{"id":1946,"date":"2018-12-11T13:35:25","date_gmt":"2018-12-11T18:35:25","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-linear-inequalities-in-two-variables\/"},"modified":"2018-12-11T13:35:25","modified_gmt":"2018-12-11T18:35:25","slug":"graph-linear-inequalities-in-two-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-linear-inequalities-in-two-variables\/","title":{"raw":"Graph Linear Inequalities in Two Variables","rendered":"Graph Linear Inequalities in Two Variables"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Verify solutions to an inequality in two variables.<\/li><li>Recognize the relation between the solutions of an inequality and its graph.<\/li><li>Graph linear inequalities in two variables<\/li><li>Solve applications using linear inequalities in two variables<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167834120877\" class=\"be-prepared\"><p id=\"fs-id1167832052568\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167834396303\" type=\"1\"><li>Graph \\(x&gt;2\\) on a number line.<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835334972\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\(4x+3&gt;23\\).<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835324646\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Translate: \\(8&lt;x&gt;3\\).<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835330450\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835361577\"><h3 data-type=\"title\">Verify Solutions to an Inequality in Two Variables<\/h3><p id=\"fs-id1167832031143\">Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at <strong data-effect=\"bold\">linear inequalities<\/strong> in two variables which are very similar to linear equations in two variables.<\/p><p id=\"fs-id1167835236709\">Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs\u2014so that your business made a profit.<\/p><div data-type=\"note\" id=\"fs-id1167834189522\"><div data-type=\"title\">Linear Inequality<\/div><p id=\"fs-id1167834156714\">A <span data-type=\"term\">linear inequality<\/span> is an inequality that can be written in one of the following forms:<\/p><div data-type=\"equation\" id=\"fs-id1167834062043\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccccc}Ax+By&gt;C\\hfill &amp; &amp; &amp; Ax+By\\ge C\\hfill &amp; &amp; &amp; Ax+By&lt;C\\hfill &amp; &amp; &amp; Ax+By\\le C\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835174912\">Where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero.<\/p><\/div><p id=\"fs-id1167835175012\">Recall that an inequality with one variable had many solutions. For example, the solution to the inequality \\(x&gt;3\\) is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See <a href=\"#CNX_IntAlg_Figure_03_04_001\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><p id=\"fs-id1171791446120\"><\/p><div data-type=\"newline\"><br><\/div><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_001\"><span data-type=\"media\" id=\"fs-id1167831920149\" data-alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.\"><\/span><\/div><p id=\"fs-id1167835283676\">Similarly, linear inequalities in two variables have many solutions. Any ordered pair \\(\\left(x,y\\right)\\) that makes an inequality true when we substitute in the values is a <span data-type=\"term\">solution to a linear inequality<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835609689\"><div data-type=\"title\">Solution to a Linear Inequality<\/div><p id=\"fs-id1167835233801\">An ordered pair \\(\\left(x,y\\right)\\) is a <strong data-effect=\"bold\">solution to a linear inequality<\/strong> if the inequality is true when we substitute the values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167835253977\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835174795\"><div data-type=\"problem\" id=\"fs-id1167835349099\"><p id=\"fs-id1167835322096\">Determine whether each ordered pair is a solution to the inequality \\(y&gt;x+4:\\)<\/p><p><span class=\"token\">\u24d0<\/span>\\(\\left(0,0\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(1,6\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(2,6\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(-5,-15\\right)\\)<span class=\"token\">\u24d4<\/span>\\(\\left(-8,12\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835325302\"><p id=\"fs-id1167834062836\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table class=\"unnumbered unstyled\" summary=\"Substitute 0 for x and 0 for y. Is 0 greater than 0 plus 4? Simplify. 0 is not greater than 4 so (0, 0) is not a solution to y is greater than x plus 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(0,0\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835193138\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834179753\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span>\\(\\phantom{\\rule{1em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835210142\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">So, \\(\\left(0,0\\right)\\) is not a solution to \\(y&gt;x+4.\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835416843\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167831824686\" class=\"unnumbered unstyled\" summary=\"Substitute 1 for x and 6 for y. Is 6 greater than 1 plus 4? Simplify. 6 is greater than 5 so (1, 6) is a solution to y is greater than x plus 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(1,6\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835281916\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span>\\(\\phantom{\\rule{1em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830700650\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834408398\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">So, \\(\\left(1,6\\right)\\) is a solution to \\(y&gt;x+4.\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835379272\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834194741\" class=\"unnumbered unstyled\" summary=\"Substitute 2 for x and 6 for y. Is 6 greater than 2 plus 4? Simplify. 6 is not greater than 6 so (2, 6) is not a solution to y is greater than x plus 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(2,6\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834526122\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span>\\(\\phantom{\\rule{1em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831891015\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835524349\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">So, \\(\\left(2,6\\right)\\) is not a solution to \\(y&gt;x+4.\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835287867\"><span class=\"token\">\u24d3<\/span><\/p><div data-type=\"newline\"><br><\/div><table class=\"unnumbered unstyled\" summary=\"Substitute negative 5 for x and negative 15 for y. Is negative 15 greater than negative 5 plus 4? Simplify. Negative 15 is not greater than negative 1 so (negative 5, negative 15) is not a solution to y is greater than x plus 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(-5,-15\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832053670\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831836128\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831887916\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830866197\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">So, \\(\\left(-5,-15\\right)\\) is not a solution to \\(y&gt;x+4.\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834367192\"><span class=\"token\">\u24d4<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835359748\" class=\"unnumbered unstyled\" summary=\"Substitute negative 8 for x and 12 for y. Is 12 greater than negative 8 plus 4? Simplify. 12 is greater than negative 4 so (negative 8, 12) is a solution to y is greater than x plus 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(-8,12\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835174228\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233360\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span>\\(\\phantom{\\rule{0.3em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835352373\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">So, \\(\\left(-8,12\\right)\\) is a solution to \\(y&gt;x+4.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835197537\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832153132\"><div data-type=\"problem\" id=\"fs-id1167834537361\"><p id=\"fs-id1167835317682\">Determine whether each ordered pair is a solution to the inequality \\(y&gt;x-3:\\)<\/p><p id=\"fs-id1167831919494\"><span class=\"token\">\u24d0<\/span>\\(\\left(0,0\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(4,9\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(-2,1\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(-5,-3\\right)\\)<span class=\"token\">\u24d4<\/span>\\(\\left(5,1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835360880\"><p id=\"fs-id1167835332128\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> yes <span class=\"token\">\u24d3<\/span> yes <span class=\"token\">\u24d4<\/span> no<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835376317\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835234515\"><div data-type=\"problem\" id=\"fs-id1167835346097\"><p id=\"fs-id1167835400395\">Determine whether each ordered pair is a solution to the inequality \\(y&lt;x+1:\\)<\/p><p id=\"fs-id1167835192223\"><span class=\"token\">\u24d0<\/span>\\(\\left(0,0\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(8,6\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(-2,-1\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(3,4\\right)\\)<span class=\"token\">\u24d4<\/span>\\(\\left(-1,-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835310230\"><p id=\"fs-id1167832043379\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> no<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span> yes<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835367015\"><h3 data-type=\"title\">Recognize the Relation Between the Solutions of an Inequality and its Graph<\/h3><p>Now, we will look at how the solutions of an inequality relate to its graph.<\/p><p id=\"fs-id1167835497551\">Let\u2019s think about the number line in shown previously again. The point \\(x=3\\) separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See <a href=\"#CNX_IntAlg_Figure_03_04_007\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_007\"><div class=\"bc-figcaption figcaption\">The solution to \\(x&gt;3\\) is the shaded part of the number line to the right of \\(x=3.\\)<\/div><span data-type=\"media\" id=\"fs-id1167835534329\" data-alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled \u201cnumbers greater than 3\u201d. The part of the number line to the left of 3 is labeled \u201cnumbers less than 3\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled \u201cnumbers greater than 3\u201d. The part of the number line to the left of 3 is labeled \u201cnumbers less than 3\u201d.\"><\/span><\/div><p id=\"fs-id1167834063445\">Similarly, the line \\(y=x+4\\) separates the plane into two regions. On one side of the line are points with \\(y&lt;x+4.\\) On the other side of the line are the points with \\(y&gt;x+4.\\) We call the line \\(y=x+4\\) a <span data-type=\"term\">boundary line<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835230438\"><div data-type=\"title\">Boundary Line<\/div><p id=\"fs-id1167830962142\">The line with equation \\(Ax+By=C\\) is the <strong data-effect=\"bold\">boundary line<\/strong> that separates the region where \\(Ax+By&gt;C\\) from the region where \\(Ax+By&lt;C.\\)<\/p><\/div><p>For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not <em data-effect=\"italics\">a<\/em> is included in the solution:<\/p><span data-type=\"media\" id=\"fs-id1167835416462\" data-alt=\"Two number lines are shown with the middle labeled with the number \u201ca\u201d. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled \u201cx is less than a\u201d and the number a is marked with an open parenthesis. The second number line is labeled \u201cx is less than or equal to a\u201d and the number a is marked with an open bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two number lines are shown with the middle labeled with the number \u201ca\u201d. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled \u201cx is less than a\u201d and the number a is marked with an open parenthesis. The second number line is labeled \u201cx is less than or equal to a\u201d and the number a is marked with an open bracket.\"><\/span><p id=\"fs-id1167835325265\">Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.<\/p><div data-type=\"equation\" id=\"fs-id1167834222348\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}Ax+By&lt;C\\hfill &amp; &amp; &amp; &amp; &amp; Ax+By\\le C\\hfill \\\\ Ax+By&gt;C\\hfill &amp; &amp; &amp; &amp; &amp; Ax+By\\ge C\\hfill \\\\ \\text{Boundary line is}\\phantom{\\rule{0.2em}{0ex}}Ax+By=C\\hfill &amp; &amp; &amp; &amp; &amp; \\text{Boundary line is}\\phantom{\\rule{0.2em}{0ex}}Ax+By=C\\hfill \\\\ \\text{Boundary line is not included in solution.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{Boundary line is included in solution.}\\hfill \\\\ \\mathbf{\\text{Boundary line is dashed.}}\\hfill &amp; &amp; &amp; &amp; &amp; \\mathbf{\\text{Boundary line is solid.}}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835319837\">Now, let\u2019s take a look at what we found in <a href=\"#fs-id1167835253977\" class=\"autogenerated-content\">(Figure)<\/a>. We\u2019ll start by graphing the line \\(y=x+4,\\) and then we\u2019ll plot the five points we tested, as shown in the graph. See <a href=\"#CNX_IntAlg_Figure_03_04_009\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_009\"><span data-type=\"media\" data-alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).\"><\/span><\/div><p id=\"fs-id1167834346594\">In <a href=\"#fs-id1167835253977\" class=\"autogenerated-content\">(Figure)<\/a> we found that some of the points were solutions to the inequality \\(y&gt;x+4\\) and some were not.<\/p><p id=\"fs-id1167832096778\">Which of the points we plotted are solutions to the inequality \\(y&gt;x+4?\\)<\/p><p id=\"fs-id1167834422752\">The points \\(\\left(1,6\\right)\\) and \\(\\left(-8,12\\right)\\) are solutions to the inequality \\(y&gt;x+4.\\) Notice that they are both on the same side of the boundary line \\(y=x+4.\\)<\/p><p id=\"fs-id1167831115420\">The two points \\(\\left(0,0\\right)\\) and \\(\\left(-5,-15\\right)\\) are on the other side of the boundary line \\(y=x+4,\\) and they are not solutions to the inequality \\(y&gt;x+4.\\) For those two points, \\(y&lt;x+4.\\)<\/p><p>What about the point \\(\\left(2,6\\right)?\\) Because \\(6=2+4,\\) the point is a solution to the equation \\(y=x+4,\\) but not a solution to the inequality \\(y&gt;x+4.\\) So the point \\(\\left(2,6\\right)\\) is on the boundary line.<\/p><p id=\"fs-id1167835479170\">Let\u2019s take another point above the boundary line and test whether or not it is a solution to the inequality \\(y&gt;x+4.\\) The point \\(\\left(0,10\\right)\\) clearly looks to above the boundary line, doesn\u2019t it? Is it a solution to the inequality?<\/p><div data-type=\"equation\" id=\"fs-id1167835421108\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill y&amp; &gt;\\hfill &amp; x+4\\hfill \\\\ \\hfill 10&amp; \\stackrel{?}{&gt;}\\hfill &amp; 0+4\\hfill \\\\ \\hfill 10&amp; &gt;\\hfill &amp; 4\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167826996089\">So, \\(\\left(0,10\\right)\\) is a solution to \\(y&gt;x+4.\\)<\/p><p id=\"fs-id1167835338082\">Any point you choose above the boundary line is a solution to the inequality \\(y&gt;x+4.\\) All points above the boundary line are solutions.<\/p><p>Similarly, all points below the boundary line, the side with \\(\\left(0,0\\right)\\) and \\(\\left(-5,-15\\right),\\) are not solutions to \\(y&gt;x+4,\\) as shown in <a href=\"#CNX_IntAlg_Figure_03_04_010\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_010\"><span data-type=\"media\" data-alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.\"><\/span><\/div><p id=\"fs-id1167835318758\">The graph of the inequality \\(y&gt;x+4\\) is shown in below.<\/p><p id=\"fs-id1167831825742\">The line \\(y=x+4\\) divides the plane into two regions. The shaded side shows the solutions to the inequality \\(y&gt;x+4.\\)<\/p><p id=\"fs-id1167835180482\">The points on the boundary line, those where \\(y=x+4,\\) are not solutions to the inequality \\(y&gt;x+4,\\) so the line itself is not part of the solution. We show that by making the line dashed, not solid.<\/p><span data-type=\"media\" id=\"fs-id1167835511460\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\"><\/span><div data-type=\"example\" id=\"fs-id1167834133970\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835370458\"><div data-type=\"problem\"><p id=\"fs-id1167835326585\">The boundary line shown in this graph is \\(y=2x-1.\\) Write the inequality shown by the graph.<\/p><span data-type=\"media\" id=\"fs-id1167834222478\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167828327084\"><p id=\"fs-id1167835264115\">The line \\(y=2x-1\\) is the boundary line. On one side of the line are the points with \\(y&gt;2x-1\\) and on the other side of the line are the points with \\(y&lt;2x-1.\\)<\/p><p id=\"fs-id1167832059873\">Let\u2019s test the point \\(\\left(0,0\\right)\\) and see which inequality describes its position relative to the boundary line.<\/p><p id=\"fs-id1167831891913\">At \\(\\left(0,0\\right),\\) which inequality is true: \\(y&gt;2x-1\\) or \\(y&lt;2x-1?\\)<\/p><div data-type=\"equation\" id=\"fs-id1167831921823\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}y&gt;2x-1\\hfill &amp; &amp; &amp; &amp; &amp; y&lt;2x-1\\hfill \\\\ 0\\stackrel{?}{&gt;}2\u00b70-1\\hfill &amp; &amp; &amp; &amp; &amp; 0\\stackrel{?}{&lt;}2\u00b70-1\\hfill \\\\ 0&gt;-1\\phantom{\\rule{0.2em}{0ex}}\\text{True}\\hfill &amp; &amp; &amp; &amp; &amp; 0&lt;-1\\phantom{\\rule{0.2em}{0ex}}\\text{False}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835232888\">Since, \\(y&gt;2x-1\\) is true, the side of the line with \\(\\left(0,0\\right),\\) is the solution. The shaded region shows the solution of the inequality \\(y&gt;2x-1.\\)<\/p><p id=\"fs-id1167835366721\">Since the boundary line is graphed with a solid line, the inequality includes the equal sign.<\/p><p id=\"fs-id1167835333508\">The graph shows the inequality \\(y\\ge 2x-1.\\)<\/p><p id=\"fs-id1167834228494\">We could use any point as a test point, provided it is not on the line. Why did we choose \\(\\left(0,0\\right)?\\) Because it\u2019s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that \\(y&lt;2x-1.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826828652\"><div data-type=\"problem\" id=\"fs-id1167832054225\"><p id=\"fs-id1167834195241\">Write the inequality shown by the graph with the boundary line \\(y=-2x+3.