=\"fs-id1167835352832\" data-alt=\"The solution is negative 1 is less than x which is less than 2. Its graph has an open circle at negative 1 an open circle at 2 and is shaded between. Its interval notation is negative 1 to 2 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_337_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than 2. Its graph has an open circle at negative 1 an open circle at 2 and is shaded between. Its interval notation is negative 1 to 2 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834372491\">\n<div data-type=\"problem\" id=\"fs-id1167834372493\">\n<p id=\"fs-id1167835395570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4ab93302ce05d0fbf38b0f127ae0140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#60;&#53;&#120;&#43;&#50;&#92;&#108;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"141\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826869972\">\n<div data-type=\"problem\" id=\"fs-id1167826869974\">\n<p id=\"fs-id1167826864186\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f941dc0cb2748d8abb50953967c68acb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#92;&#108;&#101;&#32;&#52;&#120;&#45;&#50;&#60;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"140\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834079422\"><span data-type=\"media\" id=\"fs-id1167834079426\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than or 0. Its graph has a closed circle at negative 1 and an open circle at 0 and is shaded between the closed and open circles. Its interval notation is negative 1 to 0 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_06_339_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than or 0. Its graph has a closed circle at negative 1 and an open circle at 0 and is shaded between the closed and open circles. Its interval notation is negative 1 to 0 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167835233619\"><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cor\u201d<\/strong><\/p>\n<p id=\"fs-id1167834196649\">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-id1167835350578\">\n<div data-type=\"problem\" id=\"fs-id1167835350580\">\n<p id=\"fs-id1167835350582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0980efedcb8304ee7cf7124a216fb502_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/> or <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>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120546\">\n<div data-type=\"problem\" id=\"fs-id1167831881001\">\n<p id=\"fs-id1167831881003\"><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;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b2daf07fe69e53ef183aac104448cff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835192948\"><span data-type=\"media\" id=\"fs-id1167832043596\" data-alt=\"The solution is x is less than or equal to negative 4 or x is greater than negative 3. The graph of the solutions on a number line has a closed circle at negative 4 and shading to the left and an open circle at negative 3 with shading to the right. The interval notation is the union of negative infinity to negative 4 within a parenthesis and a bracket and negative 3 and infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_341_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative 4 or x is greater than negative 3. The graph of the solutions on a number line has a closed circle at negative 4 and shading to the left and an open circle at negative 3 with shading to the right. The interval notation is the union of negative infinity to negative 4 within a parenthesis and a bracket and negative 3 and infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834146970\">\n<div data-type=\"problem\" id=\"fs-id1167834146972\">\n<p id=\"fs-id1167834146974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88de95669c546ef6d18c28d0579fbe0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47e6d9064fbb395929057ee769b1ef2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832026058\">\n<div data-type=\"problem\" id=\"fs-id1167832026060\">\n<p id=\"fs-id1167834061807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eea69ff1492329059f7b27f0da1f8307_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bdacfb2a15271cdd2fb04176f683713_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183567\"><span data-type=\"media\" id=\"fs-id1167832075536\" data-alt=\"The solution is x is less than 0 or x is greater than or equal to 2. The graph of the solutions on a number line has an open circle at 0 and shading to the left and a closed circle at 4 with shading to the right. The interval notation is the union of negative infinity to 0 within parentheses and 4 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_06_343_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 0 or x is greater than or equal to 2. The graph of the solutions on a number line has an open circle at 0 and shading to the left and a closed circle at 4 with shading to the right. The interval notation is the union of negative infinity to 0 within parentheses and 4 to infinity within a bracket and parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997912\">\n<div data-type=\"problem\" id=\"fs-id1167826997915\">\n<p id=\"fs-id1167826997917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-543ced205cc1e9b99230d6066fc9cbca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#51;&#120;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/> or<\/p>\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-0aab3d56014de64c179884097b164391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#50;&#120;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835234301\">\n<div data-type=\"problem\" id=\"fs-id1167835511148\">\n<p id=\"fs-id1167835511150\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd5f43e40b97a20af35c93bedc3c8275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#45;&#51;&#120;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\" \/> or<\/p>\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-f5cedba0fdf8977e04caf7bff3fc5925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#49;&#92;&#108;&#101;&#32;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835368318\"><span data-type=\"media\" id=\"fs-id1167830704458\" data-alt=\"The solution is x is less than or equal to negative 2 or x is greater than or equal to 2. The graph of the solutions on a number line has a closed circle at negative 2 