\\)<\/p><span data-type=\"media\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835339070\"><p id=\"fs-id1167834184145\">\\(y\\ge -2x+3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835352945\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835338956\"><div data-type=\"problem\" id=\"fs-id1167835303775\"><p id=\"fs-id1167834183575\">Write the inequality shown by the graph with the boundary line \\(y=\\frac{1}{2}x-4.\\)<\/p><span data-type=\"media\" id=\"fs-id1167830865778\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835377281\"><p id=\"fs-id1167830757629\">\\(y\\le \\frac{1}{2}x-4\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167832054721\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167827987951\"><div data-type=\"problem\" id=\"fs-id1167835369284\"><p id=\"fs-id1167834066318\">The boundary line shown in this graph is \\(2x+3y=6.\\) Write the inequality shown by the graph.<\/p><span data-type=\"media\" id=\"fs-id1167834532434\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834505860\"><p id=\"fs-id1167834306733\">The line \\(2x+3y=6\\) is the boundary line. On one side of the line are the points with \\(2x+3y&gt;6\\) and on the other side of the line are the points with \\(2x+3y&lt;6.\\)<\/p><p id=\"fs-id1167835350607\">Let\u2019s test the point \\(\\left(0,0\\right)\\) and see which inequality describes its side of the boundary line.<\/p><p id=\"fs-id1167834132988\">At \\(\\left(0,0\\right),\\) which inequality is true: \\(2x+3y&gt;6\\) or \\(2x+3y&lt;6?\\)<\/p><div data-type=\"equation\" id=\"fs-id1167831911258\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\begin{array}{ccc}\\hfill 2x+3y&amp; &gt;\\hfill &amp; 6\\hfill \\\\ \\hfill 2\\left(0\\right)+3\\left(0\\right)&amp; \\stackrel{?}{&gt;}\\hfill &amp; 6\\hfill \\\\ \\hfill 0&amp; &gt;\\hfill &amp; 6\\phantom{\\rule{0.2em}{0ex}}\\text{False}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccc}\\hfill 2x+3y&amp; &lt;\\hfill &amp; 6\\hfill \\\\ \\hfill 2\\left(0\\right)+3\\left(0\\right)&amp; \\stackrel{?}{&lt;}\\hfill &amp; 6\\hfill \\\\ \\hfill 0&amp; &lt;\\hfill &amp; 6\\phantom{\\rule{0.2em}{0ex}}\\text{True}\\hfill \\end{array}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835323826\">So the side with \\(\\left(0,0\\right)\\) is the side where \\(2x+3y&lt;6.\\)<\/p><p id=\"fs-id1167835361279\">(You may want to pick a point on the other side of the boundary line and check that \\(2x+3y&gt;6.\\))<\/p><p id=\"fs-id1167835355907\">Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.<\/p><p id=\"fs-id1167835281489\">The shaded region shows the solution to the inequality \\(2x+3y&lt;6.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834133165\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832042396\"><div data-type=\"problem\" id=\"fs-id1167826967255\"><p id=\"fs-id1167832058079\">Write the inequality shown by the shaded region in the graph with the boundary line \\(x-4y=8.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835384392\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167830836855\"><p id=\"fs-id1167834516596\">\\(x-4y\\le 8\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835189046\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835375292\"><div data-type=\"problem\" id=\"fs-id1167834516154\"><p id=\"fs-id1167835268997\">Write the inequality shown by the shaded region in the graph with the boundary line \\(3x-y=6.\\)<\/p><span data-type=\"media\" id=\"fs-id1167831896349\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835369697\"><p id=\"fs-id1167834195300\">\\(3x-y\\ge 6\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834431130\"><h3 data-type=\"title\">Graph Linear Inequalities in Two Variables<\/h3><p id=\"fs-id1167835173654\">Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.<\/p><div data-type=\"example\" id=\"fs-id1167834059141\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph a Linear Equation in Two Variables<\/div><div data-type=\"exercise\" id=\"fs-id1167834186008\"><div data-type=\"problem\" id=\"fs-id1167834473385\"><p id=\"fs-id1167835334366\">Graph the linear inequality \\(y\\ge \\frac{3}{4}x-2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835327818\"><span data-type=\"media\" id=\"fs-id1167832066478\" data-alt=\"Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_018a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).\"><\/span><span data-type=\"media\" id=\"fs-id1167834432374\" data-alt=\"Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_018b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.\"><\/span><span data-type=\"media\" id=\"fs-id1167835180882\" data-alt=\"Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_018c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832075476\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832087004\"><div data-type=\"problem\" id=\"fs-id1167835330003\"><p id=\"fs-id1167835519946\">Graph the linear inequality \\(y&gt;\\frac{5}{2}x-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834463224\"><p id=\"fs-id1167831031013\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831031015\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (2, 1), and (4, 6). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (2, 1), and (4, 6). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region and on the boundary line, represent the solutions to \\(y&gt;\\frac{5}{2}x-4.\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831239812\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830838406\"><div data-type=\"problem\" id=\"fs-id1167830838408\"><p id=\"fs-id1167835419768\">Graph the linear inequality \\(y&lt;\\frac{2}{3}x-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835237898\"><p id=\"fs-id1167835237901\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835339047\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 5), (3, negative 3), and (5, negative 1). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 5), (3, negative 3), and (5, negative 1). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region, but not those on the boundary line, represent the solutions to \\(y&lt;\\frac{2}{3}x-5.\\)<\/div><\/div><\/div><p id=\"fs-id1167834394395\">The steps we take to graph a linear inequality are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167835367051\" class=\"howto\"><div data-type=\"title\">Graph a linear inequality in two variables.<\/div><ol id=\"fs-id1167826996926\" type=\"1\" class=\"stepwise\"><li>Identify and graph the boundary line. <ul id=\"fs-id1167834547281\" data-bullet-style=\"bullet\"><li>If the inequality is \\(\\le \\text{or}\\ge ,\\) the boundary line is solid.<\/li><li>If the inequality is \\(&lt;\\text{or}&gt;,\\) the boundary line is dashed.<\/li><\/ul><\/li><li>Test a point that is not on the boundary line. Is it a solution of the inequality?<\/li><li>Shade in one side of the boundary line. <ul id=\"fs-id1167831823898\" data-bullet-style=\"bullet\"><li>If the test point is a solution, shade in the side that includes the point.<\/li><li>If the test point is not a solution, shade in the opposite side.<\/li><\/ul><\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167835380178\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835380180\"><div data-type=\"problem\" id=\"fs-id1167835365170\"><p id=\"fs-id1167834129902\">Graph the linear inequality \\(x-2y&lt;5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835377740\"><p id=\"fs-id1167835355962\">First, we graph the boundary line \\(x-2y=5.\\) The inequality is \\(&lt;\\) so we draw a dashed line.<\/p><span data-type=\"media\" id=\"fs-id1167830701265\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).\"><\/span><p>Then, we test a point. We\u2019ll use \\(\\left(0,0\\right)\\) again because it is easy to evaluate and it is not on the boundary line.<\/p><p id=\"fs-id1167828420183\">Is \\(\\left(0,0\\right)\\) a solution of \\(x-2y&lt;5?\\)<\/p><span data-type=\"media\" id=\"fs-id1167835420922\" data-alt=\"Is 0 minus 2 times 0 less than 5? Is 0 minus 0 less than 5? 0 is less than 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Is 0 minus 2 times 0 less than 5? Is 0 minus 0 less than 5? 0 is less than 5.\"><\/span><p id=\"fs-id1167835511602\">The point \\(\\left(0,0\\right)\\) is a solution of \\(x-2y&lt;5,\\) so we shade in that side of the boundary line.<\/p><span data-type=\"media\" id=\"fs-id1167835253751\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p id=\"fs-id1167831911233\">All points in the shaded region, but not those on the boundary line, represent the solutions to \\(x-2y&lt;5.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835163492\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831891076\"><div data-type=\"problem\" id=\"fs-id1167835253904\"><p id=\"fs-id1167835368944\">Graph the linear inequality: \\(2x-3y&lt;6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834138029\"><p id=\"fs-id1167834464384\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834464386\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 2), (3, 0), and (6, 2). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 2), (3, 0), and (6, 2). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region, but not those on the boundary line, represent the solutions to \\(2x-3y&lt;6.\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835410949\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835264380\"><div data-type=\"problem\" id=\"fs-id1167831883737\"><p id=\"fs-id1167831880097\">Graph the linear inequality: \\(2x-y&gt;3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835346062\"><p id=\"fs-id1167835514065\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167832119046\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 1), and (2, 1). The line divides the x y-coordinate plane into two halves. The bottom right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 1), and (2, 1). The line divides the x y-coordinate plane into two halves. The bottom right half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region, but not those on the boundary line, represent the solutions to \\(2x-y&gt;3.\\)<\/div><\/div><\/div><p id=\"fs-id1167835331510\">What if the boundary line goes through the origin? Then, we won\u2019t be able to use \\(\\left(0,0\\right)\\) as a test point. No problem\u2014we\u2019ll just choose some other point that is not on the boundary line.<\/p><div data-type=\"example\" id=\"fs-id1167835229236\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834469610\"><div data-type=\"problem\" id=\"fs-id1167834469612\"><p id=\"fs-id1167834191414\">Graph the linear inequality: \\(y\\le \\text{\u200b}-4x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835280571\"><p id=\"fs-id1167835280573\">First, we graph the boundary line \\(y=-4x.\\) It is in slope\u2013intercept form, with \\(m=-4\\) and \\(b=0.\\) The inequality is \\(\\le \\) so we draw a solid line.<\/p><span data-type=\"media\" id=\"fs-id1167835376108\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).\"><\/span><p id=\"fs-id1167835230300\">Now we need a test point. We can see that the point \\(\\left(1,0\\right)\\) is not on the boundary line.<\/p><p id=\"fs-id1167835374272\">Is \\(\\left(1,0\\right)\\) a solution of \\(y\\le -4x?\\)<\/p><span data-type=\"media\" id=\"fs-id1167832041694\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><p id=\"fs-id1167835395578\">The point \\(\\left(1,0\\right)\\) is not a solution to \\(y\\le \\text{\u200b}-4x,\\) so we shade in the opposite side of the boundary line.<\/p><span data-type=\"media\" id=\"fs-id1167831985833\" data-alt=\"Is 0 less than or equal to negative 4 times 1? 0 is not less than or equal to negative 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Is 0 less than or equal to negative 4 times 1? 0 is not less than or equal to negative 4.\"><\/span><p id=\"fs-id1167835244104\">All points in the shaded region and on the boundary line represent the solutions to \\(y\\le \\text{\u200b}-4x.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835367619\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835171005\"><div data-type=\"problem\" id=\"fs-id1167835171007\"><p id=\"fs-id1167832052015\">Graph the linear inequality: \\(y&gt;-3x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834114308\"><p id=\"fs-id1167835640409\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835640412\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 3), (0, 0), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 3), (0, 0), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region, but not those on the boundary line, represent the solutions to \\(y&gt;-3x.\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834085021\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835241372\"><div data-type=\"problem\" id=\"fs-id1167835346336\"><p id=\"fs-id1167835346339\">Graph the linear inequality: \\(y\\ge -2x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831025381\"><p id=\"fs-id1167835191448\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835191450\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 2), (0, 0), and (1, negative 2). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 2), (0, 0), and (1, negative 2). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region and on the boundary line, represent the solutions to \\(y\\ge -2x.\\)<\/div><\/div><\/div><p id=\"fs-id1167835210529\">Some linear inequalities have only one variable. They may have an <em data-effect=\"italics\">x<\/em> but no <em data-effect=\"italics\">y<\/em>, or a <em data-effect=\"italics\">y<\/em> but no <em data-effect=\"italics\">x<\/em>. In these cases, the boundary line will be either a vertical or a horizontal line.<\/p><p id=\"fs-id1167835349500\">Recall that:<\/p><div data-type=\"equation\" id=\"fs-id1167831106740\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}x=a\\hfill &amp; &amp; &amp; &amp; &amp; \\text{vertical line}\\hfill \\\\ y=b\\hfill &amp; &amp; &amp; &amp; &amp; \\text{horizontal line}\\hfill \\end{array}\\)<\/div><div data-type=\"example\" id=\"fs-id1167835200429\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835200431\"><div data-type=\"problem\" id=\"fs-id1167835346154\"><p id=\"fs-id1167835346156\">Graph the linear inequality: \\(y&gt;3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834387736\"><p id=\"fs-id1167834387738\">First, we graph the boundary line \\(y=3.\\) It is a horizontal line. The inequality is \\(&gt;\\) so we draw a dashed line.<\/p><p id=\"fs-id1167830704322\">We test the point \\(\\left(0,0\\right).\\)<\/p><div data-type=\"equation\" id=\"fs-id1167834195455\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{}\\\\ y&gt;3\\hfill \\\\ 0\\overline{)&gt;}3\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167831922006\">So, \\(\\left(0,0\\right)\\) is not a solution to \\(y&gt;3.\\)<\/p><p id=\"fs-id1167835321005\">So we shade the side that does not include \\(\\left(0,0\\right)\\) as shown in this graph.<\/p><span data-type=\"media\" id=\"fs-id1167832128683\" data-alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p id=\"fs-id1167834193406\">All points in the shaded region, but not those on the boundary line, represent the solutions to \\(y&gt;3.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834473439\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835366245\"><div data-type=\"problem\" id=\"fs-id1167835366247\"><p id=\"fs-id1167832055097\">Graph the linear inequality: \\(y&lt;5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835563964\"><p id=\"fs-id1167835514390\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835514392\" data-alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal dashed line is drawn through the points (negative 1, 5), (0, 5), and (1, 5). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal dashed line is drawn through the points (negative 1, 5), (0, 5), and (1, 5). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region, but not those on the boundary line, represent the solutions to \\(y&lt;5.\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826781485\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832058169\"><div data-type=\"problem\" id=\"fs-id1167835334494\"><p id=\"fs-id1167835334496\">Graph the linear inequality: \\(y\\le -1.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167831919784\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831919786\" data-alt=\"This figure has the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal line is drawn through the points (negative 1, negative 1), (0, negative 1), and (1, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal line is drawn through the points (negative 1, negative 1), (0, negative 1), and (1, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> All points in the shaded region and on the boundary line represent the solutions to \\(y\\le -1.\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835524373\"><h3 data-type=\"title\">Solve Applications using Linear Inequalities in Two Variables<\/h3><p id=\"fs-id1167830706031\">Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how thay might be used.<\/p><div data-type=\"example\" id=\"fs-id1167835363723\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835254481\"><div data-type=\"problem\" id=\"fs-id1167835254483\"><p id=\"fs-id1167832067984\">Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least ?240 a week. Her job in food service pays ?10 an hour and her tutoring job on campus pays ?15 an hour. How many hours does Hilaria need to work at each job to earn at least ?240?<\/p><p id=\"fs-id1167835327602\"><span class=\"token\">\u24d0<\/span> Let \\(x\\) be the number of hours she works at the job in food service and let <em data-effect=\"italics\">y<\/em> be the number of hours she works tutoring. Write an inequality that would model this situation.<\/p><p id=\"fs-id1167835325479\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167834433649\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs \\(\\left(x,y\\right)\\) that would be solutions to the inequality. Then, explain what that means for Hilaria.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835370005\"><p id=\"fs-id1167835370007\"><span class=\"token\">\u24d0<\/span> We let <em data-effect=\"italics\">x<\/em> be the number of hours she works at the job in food service and let <em data-effect=\"italics\">y<\/em> be the number of hours she works tutoring.<\/p><p id=\"fs-id1167831872422\">She earns ?10 per hour at the job in food service and ?15 an hour tutoring. At each job, the number of hours multiplied by the hourly wage will gives the amount earned at that job.<\/p><span data-type=\"media\" id=\"fs-id1167834219631\" data-alt=\"10 x plus 15 y is greater than 240. The \u201c10 x\u201d is labeled \u201cAmount earned at the food service job\u201d. The \u201c15 y\u201d is labeled \u201cthe amount earned tutoring\u201d. The \u201cis greater than 240\u201d is labeled \u201cis at least 240\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"10 x plus 15 y is greater than 240. The \u201c10 x\u201d is labeled \u201cAmount earned at the food service job\u201d. The \u201c15 y\u201d is labeled \u201cthe amount earned tutoring\u201d. The \u201cis greater than 240\u201d is labeled \u201cis at least 240\u201d.