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 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_06_345_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative 2 or x is greater than or equal to 2. The graph of the solutions on a number line has a closed circle at negative 2 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831106365\">\n<div data-type=\"problem\" id=\"fs-id1167835329690\">\n<p id=\"fs-id1167835329693\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-064d0042f344b190ec108e2fec9c6663_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/> or<\/p>\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-46c603fc4919a314d4cec36bd475d40e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#53;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"81\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830868590\">\n<div data-type=\"problem\" id=\"fs-id1167830868592\">\n<p id=\"fs-id1167832056301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef013c66dc93501f74430e81640da00c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/> or<\/p>\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-7e8a1f30094af704b193c42f89b7bde1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#49;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835514235\"><span data-type=\"media\" id=\"fs-id1167835514239\" data-alt=\"The solution is x is less than two-thirds or x is greater than 1. The graph of the solutions on a number line has an open circle at two-thirds and shading to the left and an open circle at 1 with shading to the right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 and infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_347_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than two-thirds or x is greater than 1. The graph of the solutions on a number line has an open circle at two-thirds and shading to the left and an open circle at 1 with shading to the right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 and infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834252380\">\n<div data-type=\"problem\" id=\"fs-id1167834464423\">\n<p id=\"fs-id1167834464426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1f4cfd2417cd35c038bda4044c601e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e09e121a0a310b5372e1daa545dca010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830699114\">\n<div data-type=\"problem\" id=\"fs-id1167830699116\">\n<p id=\"fs-id1167830699118\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23876600607f42519cf1618a0fc8b82a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#51;&#62;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b92a2e89277058766117bc2c7cd504ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830865772\"><span data-type=\"media\" id=\"fs-id1167834193250\" data-alt=\"The solution is x is less than 3 or x is greater than 12. The graph of the solutions on a number line has an open circle at 3 and shading to the left and an open circle at 12 with shading to the right. The interval notation is the union of negative infinity to 3 within parentheses and 12 and infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_349_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 3 or x is greater than 12. The graph of the solutions on a number line has an open circle at 3 and shading to the left and an open circle at 12 with shading to the right. The interval notation is the union of negative infinity to 3 within parentheses and 12 and infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827958539\">\n<div data-type=\"problem\" id=\"fs-id1167827958541\">\n<p id=\"fs-id1167827958543\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67510b82356acc75b1f752ad13e65692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#50;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: -1px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65f56efcf76e0582cc736b947c97dd67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#55;\" 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-id1167831107016\">\n<div data-type=\"problem\" id=\"fs-id1167835509690\">\n<p id=\"fs-id1167835509692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0dc64f4a9a1c6608d2114a88b8b3cf02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/> or<\/p>\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-328314acedf3adc978c9005194c78096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830704314\"><span data-type=\"media\" id=\"fs-id1167835489053\" data-alt=\"The solution 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_06_351_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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 data-type=\"exercise\" id=\"fs-id1167834372539\">\n<div data-type=\"problem\" id=\"fs-id1167834372541\">\n<p id=\"fs-id1167834372543\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd01cb283861fe29e3b1138a51343803_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or<\/p>\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-e3af2539959a35abc6f25916ef212316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834429124\">\n<div data-type=\"problem\" id=\"fs-id1167834429126\">\n<p id=\"fs-id1167835423006\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32287aace8e9e46572c451889db9c557_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#43;&#50;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> or<\/p>\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-f8af1160713cdf99360262dedeb68776_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"113\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832151191\"><span data-type=\"media\" id=\"fs-id1167830836979\" data-alt=\"The solution 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_06_353_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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<p id=\"fs-id1167834227956\"><strong data-effect=\"bold\">Mixed practice<\/strong><\/p>\n<p id=\"fs-id1167834227961\">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-id1167831923362\">\n<div data-type=\"problem\" id=\"fs-id1167831923364\">\n<p id=\"fs-id1167831923366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-778e38fbb866254fb7856290c9e183dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#55;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -3px;\" \/> and<\/p>\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-ca502007375fd54c6817474518047485_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#92;&#103;&#101;&#32;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832015546\">\n<div