\"><\/span><p id=\"fs-id1167834517584\"><span class=\"token\">\u24d1<\/span> To graph the inequality, we put it in slope\u2013intercept form.<\/p><div data-type=\"equation\" id=\"fs-id1167834339954\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill 10x+15y&amp; \\ge \\hfill &amp; 240\\hfill \\\\ \\hfill 15y&amp; \\ge \\hfill &amp; -10x+240\\hfill \\\\ \\hfill y&amp; \\ge \\hfill &amp; -\\frac{2}{3}x+16\\hfill \\end{array}\\)<\/div><span data-type=\"media\" id=\"fs-id1167831896699\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_027_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p id=\"fs-id1167835306260\"><span class=\"token\">\u24d2<\/span> From the graph, we see that the ordered pairs \\(\\left(15,10\\right),\\left(0,16\\right),\\left(24,0\\right)\\) represent three of infinitely many solutions. Check the values in the inequality.<\/p><span data-type=\"media\" id=\"fs-id1167835299865\" data-alt=\"First we test the point (15, 10) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 15 plus 15 times 10 greater than or equal to 240? Since 300 is greater than or equal to 240 (15, 10) is a solution. Next we test the point (0, 16) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 0 plus 15 times 16 greater than or equal to 240? Since 240 is greater than or equal to 240 (0, 16) is a solution. Then we test the point (24, 0) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 24 plus 15 times 0 greater than or equal to 240? Since 240 is greater than or equal to 240 (24, 0) is a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_028_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"First we test the point (15, 10) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 15 plus 15 times 10 greater than or equal to 240? Since 300 is greater than or equal to 240 (15, 10) is a solution. Next we test the point (0, 16) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 0 plus 15 times 16 greater than or equal to 240? Since 240 is greater than or equal to 240 (0, 16) is a solution. Then we test the point (24, 0) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 24 plus 15 times 0 greater than or equal to 240? Since 240 is greater than or equal to 240 (24, 0) is a solution.\"><\/span><p id=\"fs-id1167826987862\">For Hilaria, it means that to earn at least ?240, she can work 15 hours tutoring and 10 hours at her fast-food job, earn all her money tutoring for 16 hours, or earn all her money while working 24 hours at the job in food service.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832015628\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834284498\"><div data-type=\"problem\" id=\"fs-id1167835348793\"><p id=\"fs-id1167835348795\">Hugh works two part time jobs. One at a grocery store that pays ?10 an hour and the other is babysitting for ?13 hour. Between the two jobs, Hugh wants to earn at least ?260 a week. How many hours does Hugh need to work at each job to earn at least ?260?<\/p><p id=\"fs-id1167834063929\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours he works at the grocery store and let <em data-effect=\"italics\">y<\/em> be the number of hours he works babysitting. Write an inequality that would model this situation.<\/p><p id=\"fs-id1167835418163\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167826783577\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>) that would be solutions to the inequality. Then, explain what that means for Hugh.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835368581\"><p id=\"fs-id1167835368583\"><span class=\"token\">\u24d0<\/span>\\(10x+13y\\ge 260\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835307410\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 30. A line is drawn through the points (0, 20), (13, 10), and (26, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 30. A line is drawn through the points (0, 20), (13, 10), and (26, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835167616\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835167619\"><div data-type=\"problem\" id=\"fs-id1167835194595\"><p id=\"fs-id1167835194597\">Veronica works two part time jobs in order to earn enough money to meet her obligations of at least ?280 a week. Her job at the day spa pays ?10 an hour and her administrative assistant job on campus pays ?17.50 an hour. How many hours does Veronica need to work at each job to earn at least ?280?<\/p><p id=\"fs-id1167835189599\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours she works at the day spa and let <em data-effect=\"italics\">y<\/em> be the number of hours she works as administrative assistant. Write an inequality that would model this situation.<\/p><p><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167834431575\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>) that would be solutions to the inequality. Then, explain what that means for Veronica<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834309537\"><p id=\"fs-id1167835513671\"><span class=\"token\">\u24d0<\/span>\\(10x+17.5y\\ge 280\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835325137\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16) and (28, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16) and (28, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831891876\" class=\"media-2\"><p id=\"fs-id1167830865844\">Access this online resource for additional instruction and practice with graphing linear inequalities in two variables.<\/p><ul id=\"fs-id1167831956377\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37linineqgraphs\">Graphing Linear Inequalities in Two Variables<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167827956786\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835366934\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to graph a linear inequality in two variables.<\/strong><ol id=\"fs-id1167835262448\" type=\"1\" class=\"stepwise\"><li>Identify and graph the boundary line.<div data-type=\"newline\"><br><\/div> If the inequality is \\(\\le \\text{or}\\ge ,\\) the boundary line is solid.<div data-type=\"newline\"><br><\/div> If the inequality is \\(&lt;\\phantom{\\rule{0.2em}{0ex}}\\text{or}&gt;,\\) the boundary line is dashed.<\/li><li>Test a point that is not on the boundary line. Is it a solution of the inequality?<\/li><li>Shade in one side of the boundary line.<div data-type=\"newline\"><br><\/div> If the test point is a solution, shade in the side that includes the point.<div data-type=\"newline\"><br><\/div> If the test point is not a solution, shade in the opposite side.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834431207\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834431210\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167832060209\"><strong data-effect=\"bold\">Verify Solutions to an Inequality in Two Variables<\/strong><\/p><p id=\"fs-id1167832054111\">In the following exercises, determine whether each ordered pair is a solution to the given inequality.<\/p><div data-type=\"exercise\" id=\"fs-id1167835376934\"><div data-type=\"problem\" id=\"fs-id1167835376937\"><p id=\"fs-id1167832044125\">Determine whether each ordered pair is a solution to the inequality \\(y&gt;x-1:\\)<\/p><p id=\"fs-id1167835350000\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(0,1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-4,-1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(4,2\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(3,0\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(-2,-3\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835231057\"><p id=\"fs-id1167835171272\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> no <span class=\"token\">\u24d4<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167832053265\"><p id=\"fs-id1167834161636\">Determine whether each ordered pair is a solution to the inequality \\(y&gt;x-3:\\)<\/p><p id=\"fs-id1167835609385\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(0,0\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(2,1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(-1,-5\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(-6,-3\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(1,0\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831086840\"><div data-type=\"problem\" id=\"fs-id1167831086843\"><p id=\"fs-id1167832042712\">Determine whether each ordered pair is a solution to the inequality \\(y&lt;3x+2:\\)<\/p><p id=\"fs-id1167834505590\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(0,3\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-3,-2\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(-2,0\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(0,0\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(-1,4\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835263066\"><p id=\"fs-id1167835263068\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> yes <span class=\"token\">\u24d3<\/span> yes <span class=\"token\">\u24d4<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831023966\"><div data-type=\"problem\" id=\"fs-id1167835336368\"><p id=\"fs-id1167835336371\">Determine whether each ordered pair is a solution to the inequality \\(y&lt;-2x+5:\\)<\/p><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(-3,0\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(1,6\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(-6,-2\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(0,1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(5,-4\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826997319\"><div data-type=\"problem\" id=\"fs-id1167835201298\"><p id=\"fs-id1167835201301\">Determine whether each ordered pair is a solution to the inequality \\(3x-4y&gt;4:\\)<\/p><p id=\"fs-id1167835419834\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(5,1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-2,6\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(3,2\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(10,-5\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(0,0\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167832151136\"><p id=\"fs-id1167832151138\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> no <span class=\"token\">\u24d4<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834327594\"><p id=\"fs-id1167834327596\">Determine whether each ordered pair is a solution to the inequality \\(2x+3y&gt;2:\\)<\/p><p id=\"fs-id1167834222006\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(1,1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(4,-3\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(0,0\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(-8,12\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(3,0\\right)\\)<\/div><\/div><p id=\"fs-id1167834058748\"><strong data-effect=\"bold\">Recognize the Relation Between the Solutions of an Inequality and its Graph<\/strong><\/p><p id=\"fs-id1167831086587\">In the following exercises, write the inequality shown by the shaded region.<\/p><div data-type=\"exercise\" id=\"fs-id1167831086590\"><div data-type=\"problem\" id=\"fs-id1167831891686\"><p id=\"fs-id1167831891688\">Write the inequality shown by the graph with the boundary line \\(y=3x-4.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835329737\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835336434\"><p id=\"fs-id1167835239120\">\\(y\\le 3x-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831909967\"><div data-type=\"problem\" id=\"fs-id1167831909970\"><p id=\"fs-id1167835621626\">Write the inequality shown by the graph with the boundary line \\(y=2x-4.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835357587\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834058839\"><div data-type=\"problem\" id=\"fs-id1167834058841\"><p id=\"fs-id1167835331951\">Write the inequality shown by the graph with the boundary line \\(y=-\\frac{1}{2}x+1.\\)<\/p><span data-type=\"media\" id=\"fs-id1167831852242\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 1), (2, 0), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 1), (2, 0), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834192173\"><p id=\"fs-id1167834192176\">\\(y\\le -\\frac{1}{2}x+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831197142\"><div data-type=\"problem\" id=\"fs-id1167831197144\"><p id=\"fs-id1167835369782\">Write the inequality shown by the graph with the boundary line \\(y=-\\frac{1}{3}x-2.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835216954\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (3, negative 3), and (6, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (3, negative 3), and (6, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834308117\"><div data-type=\"problem\" id=\"fs-id1167834308119\"><p id=\"fs-id1167831228827\">Write the inequality shown by the shaded region in the graph with the boundary line \\(x+y=5.\\)<\/p><span data-type=\"media\" id=\"fs-id1167832053362\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 5), (1, 4), and (5, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 5), (1, 4), and (5, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835610067\"><p id=\"fs-id1167835610070\">\\(x+y\\ge 5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835528518\"><div data-type=\"problem\" id=\"fs-id1167835217818\"><p id=\"fs-id1167835217820\">Write the inequality shown by the shaded region in the graph with the boundary line \\(x+y=3.\\)<\/p><span data-type=\"media\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835615661\"><div data-type=\"problem\" id=\"fs-id1167835422446\"><p id=\"fs-id1167835422449\">Write the inequality shown by the shaded region in the graph with the boundary line \\(3x-y=6.\\)<\/p><span data-type=\"media\" id=\"fs-id1167831913470\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835287946\"><p id=\"fs-id1167828240723\">\\(3x-y\\le 6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479850\"><div data-type=\"problem\" id=\"fs-id1167835300614\"><p id=\"fs-id1167835300617\">Write the inequality shown by the shaded region in the graph with the boundary line \\(2x-y=4.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835238193\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><p id=\"fs-id1167835302992\"><strong data-effect=\"bold\">Graph Linear Inequalities in Two Variables<\/strong><\/p><p id=\"fs-id1167835423443\">In the following exercises, graph each linear inequality.<\/p><div data-type=\"exercise\" id=\"fs-id1167834395424\"><div data-type=\"problem\" id=\"fs-id1167834395427\"><p id=\"fs-id1167831923108\">Graph the linear inequality: \\(y&gt;\\frac{2}{3}x-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834495534\"><span data-type=\"media\" id=\"fs-id1167834395373\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834308235\"><div data-type=\"problem\" id=\"fs-id1167834308238\"><p id=\"fs-id1167835350862\">Graph the linear inequality: \\(y&lt;\\frac{3}{5}x+2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835254401\"><div data-type=\"problem\" id=\"fs-id1167835254403\"><p id=\"fs-id1167831821812\">Graph the linear inequality: \\(y\\le -\\frac{1}{2}x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831883445\"><span data-type=\"media\" id=\"fs-id1167831883449\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826813777\"><div data-type=\"problem\" id=\"fs-id1167826813780\"><p id=\"fs-id1167826813782\">Graph the linear inequality: \\(y\\ge -\\frac{1}{3}x-2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834345405\"><div data-type=\"problem\" id=\"fs-id1167834345408\"><p id=\"fs-id1167834345410\">Graph the linear inequality: \\(x-y\\le 3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835339974\"><span data-type=\"media\" id=\"fs-id1167835244701\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831912084\"><div data-type=\"problem\" id=\"fs-id1167834131362\"><p id=\"fs-id1167834131364\">Graph the linear inequality: \\(x-y\\ge -2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830865770\"><div data-type=\"problem\"><p id=\"fs-id1167834179696\">Graph the linear inequality: \\(4x+y&gt;-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834183954\"><span data-type=\"media\" id=\"fs-id1167832053177\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835341770\"><div data-type=\"problem\" id=\"fs-id1167835336678\"><p id=\"fs-id1167835336680\">Graph the linear inequality: \\(x+5y&lt;-5.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830702903\"><div data-type=\"problem\" id=\"fs-id1167830702905\"><p id=\"fs-id1167834526263\">Graph the linear inequality: \\(3x+2y\\ge -6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834567460\"><span data-type=\"media\" id=\"fs-id1167835479044\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834061947\"><div data-type=\"problem\" id=\"fs-id1167835376646\"><p id=\"fs-id1167835376648\">Graph the linear inequality: \\(4x+2y\\ge -8.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834431761\"><div data-type=\"problem\" id=\"fs-id1167834431763\"><p id=\"fs-id1167834431765\">Graph the linear inequality: \\(y&gt;4x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835369505\"><span data-type=\"media\" id=\"fs-id1167834300213\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835280009\"><div data-type=\"problem\" id=\"fs-id1167835280011\"><p id=\"fs-id1167831835385\">Graph the linear inequality: \\(y\\le -3x.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834526549\"><div data-type=\"problem\" id=\"fs-id1167834526551\"><p id=\"fs-id1167834526553\">Graph the linear inequality: \\(y&lt;-10.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832054972\"><span data-type=\"media\" id=\"fs-id1167835305019\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, 3), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, 3), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835331419\"><div data-type=\"problem\" id=\"fs-id1167835331421\"><p id=\"fs-id1167835377774\">Graph the linear inequality: \\(y\\ge 2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830865783\"><div data-type=\"problem\" id=\"fs-id1167830865785\"><p id=\"fs-id1167830964366\">Graph the linear inequality: \\(x\\le 5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835352901\"><span data-type=\"media\" id=\"fs-id1167832150918\" data-alt=\"This figure has the graph of a straight vertical dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A vertical dashed line is drawn through the points (5, negative 1), (5, 0), and (5, 1). The line divides the x y-coordinate plane into two halves. The left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight vertical dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A vertical dashed line is drawn through the points (5, negative 1), (5, 0), and (5, 1). The line divides the x y-coordinate plane into two halves. The left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835349534\"><div data-type=\"problem\" id=\"fs-id1167835349536\"><p id=\"fs-id1167832052493\">Graph the linear inequality: \\(x\\ge 0.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835381719\"><div data-type=\"problem\" id=\"fs-id1167835381721\"><p id=\"fs-id1167834431115\">Graph the linear inequality: \\(x-y&lt;4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835180853\"><span data-type=\"media\" id=\"fs-id1167835180858\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835235204\"><div data-type=\"problem\" id=\"fs-id1167835235206\"><p id=\"fs-id1167835235209\">Graph the linear inequality: \\(x-y&lt;-3.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835234653\"><div data-type=\"problem\" id=\"fs-id1167835234655\"><p id=\"fs-id1167835234658\">Graph the linear inequality: \\(y\\ge \\frac{3}{2}x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827956861\"><span data-type=\"media\" id=\"fs-id1167835240348\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834064077\"><div data-type=\"problem\" id=\"fs-id1167835358355\"><p id=\"fs-id1167835358357\">Graph the linear inequality: \\(y\\le \\frac{5}{4}x.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835589641\"><div data-type=\"problem\" id=\"fs-id1167835274944\"><p id=\"fs-id1167835274946\">Graph the linear inequality: \\(y&gt;-2x+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834300617\"><span data-type=\"media\" id=\"fs-id1167834300621\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 1), (1, negative 1), and (2, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 1), (1, negative 1), and (2, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826996648\"><div data-type=\"problem\" id=\"fs-id1167826996650\"><p id=\"fs-id1167826996653\">Graph the linear inequality: \\(y&lt;-3x-4.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835369221\"><div data-type=\"problem\" id=\"fs-id1167835369223\"><p id=\"fs-id1167835338055\">Graph the linear inequality: \\(2x+y\\ge -4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835166485\"><span data-type=\"media\" id=\"fs-id1167834463977\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_334_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835217909\"><div data-type=\"problem\" id=\"fs-id1167835217911\"><p id=\"fs-id1167834533381\">Graph the linear inequality: \\(x+2y\\le -2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831894741\"><div data-type=\"problem\" id=\"fs-id1167831239471\"><p id=\"fs-id1167831239473\">Graph the linear inequality: \\(2x-5y&gt;10.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835240479\"><span data-type=\"media\" id=\"fs-id1167835284949\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831922737\"><div data-type=\"problem\" id=\"fs-id1167831922739\"><p id=\"fs-id1167834214189\">Graph the linear inequality: \\(4x-3y&gt;12.\\)<\/p><\/div><\/div><p id=\"fs-id1167835284781\"><strong data-effect=\"bold\">Solve Applications using Linear Inequalities in Two Variables<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1167832015642\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832015644\"><p id=\"fs-id1167835235704\">Harrison works two part time jobs. One at a gas station that pays ?11 an hour and the other is IT troubleshooting for \\(\\text{?}16.50\\) an hour. Between the two jobs, Harrison wants to earn at least ?330 a week. How many hours does Harrison need to work at each job to earn at least ?330?<\/p><p id=\"fs-id1167835362766\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours he works at the gas station and let <em data-effect=\"italics\">y<\/em> be the number of (hours he works troubleshooting. Write an inequality that would model this situation.<\/p><p id=\"fs-id1167834131200\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167826997211\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs \\(\\left(x,y\\right)\\) that would be solutions to the inequality. Then, explain what that means for Harrison.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834196585\"><p id=\"fs-id1167834196587\"><span class=\"token\">\u24d0<\/span>\\(11x+16.5y\\ge 330\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831228835\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831191375\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832015790\"><p id=\"fs-id1167832015792\">Elena needs to earn at least ?450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays ?9 an hour and the other as an intern in a genetics lab for ?22.50 per hour. How many hours does Elena need to work at each job to earn at least ?450 per week?<\/p><p id=\"fs-id1167834429285\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours she works teaching swimming and let <em data-effect=\"italics\">y<\/em> be the number of hours she works as an intern. Write an inequality that would model this situation.<\/p><p id=\"fs-id1167835235765\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167834472406\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs \\(\\left(x,y\\right)\\) that would be solutions to the inequality. Then, explain what that means for Elena.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835234207\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835244283\"><p id=\"fs-id1167835244285\">The doctor tells Laura she needs to exercise enough to burn 500 calories each day. She prefers to either run or bike and burns 15 calories per minute while running and 10 calories a minute while biking.<\/p><p id=\"fs-id1167831887210\"><span class=\"token\">\u24d0<\/span> If <em data-effect=\"italics\">x<\/em> is the number of minutes that Laura runs and <em data-effect=\"italics\">y<\/em> is the number minutes she bikes, find the inequality that models the situation.<\/p><p id=\"fs-id1167831890641\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167834495426\"><span class=\"token\">\u24d2<\/span> List three solutions to the inequality. What options do the solutions provide Laura?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835263230\"><p id=\"fs-id1167834135022\"><span class=\"token\">\u24d0<\/span>\\(15x+10y\\ge 500\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834489866\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 60. A line is drawn through the points (0, 50) and (20, 20). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 60. A line is drawn through the points (0, 50) and (20, 20). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835202221\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835202224\"><p id=\"fs-id1167835202226\">Armando\u2019s workouts consist of kickboxing and swimming. While kickboxing, he burns 10 calories per minute and he burns 7 calories a minute while swimming. He wants to burn 600 calories each day.<\/p><p id=\"fs-id1167834345659\"><span class=\"token\">\u24d0<\/span> If <em data-effect=\"italics\">x<\/em> is the number of minutes that Armando will kickbox and <em data-effect=\"italics\">y<\/em> is the number minutes he will swim, find the inequality that will help Armando create a workout for today.<\/p><p id=\"fs-id1167834515425\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p><p id=\"fs-id1167835365741\"><span class=\"token\">\u24d2<\/span> List three solutions to the inequality. What options do the solutions provide Armando?<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835333874\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167830701036\"><div data-type=\"problem\" id=\"fs-id1167832054672\"><p id=\"fs-id1167832054674\">Lester thinks that the solution of any inequality with a \\(&gt;\\) sign is the region above the line and the solution of any inequality with a \\(&lt;\\) sign is the region below the line. Is Lester correct? Explain why or why not.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835338740\"><p>Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835335460\"><div data-type=\"problem\" id=\"fs-id1167834229208\"><p id=\"fs-id1167834229210\">Explain why, in some graphs of linear inequalities, the boundary line is solid but in other graphs it is dashed.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167826807779\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167826857196\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167831943967\" data-alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cverify solutions to an inequality in two variables.\u201d, \u201crecognize the relation between the solutions of an inequality and its graph\u201d, and \u201cgraph linear inequalities\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_209_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cverify solutions to an inequality in two variables.\u201d, \u201crecognize the relation between the solutions of an inequality and its graph\u201d, and \u201cgraph linear inequalities\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><\/span><p id=\"fs-id1167835328724\"><span class=\"token\">\u24d1<\/span> On a scale of 1\u201310, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167834061630\"><dt>boundary line<\/dt><dd id=\"fs-id1167834061636\">The line with equation \\(Ax+By=C\\) is the boundary line that separates the region where \\(Ax+By&gt;C\\) from the region where \\(Ax+By&lt;C.\\)<\/dd><\/dl><dl id=\"fs-id1167835310131\"><dt>linear inequality<\/dt><dd id=\"fs-id1167826779362\">A linear inequality is an inequality that can be written in one of the following forms: \\(Ax+By&gt;C,\\phantom{\\rule{0.4em}{0ex}}Ax+By\\ge C,\\phantom{\\rule{0.4em}{0ex}}Ax+By&lt;C,\\) or \\(Ax+By\\le C,\\) where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero.<\/dd><\/dl><dl id=\"fs-id1167834328459\"><dt>solution to a linear inequality<\/dt><dd id=\"fs-id1167834328465\">An ordered pair \\(\\left(x,y\\right)\\) is a solution to a linear inequality if the inequality is true when we substitute the values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/dd><\/dl><\/div>\n","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>Verify solutions to an inequality in two variables.<\/li>\n<li>Recognize the relation between the solutions of an inequality and its graph.<\/li>\n<li>Graph linear inequalities in two variables<\/li>\n<li>Solve applications using linear inequalities in two variables<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834120877\" class=\"be-prepared\">\n<p id=\"fs-id1167832052568\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167834396303\" type=\"1\">\n<li>Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0d114ca900168936b5c270433aff883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> on a number line.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835334972\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f408c94aa283c82467de5d4726945703_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#51;&#62;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"91\" style=\"vertical-align: -2px;\" \/>.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835324646\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Translate: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca8086a00e2881052cb89e1abb780fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#60;&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"75\" style=\"vertical-align: 0px;\" \/>.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835330450\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835361577\">\n<h3 data-type=\"title\">Verify Solutions to an Inequality in Two Variables<\/h3>\n<p id=\"fs-id1167832031143\">Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at <strong data-effect=\"bold\">linear inequalities<\/strong> in two variables which are very similar to linear equations in two variables.<\/p>\n<p id=\"fs-id1167835236709\">Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs\u2014so that your business made a profit.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834189522\">\n<div data-type=\"title\">Linear Inequality<\/div>\n<p id=\"fs-id1167834156714\">A <span data-type=\"term\">linear inequality<\/span> is an inequality that can be written in one of the following forms:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834062043\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5b175abd4884de81c77ca6892e259cc_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;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#65;&#120;&#43;&#66;&#121;&#62;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#65;&#120;&#43;&#66;&#121;&#92;&#103;&#101;&#32;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#65;&#120;&#43;&#66;&#121;&#60;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#65;&#120;&#43;&#66;&#121;&#92;&#108;&#101;&#32;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"571\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167835174912\">Where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero.<\/p>\n<\/div>\n<p id=\"fs-id1167835175012\">Recall that an inequality with one variable had many solutions. For example, the solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See <a href=\"#CNX_IntAlg_Figure_03_04_001\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1171791446120\">\n<div data-type=\"newline\"><\/div>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_001\"><span data-type=\"media\" id=\"fs-id1167831920149\" data-alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.\" \/><\/span><\/div>\n<p id=\"fs-id1167835283676\">Similarly, linear inequalities in two variables have many solutions. Any ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that makes an inequality true when we substitute in the values is a <span data-type=\"term\">solution to a linear inequality<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835609689\">\n<div data-type=\"title\">Solution to a Linear Inequality<\/div>\n<p id=\"fs-id1167835233801\">An ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> is a <strong data-effect=\"bold\">solution to a linear inequality<\/strong> if the inequality is true when we substitute the values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835253977\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835174795\">\n<div data-type=\"problem\" id=\"fs-id1167835349099\">\n<p id=\"fs-id1167835322096\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7798b20e6c6b78d36b034745d5b6d95c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd564d67f67eb6682aa330fabe50d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80be894898c77ffda9d4d5843e6791f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0045a8d8e56559d431f457a0e92f7ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85f1c64706ea3d5841145bbd43ef0ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835325302\">\n<p id=\"fs-id1167834062836\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table class=\"unnumbered unstyled\" summary=\"Substitute 0 for x and 0 for y. Is 0 greater than 0 plus 4? Simplify. 0 is not greater than 4 so (0, 0) is not a solution to y is greater than x plus 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835193138\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834179753\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e000faf1f3ff373bd4c89a5102c331a_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;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002c_img.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=\"left\"><span data-type=\"media\" id=\"fs-id1167835210142\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_002d_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 data-valign=\"top\" data-align=\"left\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is not a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835416843\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167831824686\" class=\"unnumbered unstyled\" summary=\"Substitute 1 for x and 6 for y. Is 6 greater than 1 plus 4? Simplify. 6 is greater than 5 so (1, 6) is a solution to y is greater than x plus 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd564d67f67eb6682aa330fabe50d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835281916\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e000faf1f3ff373bd4c89a5102c331a_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;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830700650\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003c_img.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=\"left\"><span data-type=\"media\" id=\"fs-id1167834408398\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_003d_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 data-valign=\"top\" data-align=\"left\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd564d67f67eb6682aa330fabe50d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835379272\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834194741\" class=\"unnumbered unstyled\" summary=\"Substitute 2 for x and 6 for y. Is 6 greater than 2 plus 4? Simplify. 6 is not greater than 6 so (2, 6) is not a solution to y is greater than x plus 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80be894898c77ffda9d4d5843e6791f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834526122\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e000faf1f3ff373bd4c89a5102c331a_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;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831891015\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004c_img.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=\"left\"><span data-type=\"media\" id=\"fs-id1167835524349\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_004d_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 data-valign=\"top\" data-align=\"left\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80be894898c77ffda9d4d5843e6791f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is not a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835287867\"><span class=\"token\">\u24d3<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table class=\"unnumbered unstyled\" summary=\"Substitute negative 5 for x and negative 15 for y. Is negative 15 greater than negative 5 plus 4? Simplify. Negative 15 is not greater than negative 1 so (negative 5, negative 15) is not a solution to y is greater than x plus 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0045a8d8e56559d431f457a0e92f7ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832053670\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831836128\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831887916\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005c_img.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=\"left\"><span data-type=\"media\" id=\"fs-id1167830866197\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_005d_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 data-valign=\"top\" data-align=\"left\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0045a8d8e56559d431f457a0e92f7ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> is not a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834367192\"><span class=\"token\">\u24d4<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835359748\" class=\"unnumbered unstyled\" summary=\"Substitute negative 8 for x and 12 for y. Is 12 greater than negative 8 plus 4? Simplify. 