data-type=\"problem\" id=\"fs-id1167832015548\">\n<p id=\"fs-id1167832015550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1928be6844a30f8bb1b538e4ade78d3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/> and<\/p>\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-1c01b901e26157110d786fd682fa7763_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835415130\"><span data-type=\"media\" id=\"fs-id1167835415134\" data-alt=\"The solution is x is less than 1. Its graph has an open circle at negative 1 is shaded to the right. Its interval notation is 1 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_355_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 1. Its graph has an open circle at negative 1 is shaded to the right. Its interval notation is 1 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831832694\">\n<div data-type=\"problem\" id=\"fs-id1167831832696\">\n<p id=\"fs-id1167835343859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f70a4dc6ee0bc436c74814fdab33203d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#45;&#55;&#120;&#92;&#103;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\" \/> or<\/p>\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-195bbed939c1c63e3c74e041ed3b7ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537127\">\n<div data-type=\"problem\" id=\"fs-id1167834563618\">\n<p id=\"fs-id1167834563621\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de74e55103c8a8611f6e5fa8dff98a41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#53;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or<\/p>\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-643ddebfb02bbfbf4275f1358e5419ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"113\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834185831\"><span data-type=\"media\" id=\"fs-id1167834448967\" data-alt=\"The solution 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_06_357_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution 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 data-type=\"exercise\" id=\"fs-id1167826880124\">\n<div data-type=\"problem\" id=\"fs-id1167826880126\">\n<p id=\"fs-id1167826880129\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e9b9fd0cdc5b8f50b8da744738b8779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#50;&#120;&#45;&#49;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535529\">\n<div data-type=\"problem\" id=\"fs-id1167835377724\">\n<p id=\"fs-id1167835377726\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9161fd469ba978443068ed78ab6015d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#60;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"129\" style=\"vertical-align: -6px;\" \/> and<\/p>\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-7dbbc9d0b583cd5d875757e6b421d46b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"83\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834274010\"><span data-type=\"media\" id=\"fs-id1167834274014\" 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_06_359_img_new.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 data-type=\"exercise\" id=\"fs-id1167834184189\">\n<div data-type=\"problem\" id=\"fs-id1167834515957\">\n<p id=\"fs-id1167834515959\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ba883cc77a2034a943eb28c9eb7c4ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#50;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: -1px;\" \/> or<\/p>\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-1b7d6673cf3375d7839a7350461e6275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#49;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835356561\">\n<p id=\"fs-id1167832054573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-472166ee7d13a85b1e1a9c0d937a1aa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: -3px;\" \/> and<\/p>\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-6c33d0d8d0a3f218ed166c00a495941e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#49;&#62;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"96\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167826978569\" data-alt=\"The solution is negative 1 is less than x which is less than or equal to two-thirds. Its graph has an open circle at negative 1 and a closed circle at two-thirds and is shaded between the open and closed circles. Its interval notation is negative 1 to two-thirds 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_06_361_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than or equal to two-thirds. Its graph has an open circle at negative 1 and a closed circle at two-thirds and is shaded between the open and closed circles. Its interval notation is negative 1 to two-thirds within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831880376\">\n<div data-type=\"problem\" id=\"fs-id1167831880378\">\n<p id=\"fs-id1167831880381\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16808ab3ef4a8fd3ce89593a2aaeded6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/> and<\/p>\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-caa774d77b813989033c5a9824528e2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835173744\">\n<div data-type=\"problem\" id=\"fs-id1167835173746\">\n<p id=\"fs-id1167835173748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3ad408853603c7d1ed9e734f3ef9338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#51;&#120;&#45;&#50;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826870092\"><span data-type=\"media\" id=\"fs-id1167834563744\" data-alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and a closed circle at 2 and is shaded between the closed circles. Its interval notation is negative 1 to 4 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_363_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and a closed circle at 2 and is shaded between the closed circles. Its interval notation is negative 1 to 4 within brackets.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167835350832\"><strong data-effect=\"bold\">Solve Applications with Compound Inequalities<\/strong><\/p>\n<p id=\"fs-id1167835350838\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831911300\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831911302\">\n<p id=\"fs-id1167831911304\">Penelope is playing a number game with her sister June. Penelope is thinking of a number and wants June to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Penelope might be thinking of.