12 is greater than negative 4 so (negative 8, 12) is a solution to y is greater than x plus 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85f1c64706ea3d5841145bbd43ef0ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835174228\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233360\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d3dbc4c42841c8465c416ec545ec4c6_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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006c_img.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=\"left\"><span data-type=\"media\" id=\"fs-id1167835352373\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_006d_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 data-valign=\"top\" data-align=\"left\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85f1c64706ea3d5841145bbd43ef0ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> is a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835197537\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832153132\">\n<div data-type=\"problem\" id=\"fs-id1167834537361\">\n<p id=\"fs-id1167835317682\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02bf33495873518ce8011517e13fb983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#45;&#51;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167831919494\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6971fcc103f3082fe9f67f649219d6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb242707b07d2123762ae0b5253ad3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b932df7404bba7c4812c40fc6d0d4b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835360880\">\n<p id=\"fs-id1167835332128\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> yes <span class=\"token\">\u24d3<\/span> yes <span class=\"token\">\u24d4<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835376317\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835234515\">\n<div data-type=\"problem\" id=\"fs-id1167835346097\">\n<p id=\"fs-id1167835400395\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ef58d8081aa86cbb65f6396cef37f3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#120;&#43;&#49;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835192223\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cc39b059b690d1072109ca1af8bc331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef9b0ead245d0b508218b1cb59d48442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aecc845ff8fcecb422091e6436adca8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8235f4c83db17d3a3454713a44752b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835310230\">\n<p id=\"fs-id1167832043379\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> no<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span> yes<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835367015\">\n<h3 data-type=\"title\">Recognize the Relation Between the Solutions of an Inequality and its Graph<\/h3>\n<p>Now, we will look at how the solutions of an inequality relate to its graph.<\/p>\n<p id=\"fs-id1167835497551\">Let\u2019s think about the number line in shown previously again. The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See <a href=\"#CNX_IntAlg_Figure_03_04_007\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_007\">\n<div class=\"bc-figcaption figcaption\">The solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b36e158b8f20164d896b9a7d17fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> is the shaded part of the number line to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf60bc9fcf312a246a055c15ee98033c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835534329\" data-alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled \u201cnumbers greater than 3\u201d. The part of the number line to the left of 3 is labeled \u201cnumbers less than 3\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled \u201cnumbers greater than 3\u201d. The part of the number line to the left of 3 is labeled \u201cnumbers less than 3\u201d.\" \/><\/span><\/div>\n<p id=\"fs-id1167834063445\">Similarly, the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bf5d7a264c371a4ffdc73e723eb682f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/> separates the plane into two regions. On one side of the line are points with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcb36535919d40806e4973563530add9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> On the other side of the line are the points with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> We call the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bf5d7a264c371a4ffdc73e723eb682f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/> a <span data-type=\"term\">boundary line<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835230438\">\n<div data-type=\"title\">Boundary Line<\/div>\n<p id=\"fs-id1167830962142\">The line with equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0520a31036e9a951aea74693c8b23cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> is the <strong data-effect=\"bold\">boundary line<\/strong> that separates the region where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23b6e4813cdddb52957346c2cd079311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#62;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> from the region where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-760db367ad2bb780c3331d9cefc0ff43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#60;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<p>For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not <em data-effect=\"italics\">a<\/em> is included in the solution:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835416462\" data-alt=\"Two number lines are shown with the middle labeled with the number \u201ca\u201d. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled \u201cx is less than a\u201d and the number a is marked with an open parenthesis. The second number line is labeled \u201cx is less than or equal to a\u201d and the number a is marked with an open bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two number lines are shown with the middle labeled with the number \u201ca\u201d. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled \u201cx is less than a\u201d and the number a is marked with an open parenthesis. The second number line is labeled \u201cx is less than or equal to a\u201d and the number a is marked with an open bracket.\" \/><\/span><\/p>\n<p id=\"fs-id1167835325265\">Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834222348\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45886c6f60d2f9d68fd98729749ac29a_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;&#99;&#99;&#125;&#65;&#120;&#43;&#66;&#121;&#60;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#65;&#120;&#43;&#66;&#121;&#92;&#108;&#101;&#32;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#65;&#120;&#43;&#66;&#121;&#62;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#65;&#120;&#43;&#66;&#121;&#92;&#103;&#101;&#32;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#117;&#110;&#100;&#97;&#114;&#121;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#117;&#110;&#100;&#97;&#114;&#121;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#117;&#110;&#100;&#97;&#114;&#121;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#32;&#110;&#111;&#116;&#32;&#105;&#110;&#99;&#108;&#117;&#100;&#101;&#100;&#32;&#105;&#110;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#117;&#110;&#100;&#97;&#114;&#121;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#32;&#105;&#110;&#99;&#108;&#117;&#100;&#101;&#100;&#32;&#105;&#110;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#117;&#110;&#100;&#97;&#114;&#121;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#32;&#100;&#97;&#115;&#104;&#101;&#100;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#117;&#110;&#100;&#97;&#114;&#121;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#32;&#115;&#111;&#108;&#105;&#100;&#46;&#125;&#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=\"103\" width=\"695\" style=\"vertical-align: -47px;\" \/><\/div>\n<p id=\"fs-id1167835319837\">Now, let\u2019s take a look at what we found in <a href=\"#fs-id1167835253977\" class=\"autogenerated-content\">(Figure)<\/a>. We\u2019ll start by graphing the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd156e62a0f7603729e741c1e4e96c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> and then we\u2019ll plot the five points we tested, as shown in the graph. See <a href=\"#CNX_IntAlg_Figure_03_04_009\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_009\"><span data-type=\"media\" data-alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).\" \/><\/span><\/div>\n<p id=\"fs-id1167834346594\">In <a href=\"#fs-id1167835253977\" class=\"autogenerated-content\">(Figure)<\/a> we found that some of the points were solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02690019f51a80ed91235ac29667481d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/> and some were not.<\/p>\n<p id=\"fs-id1167832096778\">Which of the points we plotted are solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b8e9e197015fe13f63e48211862c4fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834422752\">The points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd564d67f67eb6682aa330fabe50d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85f1c64706ea3d5841145bbd43ef0ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> are solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> Notice that they are both on the same side of the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe4821f581d60afbcaa436166a1fbeca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167831115420\">The two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0045a8d8e56559d431f457a0e92f7ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> are on the other side of the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd156e62a0f7603729e741c1e4e96c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> and they are not solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> For those two points, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcb36535919d40806e4973563530add9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>What about the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb3d4ef9672e6a37f0150e52719998df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/> Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b292a819dbca52971d1dba29473f4ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#61;&#50;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -4px;\" \/> the point is a solution to the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd156e62a0f7603729e741c1e4e96c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> but not a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> So the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80be894898c77ffda9d4d5843e6791f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is on the boundary line.<\/p>\n<p id=\"fs-id1167835479170\">Let\u2019s take another point above the boundary line and test whether or not it is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f16714015fdd77f25981f5ca60f79d0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> clearly looks to above the boundary line, doesn\u2019t it? Is it a solution to the inequality?<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835421108\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6f05a1dee265b5899fd8373f26a0bbb_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#62;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#38;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"104\" style=\"vertical-align: -27px;\" \/><\/div>\n<p id=\"fs-id1167826996089\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f16714015fdd77f25981f5ca60f79d0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> is a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835338082\">Any point you choose above the boundary line is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> All points above the boundary line are solutions.<\/p>\n<p>Similarly, all points below the boundary line, the side with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bae150af0c4f856efb3793b839d2a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/> are not solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f636ab9f12ad6303f1d078c26cfdb47f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> as shown in <a href=\"#CNX_IntAlg_Figure_03_04_010\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_04_010\"><span data-type=\"media\" data-alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.\" \/><\/span><\/div>\n<p id=\"fs-id1167835318758\">The graph of the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02690019f51a80ed91235ac29667481d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/> is shown in below.<\/p>\n<p id=\"fs-id1167831825742\">The line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bf5d7a264c371a4ffdc73e723eb682f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/> divides the plane into two regions. The shaded side shows the solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd0c0f61d8d5cf5120ddcd024dab826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835180482\">The points on the boundary line, those where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd156e62a0f7603729e741c1e4e96c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> are not solutions to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f636ab9f12ad6303f1d078c26cfdb47f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> so the line itself is not part of the solution. We show that by making the line dashed, not solid.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835511460\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1167834133970\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835370458\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835326585\">The boundary line shown in this graph is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d1527e9ed0aa738eeb988b8896309b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/> Write the inequality shown by the graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834222478\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167828327084\">\n<p id=\"fs-id1167835264115\">The line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b298fa54b924ab7c26b7e36b4e6d9c89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/> is the boundary line. On one side of the line are the points with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad70505f3bb17978eb2a99bcadba3de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/> and on the other side of the line are the points with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36d51fc1964f7be4ecb451e4ce64a3a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#50;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832059873\">Let\u2019s test the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and see which inequality describes its position relative to the boundary line.<\/p>\n<p id=\"fs-id1167831891913\">At <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> which inequality is true: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad70505f3bb17978eb2a99bcadba3de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55bfa2d41e6c0cdacfa85121244c012d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#50;&#120;&#45;&#49;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167831921823\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cbf5520d62b37d216e1d7b669f4e5e2a_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;&#99;&#99;&#125;&#121;&#62;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#121;&#60;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#48;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#62;&#125;&#50;&middot;&#48;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#48;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#60;&#125;&#50;&middot;&#48;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#48;&#62;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#117;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#48;&#60;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#108;&#115;&#101;&#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=\"65\" width=\"274\" style=\"vertical-align: -27px;\" \/><\/div>\n<p id=\"fs-id1167835232888\">Since, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad70505f3bb17978eb2a99bcadba3de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/> is true, the side of the line with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> is the solution. The shaded region shows the solution of the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5e53b102f2370645b348d1012e08fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#50;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835366721\">Since the boundary line is graphed with a solid line, the inequality includes the equal sign.<\/p>\n<p id=\"fs-id1167835333508\">The graph shows the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b42bb3e3efc0fe5b1574bfc4d2f9656_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#50;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834228494\">We could use any point as a test point, provided it is not on the line. Why did we choose <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-491bbdafc904b7d497da4c54912a0821_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/> Because it\u2019s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36d51fc1964f7be4ecb451e4ce64a3a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#50;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826828652\">\n<div data-type=\"problem\" id=\"fs-id1167832054225\">\n<p id=\"fs-id1167834195241\">Write the inequality shown by the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb33190b41ec6c679a06ec6c4dfa63cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835339070\">\n<p id=\"fs-id1167834184145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6245efebf4a843ebeac270e7ef0f11cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#45;&#50;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835352945\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835338956\">\n<div data-type=\"problem\" id=\"fs-id1167835303775\">\n<p id=\"fs-id1167834183575\">Write the inequality shown by the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77a40ba816c34e5fb82e0825a41d458c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167830865778\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377281\">\n<p id=\"fs-id1167830757629\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6140cee7a8f0c40e7f15b2002f9a404e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167832054721\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167827987951\">\n<div data-type=\"problem\" id=\"fs-id1167835369284\">\n<p id=\"fs-id1167834066318\">The boundary line shown in this graph is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a884936fcb4560e1b44b682f05fe984e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/> Write the inequality shown by the graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834532434\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505860\">\n<p id=\"fs-id1167834306733\">The line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07795f3421886c79416fa26cd22cc5f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/> is the boundary line. On one side of the line are the points with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1868dee734be1e5445cdddae5a48f9a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/> and on the other side of the line are the points with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a168cc334a330a7a6c71c25ce7b57a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#60;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835350607\">Let\u2019s test the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and see which inequality describes its side of the boundary line.<\/p>\n<p id=\"fs-id1167834132988\">At <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> which inequality is true: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1868dee734be1e5445cdddae5a48f9a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08db206b840184567a42bc5f040fd379_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#60;&#54;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167831911258\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0074f69aeb04d283124ea1bdcae93e87_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;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#43;&#51;&#121;&#38;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#62;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#108;&#115;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#43;&#51;&#121;&#38;&#32;&#60;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#60;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#60;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#117;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"471\" style=\"vertical-align: -27px;\" \/><\/div>\n<p id=\"fs-id1167835323826\">So the side with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is the side where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a168cc334a330a7a6c71c25ce7b57a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#60;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835361279\">(You may want to pick a point on the other side of the boundary line and check that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fdd0ae0d73117517a66a688baaba4fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#62;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/>)<\/p>\n<p id=\"fs-id1167835355907\">Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.<\/p>\n<p id=\"fs-id1167835281489\">The shaded region shows the solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a168cc334a330a7a6c71c25ce7b57a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#60;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834133165\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832042396\">\n<div data-type=\"problem\" id=\"fs-id1167826967255\">\n<p id=\"fs-id1167832058079\">Write the inequality shown by the shaded region in the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a0f099ef716d94ce09b3e4a750872da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;&#121;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835384392\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167830836855\">\n<p id=\"fs-id1167834516596\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dad93b217bc64fcd74437e216e9167b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;&#121;&#92;&#108;&#101;&#32;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835189046\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835375292\">\n<div data-type=\"problem\" id=\"fs-id1167834516154\">\n<p id=\"fs-id1167835268997\">Write the inequality shown by the shaded region in the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8ebef9b6cf79b30522cd6c4a29421a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#121;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831896349\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835369697\">\n<p id=\"fs-id1167834195300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f14d02aabf2dd9373599b84abe706613_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#121;&#92;&#103;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834431130\">\n<h3 data-type=\"title\">Graph Linear Inequalities in Two Variables<\/h3>\n<p id=\"fs-id1167835173654\">Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834059141\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph a Linear Equation in Two Variables<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834186008\">\n<div data-type=\"problem\" id=\"fs-id1167834473385\">\n<p id=\"fs-id1167835334366\">Graph the linear inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82fd172bba30016fe918e5f5241cb556_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835327818\"><span data-type=\"media\" id=\"fs-id1167832066478\" data-alt=\"Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_018a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834432374\" data-alt=\"Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_018b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835180882\" data-alt=\"Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_018c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832075476\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832087004\">\n<div data-type=\"problem\" id=\"fs-id1167835330003\">\n<p id=\"fs-id1167835519946\">Graph the linear inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a16c5fce29cef0f70461a018205f800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834463224\">\n<p id=\"fs-id1167831031013\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831031015\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (2, 1), and (4, 6). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (2, 1), and (4, 6). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region and on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a16c5fce29cef0f70461a018205f800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831239812\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830838406\">\n<div data-type=\"problem\" id=\"fs-id1167830838408\">\n<p id=\"fs-id1167835419768\">Graph the linear inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af753db008eaa87eebdeac4d74f90fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835237898\">\n<p id=\"fs-id1167835237901\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835339047\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 5), (3, negative 3), and (5, negative 1). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 5), (3, negative 3), and (5, negative 1). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af753db008eaa87eebdeac4d74f90fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834394395\">The steps we take to graph a linear inequality are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835367051\" class=\"howto\">\n<div data-type=\"title\">Graph a linear inequality in two variables.<\/div>\n<ol id=\"fs-id1167826996926\" type=\"1\" class=\"stepwise\">\n<li>Identify and graph the boundary line.\n<ul id=\"fs-id1167834547281\" data-bullet-style=\"bullet\">\n<li>If the inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36610b8031649083595c5ff00a10d1e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#103;&#101;&#32;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/> the boundary line is solid.<\/li>\n<li>If the inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05416d28213eb63c221d1b4cd7499bfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#62;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -4px;\" \/> the boundary line is dashed.<\/li>\n<\/ul>\n<\/li>\n<li>Test a point that is not on the boundary line. Is it a solution of the inequality?<\/li>\n<li>Shade in one side of the boundary line.\n<ul id=\"fs-id1167831823898\" data-bullet-style=\"bullet\">\n<li>If the test point is a solution, shade in the side that includes the point.<\/li>\n<li>If the test point is not a solution, shade in the opposite side.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835380178\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835380180\">\n<div data-type=\"problem\" id=\"fs-id1167835365170\">\n<p id=\"fs-id1167834129902\">Graph the linear inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ae62fb930f14596e32763eac6dc6a9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377740\">\n<p id=\"fs-id1167835355962\">First, we graph the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c17e48205536e3ab74b1b620c18cc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> The inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5fae947beab7b30f45d5b6603772f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/> so we draw a dashed line.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167830701265\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).\" \/><\/span><\/p>\n<p>Then, we test a point. We\u2019ll use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> again because it is easy to evaluate and it is not on the boundary line.<\/p>\n<p id=\"fs-id1167828420183\">Is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> a solution of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9254de4f0f875344e8601e8ecb08534_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835420922\" data-alt=\"Is 0 minus 2 times 0 less than 5? Is 0 minus 0 less than 5? 0 is less than 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Is 0 minus 2 times 0 less than 5? Is 0 minus 0 less than 5? 0 is less than 5.\" \/><\/span><\/p>\n<p id=\"fs-id1167835511602\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is a solution of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9ffca6abb59390931d9843fc39a1d4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> so we shade in that side of the boundary line.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835253751\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<p id=\"fs-id1167831911233\">All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ae62fb930f14596e32763eac6dc6a9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835163492\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831891076\">\n<div data-type=\"problem\" id=\"fs-id1167835253904\">\n<p id=\"fs-id1167835368944\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de79bc812e414b32a7398d005d733f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#51;&#121;&#60;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834138029\">\n<p id=\"fs-id1167834464384\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834464386\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 2), (3, 0), and (6, 2). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 2), (3, 0), and (6, 2). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de79bc812e414b32a7398d005d733f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#51;&#121;&#60;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835410949\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835264380\">\n<div data-type=\"problem\" id=\"fs-id1167831883737\">\n<p id=\"fs-id1167831880097\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5b867ea93f409606b49b9d75f7ca617_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#121;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835346062\">\n<p id=\"fs-id1167835514065\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832119046\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 1), and (2, 1). The line divides the x y-coordinate plane into two halves. The bottom right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 1), and (2, 1). The line divides the x y-coordinate plane into two halves. The bottom right half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5b867ea93f409606b49b9d75f7ca617_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#121;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835331510\">What if the boundary line goes through the origin? Then, we won\u2019t be able to use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> as a test point. No problem\u2014we\u2019ll just choose some other point that is not on the boundary line.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835229236\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834469610\">\n<div data-type=\"problem\" id=\"fs-id1167834469612\">\n<p id=\"fs-id1167834191414\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f006b71eed97d369515cc0906468454_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#45;&#52;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835280571\">\n<p id=\"fs-id1167835280573\">First, we graph the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72cff3e921e5c228ac1f0c9870e78e8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/> It is in slope\u2013intercept form, with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-754283866fe7f24eadf016df34fe5cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"62\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66a46c671318312a1bdf2d5cc09c54de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> The inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e2b165150917469474a6d203f27e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/> so we draw a solid line.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835376108\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).\" \/><\/span><\/p>\n<p id=\"fs-id1167835230300\">Now we need a test point. We can see that the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is not on the boundary line.<\/p>\n<p id=\"fs-id1167835374272\">Is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> a solution of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c028217dc551c9f7e49b14f27a5259b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#45;&#52;&#120;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832041694\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<p id=\"fs-id1167835395578\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is not a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9625772f6e89a24ae69aea65d80553f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#45;&#52;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> so we shade in the opposite side of the boundary line.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831985833\" data-alt=\"Is 0 less than or equal to negative 4 times 1? 0 is not less than or equal to negative 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Is 0 less than or equal to negative 4 times 1? 0 is not less than or equal to negative 4.\" \/><\/span><\/p>\n<p id=\"fs-id1167835244104\">All points in the shaded region and on the boundary line represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f006b71eed97d369515cc0906468454_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#45;&#52;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835367619\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835171005\">\n<div data-type=\"problem\" id=\"fs-id1167835171007\">\n<p id=\"fs-id1167832052015\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-879e9d5a479276a27c5d5e2357fe9b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#45;&#51;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834114308\">\n<p id=\"fs-id1167835640409\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835640412\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 3), (0, 0), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 3), (0, 0), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-879e9d5a479276a27c5d5e2357fe9b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#45;&#51;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834085021\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835241372\">\n<div data-type=\"problem\" id=\"fs-id1167835346336\">\n<p id=\"fs-id1167835346339\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4e97f1b9cf72cfd4d05135fd2d6564d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#45;&#50;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831025381\">\n<p id=\"fs-id1167835191448\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835191450\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 2), (0, 0), and (1, negative 2). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 2), (0, 0), and (1, negative 2). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region and on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4e97f1b9cf72cfd4d05135fd2d6564d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#45;&#50;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835210529\">Some linear inequalities have only one variable. They may have an <em data-effect=\"italics\">x<\/em> but no <em data-effect=\"italics\">y<\/em>, or a <em data-effect=\"italics\">y<\/em> but no <em data-effect=\"italics\">x<\/em>. In these cases, the boundary line will be either a vertical or a horizontal line.<\/p>\n<p id=\"fs-id1167835349500\">Recall that:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167831106740\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f990a0b0ce7c941d4e87e6dd4caff4b3_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;&#99;&#99;&#125;&#120;&#61;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#32;&#108;&#105;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#114;&#105;&#122;&#111;&#110;&#116;&#97;&#108;&#32;&#108;&#105;&#110;&#101;&#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=\"39\" width=\"235\" style=\"vertical-align: -15px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1167835200429\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835200431\">\n<div data-type=\"problem\" id=\"fs-id1167835346154\">\n<p id=\"fs-id1167835346156\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccfd953c0afa3f346842f451ac6c47fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834387736\">\n<p id=\"fs-id1167834387738\">First, we graph the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0eb02c9be9d3f22890308095c7a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> It is a horizontal line. The inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/> so we draw a dashed line.