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826802327\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826802329\">\n<p id=\"fs-id1167831103348\">Gregory is thinking of a number and he wants his sister Lauren to guess the number. His first clue is that six less than twice his number is between four and forty-two. Write a compound inequality that shows the range of numbers that Gregory might be thinking of.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831103354\">\n<p id=\"fs-id1167831103356\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6affecdff614d810e766f73c85b316f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#32;&#110;&#92;&#108;&#101;&#32;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834190496\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834190498\">\n<p id=\"fs-id1167834190501\">Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 18 feet. The perimeter of the dog run must be at least 42 feet and no more than 72 feet. Use a compound inequality to find the range of values for the width of the dog run.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835360423\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834229072\">\n<p id=\"fs-id1167834229074\">Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834229081\">\n<p id=\"fs-id1167834581467\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7b19069cf79c2973312b174e81025f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#32;&#119;&#92;&#108;&#101;&#32;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"everyday\" data-depth=\"2\" id=\"fs-id1167835355931\">\n<h4 data-type=\"title\">Everyday Math<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167827943008\">\n<div data-type=\"problem\" id=\"fs-id1167827943010\">\n<p id=\"fs-id1167827943013\"><strong data-effect=\"bold\">Blood Pressure<\/strong> A person\u2019s blood pressure is measured with two numbers. The systolic blood pressure measures the pressure of the blood on the arteries as the heart beats. The diastolic blood pressure measures the pressure while the heart is resting.<\/p>\n<p id=\"fs-id1167835342667\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be your systolic blood pressure. Research and then write the compound inequality that shows you what a normal systolic blood pressure should be for someone your age.<\/p>\n<p id=\"fs-id1167826788433\"><span class=\"token\">\u24d1<\/span> Let <em data-effect=\"italics\">y<\/em> be your diastolic blood pressure. Research and then write the compound inequality that shows you what a normal diastolic blood pressure should be for someone your age.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834214012\">\n<div data-type=\"problem\">\n<p><strong data-effect=\"bold\">Body Mass Index<\/strong> (BMI) is a measure of body fat is determined using your height and weight.<\/p>\n<p id=\"fs-id1167834319818\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be your BMI. Research and then write the compound inequality to show the BMI range for you to be considered normal weight.<\/p>\n<p id=\"fs-id1167826941158\"><span class=\"token\">\u24d1<\/span> Research a BMI calculator and determine your BMI. Is it a solution to the inequality in part (a)?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834193729\">\n<p id=\"fs-id1167834193731\"><span class=\"token\">\u24d0<\/span> answers vary <span class=\"token\">\u24d1<\/span> answers vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835336642\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835283002\">\n<div data-type=\"problem\" id=\"fs-id1167835283005\">\n<p id=\"fs-id1167835283007\">In your own words, explain the difference between the properties of equality and the properties of inequality.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835341545\">\n<div data-type=\"problem\" id=\"fs-id1167835341547\">\n<p id=\"fs-id1167835341549\">Explain the steps for solving the compound inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2886b989e1a5c5e4dd77c2d39769e126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#45;&#55;&#120;&#92;&#103;&#101;&#32;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a360f8a8ed82223c3b5ca3ce911488_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#55;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228394\">\n<p id=\"fs-id1167834228396\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167830770224\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167830770229\"><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-id1167835420960\" data-alt=\"This table has four columns and four 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 solve compound inequalities with \u201cand.\u201d In row 3, the I can was solve compound inequalities with \u201cor.\u201d In row 4, the I can was solve applications with compound inequalities.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_02_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four 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 solve compound inequalities with \u201cand.\u201d In row 3, the I can was solve compound inequalities with \u201cor.\u201d In row 4, the I can was solve applications with compound inequalities.\" \/><\/span><\/p>\n<p id=\"fs-id1167835365234\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167834377510\">\n<dt>compound inequality<\/dt>\n<dd id=\"fs-id1167834377515\">A compound inequality is made up of two inequalities connected by the word \u201cand\u201d or the word \u201cor.\u201d<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1570","chapter","type-chapter","status-publish","hentry"],"part":991,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1570","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":3,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1570\/revisions"}],"predecessor-version":[{"id":15158,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1570\/revisions\/15158"}],"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\/1570\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=1570"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=1570"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=1570"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=1570"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}