<\/p>\n<p id=\"fs-id1167830704322\">We test the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167834195455\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13198ce5ddf90f6a7a89e0f2e92cb3b0_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;&#125;&#92;&#92;&#32;&#121;&#62;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#48;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#62;&#125;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"43\" style=\"vertical-align: -27px;\" \/><\/div>\n<p id=\"fs-id1167831922006\">So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is not a solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccfd953c0afa3f346842f451ac6c47fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835321005\">So we shade the side that does not include <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> as shown in this graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832128683\" data-alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<p id=\"fs-id1167834193406\">All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccfd953c0afa3f346842f451ac6c47fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834473439\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835366245\">\n<div data-type=\"problem\" id=\"fs-id1167835366247\">\n<p id=\"fs-id1167832055097\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c13ce100cc4f6b884b962393c0c1556_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835563964\">\n<p id=\"fs-id1167835514390\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835514392\" data-alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal dashed line is drawn through the points (negative 1, 5), (0, 5), and (1, 5). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal dashed line is drawn through the points (negative 1, 5), (0, 5), and (1, 5). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region, but not those on the boundary line, represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c13ce100cc4f6b884b962393c0c1556_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826781485\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832058169\">\n<div data-type=\"problem\" id=\"fs-id1167835334494\">\n<p id=\"fs-id1167835334496\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21de94dc2eec758056217587591544f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831919784\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831919786\" data-alt=\"This figure has the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal line is drawn through the points (negative 1, negative 1), (0, negative 1), and (1, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal line is drawn through the points (negative 1, negative 1), (0, negative 1), and (1, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> All points in the shaded region and on the boundary line represent the solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21de94dc2eec758056217587591544f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835524373\">\n<h3 data-type=\"title\">Solve Applications using Linear Inequalities in Two Variables<\/h3>\n<p id=\"fs-id1167830706031\">Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how thay might be used.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835363723\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835254481\">\n<div data-type=\"problem\" id=\"fs-id1167835254483\">\n<p id=\"fs-id1167832067984\">Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least ?240 a week. Her job in food service pays ?10 an hour and her tutoring job on campus pays ?15 an hour. How many hours does Hilaria need to work at each job to earn at least ?240?<\/p>\n<p id=\"fs-id1167835327602\"><span class=\"token\">\u24d0<\/span> Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> be the number of hours she works at the job in food service and let <em data-effect=\"italics\">y<\/em> be the number of hours she works tutoring. Write an inequality that would model this situation.<\/p>\n<p id=\"fs-id1167835325479\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167834433649\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that would be solutions to the inequality. Then, explain what that means for Hilaria.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370005\">\n<p id=\"fs-id1167835370007\"><span class=\"token\">\u24d0<\/span> We let <em data-effect=\"italics\">x<\/em> be the number of hours she works at the job in food service and let <em data-effect=\"italics\">y<\/em> be the number of hours she works tutoring.<\/p>\n<p id=\"fs-id1167831872422\">She earns ?10 per hour at the job in food service and ?15 an hour tutoring. At each job, the number of hours multiplied by the hourly wage will gives the amount earned at that job.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834219631\" data-alt=\"10 x plus 15 y is greater than 240. The \u201c10 x\u201d is labeled \u201cAmount earned at the food service job\u201d. The \u201c15 y\u201d is labeled \u201cthe amount earned tutoring\u201d. The \u201cis greater than 240\u201d is labeled \u201cis at least 240\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"10 x plus 15 y is greater than 240. The \u201c10 x\u201d is labeled \u201cAmount earned at the food service job\u201d. The \u201c15 y\u201d is labeled \u201cthe amount earned tutoring\u201d. The \u201cis greater than 240\u201d is labeled \u201cis at least 240\u201d.\" \/><\/span><\/p>\n<p id=\"fs-id1167834517584\"><span class=\"token\">\u24d1<\/span> To graph the inequality, we put it in slope\u2013intercept form.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834339954\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-950899a85570fc3c09259488d81d364c_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#120;&#43;&#49;&#53;&#121;&#38;&#32;&#92;&#103;&#101;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#53;&#121;&#38;&#32;&#92;&#103;&#101;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#48;&#120;&#43;&#50;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#92;&#103;&#101;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"212\" style=\"vertical-align: -28px;\" \/><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831896699\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_027_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<p id=\"fs-id1167835306260\"><span class=\"token\">\u24d2<\/span> From the graph, we see that the ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bcc225e225e607dd912dcd62dc8620e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" style=\"vertical-align: -4px;\" \/> represent three of infinitely many solutions. Check the values in the inequality.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835299865\" data-alt=\"First we test the point (15, 10) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 15 plus 15 times 10 greater than or equal to 240? Since 300 is greater than or equal to 240 (15, 10) is a solution. Next we test the point (0, 16) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 0 plus 15 times 16 greater than or equal to 240? Since 240 is greater than or equal to 240 (0, 16) is a solution. Then we test the point (24, 0) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 24 plus 15 times 0 greater than or equal to 240? Since 240 is greater than or equal to 240 (24, 0) is a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_028_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"First we test the point (15, 10) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 15 plus 15 times 10 greater than or equal to 240? Since 300 is greater than or equal to 240 (15, 10) is a solution. Next we test the point (0, 16) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 0 plus 15 times 16 greater than or equal to 240? Since 240 is greater than or equal to 240 (0, 16) is a solution. Then we test the point (24, 0) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 24 plus 15 times 0 greater than or equal to 240? Since 240 is greater than or equal to 240 (24, 0) is a solution.\" \/><\/span><\/p>\n<p id=\"fs-id1167826987862\">For Hilaria, it means that to earn at least ?240, she can work 15 hours tutoring and 10 hours at her fast-food job, earn all her money tutoring for 16 hours, or earn all her money while working 24 hours at the job in food service.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832015628\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834284498\">\n<div data-type=\"problem\" id=\"fs-id1167835348793\">\n<p id=\"fs-id1167835348795\">Hugh works two part time jobs. One at a grocery store that pays ?10 an hour and the other is babysitting for ?13 hour. Between the two jobs, Hugh wants to earn at least ?260 a week. How many hours does Hugh need to work at each job to earn at least ?260?<\/p>\n<p id=\"fs-id1167834063929\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours he works at the grocery store and let <em data-effect=\"italics\">y<\/em> be the number of hours he works babysitting. Write an inequality that would model this situation.<\/p>\n<p id=\"fs-id1167835418163\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167826783577\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>) that would be solutions to the inequality. Then, explain what that means for Hugh.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835368581\">\n<p id=\"fs-id1167835368583\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6da3ae9d2e8a008097127e18a5712214_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#43;&#49;&#51;&#121;&#92;&#103;&#101;&#32;&#50;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835307410\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 30. A line is drawn through the points (0, 20), (13, 10), and (26, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 30. A line is drawn through the points (0, 20), (13, 10), and (26, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835167616\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835167619\">\n<div data-type=\"problem\" id=\"fs-id1167835194595\">\n<p id=\"fs-id1167835194597\">Veronica works two part time jobs in order to earn enough money to meet her obligations of at least ?280 a week. Her job at the day spa pays ?10 an hour and her administrative assistant job on campus pays ?17.50 an hour. How many hours does Veronica need to work at each job to earn at least ?280?<\/p>\n<p id=\"fs-id1167835189599\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours she works at the day spa and let <em data-effect=\"italics\">y<\/em> be the number of hours she works as administrative assistant. Write an inequality that would model this situation.<\/p>\n<p><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167834431575\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>) that would be solutions to the inequality. Then, explain what that means for Veronica<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834309537\">\n<p id=\"fs-id1167835513671\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d07e5c279711c02d774040a2be6bc816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#43;&#49;&#55;&#46;&#53;&#121;&#92;&#103;&#101;&#32;&#50;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835325137\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16) and (28, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16) and (28, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831891876\" class=\"media-2\">\n<p id=\"fs-id1167830865844\">Access this online resource for additional instruction and practice with graphing linear inequalities in two variables.<\/p>\n<ul id=\"fs-id1167831956377\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37linineqgraphs\">Graphing Linear Inequalities in Two Variables<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167827956786\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835366934\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to graph a linear inequality in two variables.<\/strong>\n<ol id=\"fs-id1167835262448\" type=\"1\" class=\"stepwise\">\n<li>Identify and graph the boundary line.\n<div data-type=\"newline\"><\/div>\n<p> If the inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36610b8031649083595c5ff00a10d1e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#103;&#101;&#32;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/> the boundary line is solid.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If the inequality is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8e11dfbcdfd6b6bb363a9a8adcdeb3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#62;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -4px;\" \/> the boundary line is dashed.<\/li>\n<li>Test a point that is not on the boundary line. Is it a solution of the inequality?<\/li>\n<li>Shade in one side of the boundary line.\n<div data-type=\"newline\"><\/div>\n<p> If the test point is a solution, shade in the side that includes the point.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If the test point is not a solution, shade in the opposite side.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834431207\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834431210\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167832060209\"><strong data-effect=\"bold\">Verify Solutions to an Inequality in Two Variables<\/strong><\/p>\n<p id=\"fs-id1167832054111\">In the following exercises, determine whether each ordered pair is a solution to the given inequality.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835376934\">\n<div data-type=\"problem\" id=\"fs-id1167835376937\">\n<p id=\"fs-id1167832044125\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-170d5ba6f4da3ecff82f71f14f22d268_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#45;&#49;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835350000\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99d11505e9b59f8e2d3351529e3354c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ef6c187a0599fedf1caa75800c24233_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb6560ec04491f4e7cf02e14e6df5ec3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28a186f2424d2e935d6aa6388441b6d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6b25c6279fe53efe7c78796a156eeed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835231057\">\n<p id=\"fs-id1167835171272\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> no <span class=\"token\">\u24d4<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167832053265\">\n<p id=\"fs-id1167834161636\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02bf33495873518ce8011517e13fb983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#120;&#45;&#51;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835609385\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d66960a9b4adfa7702388a061d743cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a7b90c027eccd1d5f6de21196682699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831086840\">\n<div data-type=\"problem\" id=\"fs-id1167831086843\">\n<p id=\"fs-id1167832042712\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d5cb6cb44fa37dd9c0adce715a1165c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#51;&#120;&#43;&#50;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834505590\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef24d382319a3ce81f280194edba003a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a17ffffadfb8456567f4803e943d15a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b098ad691d3b1e66796376e6000e2385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835263066\">\n<p id=\"fs-id1167835263068\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> yes <span class=\"token\">\u24d3<\/span> yes <span class=\"token\">\u24d4<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831023966\">\n<div data-type=\"problem\" id=\"fs-id1167835336368\">\n<p id=\"fs-id1167835336371\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fe4fa1d105b022797998773d21cf265_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#50;&#120;&#43;&#53;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ba9cc2a7f12e65a6b3de8f34bcc16e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd564d67f67eb6682aa330fabe50d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72256ab78cf21de411fa10ad0cdbeb04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99d11505e9b59f8e2d3351529e3354c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12aeb9559a0d3673fa3759c68e52b96b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997319\">\n<div data-type=\"problem\" id=\"fs-id1167835201298\">\n<p id=\"fs-id1167835201301\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f776df3b2c4d1dbedd95589134cc37e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#52;&#121;&#62;&#52;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835419834\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b932df7404bba7c4812c40fc6d0d4b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b61914c33c0ffe75331208eaa502d627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3b8ee106f7e5a95f1dd3e0ab5f16435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-064fbe5b13d7774ef232dcf1aebc3fa5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167832151136\">\n<p id=\"fs-id1167832151138\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> no <span class=\"token\">\u24d3<\/span> no <span class=\"token\">\u24d4<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834327594\">\n<p id=\"fs-id1167834327596\">Determine whether each ordered pair is a solution to the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c19cadacf38b1d71488a37bcdc4b4d44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;&#121;&#62;&#50;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834222006\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-745bd1150b5c3e65ae8bad5282a5b3b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a73c1812707ce5b0b7341d06a667a00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85f1c64706ea3d5841145bbd43ef0ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28a186f2424d2e935d6aa6388441b6d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834058748\"><strong data-effect=\"bold\">Recognize the Relation Between the Solutions of an Inequality and its Graph<\/strong><\/p>\n<p id=\"fs-id1167831086587\">In the following exercises, write the inequality shown by the shaded region.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831086590\">\n<div data-type=\"problem\" id=\"fs-id1167831891686\">\n<p id=\"fs-id1167831891688\">Write the inequality shown by the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c2d4d4090a564e1f3b5cb4539eff63f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835329737\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835336434\">\n<p id=\"fs-id1167835239120\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59af3eacf88b5ad33a850108d06f7329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#51;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831909967\">\n<div data-type=\"problem\" id=\"fs-id1167831909970\">\n<p id=\"fs-id1167835621626\">Write the inequality shown by the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-282d7952f601d30c7bedfe3b6c0f11c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835357587\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834058839\">\n<div data-type=\"problem\" id=\"fs-id1167834058841\">\n<p id=\"fs-id1167835331951\">Write the inequality shown by the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36a6283f44c69f02baadc3ece2d8b0ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831852242\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 1), (2, 0), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 1), (2, 0), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834192173\">\n<p id=\"fs-id1167834192176\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29414f49d3779d8884eed82a06389503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831197142\">\n<div data-type=\"problem\" id=\"fs-id1167831197144\">\n<p id=\"fs-id1167835369782\">Write the inequality shown by the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e57ce76d8028248de84215c0017483d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835216954\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (3, negative 3), and (6, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (3, negative 3), and (6, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834308117\">\n<div data-type=\"problem\" id=\"fs-id1167834308119\">\n<p id=\"fs-id1167831228827\">Write the inequality shown by the shaded region in the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-276342e5d1b5695a41280d0b063a09d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832053362\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 5), (1, 4), and (5, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 5), (1, 4), and (5, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835610067\">\n<p id=\"fs-id1167835610070\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb6daa81ceff7eaef4487b4f86484958_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835528518\">\n<div data-type=\"problem\" id=\"fs-id1167835217818\">\n<p id=\"fs-id1167835217820\">Write the inequality shown by the shaded region in the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c147e7fa15bc1c3c57fa3d9cdbd6dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835615661\">\n<div data-type=\"problem\" id=\"fs-id1167835422446\">\n<p id=\"fs-id1167835422449\">Write the inequality shown by the shaded region in the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8ebef9b6cf79b30522cd6c4a29421a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#121;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831913470\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835287946\">\n<p id=\"fs-id1167828240723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f2acf43726ff087ff6e6ba6a8bc5bbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#121;&#92;&#108;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479850\">\n<div data-type=\"problem\" id=\"fs-id1167835300614\">\n<p id=\"fs-id1167835300617\">Write the inequality shown by the shaded region in the graph with the boundary line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31bc73671ff7236e1e0cdaba6c80c310_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#121;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835238193\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167835302992\"><strong data-effect=\"bold\">Graph Linear Inequalities in Two Variables<\/strong><\/p>\n<p id=\"fs-id1167835423443\">In the following exercises, graph each linear inequality.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834395424\">\n<div data-type=\"problem\" id=\"fs-id1167834395427\">\n<p id=\"fs-id1167831923108\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48e376f4e75040d4047a7657cb50a2f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834495534\"><span data-type=\"media\" id=\"fs-id1167834395373\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834308235\">\n<div data-type=\"problem\" id=\"fs-id1167834308238\">\n<p id=\"fs-id1167835350862\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c008a3f8f1ecd1656586e721c0bf30a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835254401\">\n<div data-type=\"problem\" id=\"fs-id1167835254403\">\n<p id=\"fs-id1167831821812\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d189cfe21adfa670c9e7ce9cd5bb129a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831883445\"><span data-type=\"media\" id=\"fs-id1167831883449\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826813777\">\n<div data-type=\"problem\" id=\"fs-id1167826813780\">\n<p id=\"fs-id1167826813782\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4b0e7e840286d9d663e49dbff1bb88b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834345405\">\n<div data-type=\"problem\" id=\"fs-id1167834345408\">\n<p id=\"fs-id1167834345410\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f28f816b38cc809056b0770352aed24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#92;&#108;&#101;&#32;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835339974\"><span data-type=\"media\" id=\"fs-id1167835244701\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831912084\">\n<div data-type=\"problem\" id=\"fs-id1167834131362\">\n<p id=\"fs-id1167834131364\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d38c93265628a0601d2df7d2755907c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#92;&#103;&#101;&#32;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865770\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834179696\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fb563a2e8f4656945a6c07be13de4a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#121;&#62;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183954\"><span data-type=\"media\" id=\"fs-id1167832053177\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835341770\">\n<div data-type=\"problem\" id=\"fs-id1167835336678\">\n<p id=\"fs-id1167835336680\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1d0b3f92b3aac0e4b23f4fd41a49820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#53;&#121;&#60;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830702903\">\n<div data-type=\"problem\" id=\"fs-id1167830702905\">\n<p id=\"fs-id1167834526263\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5cdc606bfd789e48149dc17e5b50272_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#50;&#121;&#92;&#103;&#101;&#32;&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834567460\"><span data-type=\"media\" id=\"fs-id1167835479044\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834061947\">\n<div data-type=\"problem\" id=\"fs-id1167835376646\">\n<p id=\"fs-id1167835376648\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2e5790b15cf4fad6609aa4ba58cd127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#50;&#121;&#92;&#103;&#101;&#32;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834431761\">\n<div data-type=\"problem\" id=\"fs-id1167834431763\">\n<p id=\"fs-id1167834431765\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95df7aa2dbd4880fd80e3589e53ce082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#52;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835369505\"><span data-type=\"media\" id=\"fs-id1167834300213\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835280009\">\n<div data-type=\"problem\" id=\"fs-id1167835280011\">\n<p id=\"fs-id1167831835385\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6e37a26ba6848a897f03dc88395c500_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#45;&#51;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834526549\">\n<div data-type=\"problem\" id=\"fs-id1167834526551\">\n<p id=\"fs-id1167834526553\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f8345116f39e527987033f73be187c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832054972\"><span data-type=\"media\" id=\"fs-id1167835305019\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, 3), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, 3), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835331419\">\n<div data-type=\"problem\" id=\"fs-id1167835331421\">\n<p id=\"fs-id1167835377774\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-137b70b54bad00c92afab57abb11e391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865783\">\n<div data-type=\"problem\" id=\"fs-id1167830865785\">\n<p id=\"fs-id1167830964366\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63e5be1131c6fc0ea194180a810967e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352901\"><span data-type=\"media\" id=\"fs-id1167832150918\" data-alt=\"This figure has the graph of a straight vertical dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A vertical dashed line is drawn through the points (5, negative 1), (5, 0), and (5, 1). The line divides the x y-coordinate plane into two halves. The left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight vertical dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A vertical dashed line is drawn through the points (5, negative 1), (5, 0), and (5, 1). The line divides the x y-coordinate plane into two halves. The left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835349534\">\n<div data-type=\"problem\" id=\"fs-id1167835349536\">\n<p id=\"fs-id1167832052493\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b89da70608ff3e13508228dddf24f6e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835381719\">\n<div data-type=\"problem\" id=\"fs-id1167835381721\">\n<p id=\"fs-id1167834431115\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47648db5b7a9fbfd1f8d1452273a046a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#60;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835180853\"><span data-type=\"media\" id=\"fs-id1167835180858\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835235204\">\n<div data-type=\"problem\" id=\"fs-id1167835235206\">\n<p id=\"fs-id1167835235209\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-650a97a01a600be56a13f9411dc86494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#60;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835234653\">\n<div data-type=\"problem\" id=\"fs-id1167835234655\">\n<p id=\"fs-id1167835234658\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25e91f07f6e6551ab2b82a66a76e7168_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827956861\"><span data-type=\"media\" id=\"fs-id1167835240348\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834064077\">\n<div data-type=\"problem\" id=\"fs-id1167835358355\">\n<p id=\"fs-id1167835358357\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7040274012e648a9302ae28fb229125c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835589641\">\n<div data-type=\"problem\" id=\"fs-id1167835274944\">\n<p id=\"fs-id1167835274946\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dcf7d9057918fd5d0732ea5d7a98135_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#45;&#50;&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834300617\"><span data-type=\"media\" id=\"fs-id1167834300621\" data-alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 1), (1, negative 1), and (2, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 1), (1, negative 1), and (2, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826996648\">\n<div data-type=\"problem\" id=\"fs-id1167826996650\">\n<p id=\"fs-id1167826996653\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5060b2c5164c6a100ca41857e033566f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#51;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835369221\">\n<div data-type=\"problem\" id=\"fs-id1167835369223\">\n<p id=\"fs-id1167835338055\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-384c97999f556b7682ddec5423256dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835166485\"><span data-type=\"media\" id=\"fs-id1167834463977\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_334_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835217909\">\n<div data-type=\"problem\" id=\"fs-id1167835217911\">\n<p id=\"fs-id1167834533381\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3079737e2202a4dd4500b6872cd14159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#121;&#92;&#108;&#101;&#32;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831894741\">\n<div data-type=\"problem\" id=\"fs-id1167831239471\">\n<p id=\"fs-id1167831239473\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cbe3153b9ba88f5a96e76f98bb5f9ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#53;&#121;&#62;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835240479\"><span data-type=\"media\" id=\"fs-id1167835284949\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831922737\">\n<div data-type=\"problem\" id=\"fs-id1167831922739\">\n<p id=\"fs-id1167834214189\">Graph the linear inequality: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31fc8f8714167fcaf23e9b7dcfb4cfee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#51;&#121;&#62;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835284781\"><strong data-effect=\"bold\">Solve Applications using Linear Inequalities in Two Variables<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832015642\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832015644\">\n<p id=\"fs-id1167835235704\">Harrison works two part time jobs. One at a gas station that pays ?11 an hour and the other is IT troubleshooting for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe05b8d1d236a1fc91567b94f4f23d0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#49;&#54;&#46;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -1px;\" \/> an hour. Between the two jobs, Harrison wants to earn at least ?330 a week. How many hours does Harrison need to work at each job to earn at least ?330?<\/p>\n<p id=\"fs-id1167835362766\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours he works at the gas station and let <em data-effect=\"italics\">y<\/em> be the number of (hours he works troubleshooting. Write an inequality that would model this situation.<\/p>\n<p id=\"fs-id1167834131200\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167826997211\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that would be solutions to the inequality. Then, explain what that means for Harrison.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834196585\">\n<p id=\"fs-id1167834196587\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e01ad503b9062874bf7b9053bae1f4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#120;&#43;&#49;&#54;&#46;&#53;&#121;&#92;&#103;&#101;&#32;&#51;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831228835\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831191375\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832015790\">\n<p id=\"fs-id1167832015792\">Elena needs to earn at least ?450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays ?9 an hour and the other as an intern in a genetics lab for ?22.50 per hour. How many hours does Elena need to work at each job to earn at least ?450 per week?<\/p>\n<p id=\"fs-id1167834429285\"><span class=\"token\">\u24d0<\/span> Let <em data-effect=\"italics\">x<\/em> be the number of hours she works teaching swimming and let <em data-effect=\"italics\">y<\/em> be the number of hours she works as an intern. Write an inequality that would model this situation.<\/p>\n<p id=\"fs-id1167835235765\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167834472406\"><span class=\"token\">\u24d2<\/span> Find three ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that would be solutions to the inequality. Then, explain what that means for Elena.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835234207\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835244283\">\n<p id=\"fs-id1167835244285\">The doctor tells Laura she needs to exercise enough to burn 500 calories each day. She prefers to either run or bike and burns 15 calories per minute while running and 10 calories a minute while biking.<\/p>\n<p id=\"fs-id1167831887210\"><span class=\"token\">\u24d0<\/span> If <em data-effect=\"italics\">x<\/em> is the number of minutes that Laura runs and <em data-effect=\"italics\">y<\/em> is the number minutes she bikes, find the inequality that models the situation.<\/p>\n<p id=\"fs-id1167831890641\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167834495426\"><span class=\"token\">\u24d2<\/span> List three solutions to the inequality. What options do the solutions provide Laura?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835263230\">\n<p id=\"fs-id1167834135022\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83b55d053a0705759c23e86d5df05bfb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#120;&#43;&#49;&#48;&#121;&#92;&#103;&#101;&#32;&#53;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834489866\" data-alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 60. A line is drawn through the points (0, 50) and (20, 20). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 60. A line is drawn through the points (0, 50) and (20, 20). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Answers will vary.<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835202221\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835202224\">\n<p id=\"fs-id1167835202226\">Armando\u2019s workouts consist of kickboxing and swimming. While kickboxing, he burns 10 calories per minute and he burns 7 calories a minute while swimming. He wants to burn 600 calories each day.<\/p>\n<p id=\"fs-id1167834345659\"><span class=\"token\">\u24d0<\/span> If <em data-effect=\"italics\">x<\/em> is the number of minutes that Armando will kickbox and <em data-effect=\"italics\">y<\/em> is the number minutes he will swim, find the inequality that will help Armando create a workout for today.<\/p>\n<p id=\"fs-id1167834515425\"><span class=\"token\">\u24d1<\/span> Graph the inequality.<\/p>\n<p id=\"fs-id1167835365741\"><span class=\"token\">\u24d2<\/span> List three solutions to the inequality. What options do the solutions provide Armando?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835333874\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167830701036\">\n<div data-type=\"problem\" id=\"fs-id1167832054672\">\n<p id=\"fs-id1167832054674\">Lester thinks that the solution of any inequality with a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/> sign is the region above the line and the solution of any inequality with a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5fae947beab7b30f45d5b6603772f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/> sign is the region below the line. Is Lester correct? Explain why or why not.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835338740\">\n<p>Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835335460\">\n<div data-type=\"problem\" id=\"fs-id1167834229208\">\n<p id=\"fs-id1167834229210\">Explain why, in some graphs of linear inequalities, the boundary line is solid but in other graphs it is dashed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167826807779\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167826857196\"><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-id1167831943967\" data-alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cverify solutions to an inequality in two variables.\u201d, \u201crecognize the relation between the solutions of an inequality and its graph\u201d, and \u201cgraph linear inequalities\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_04_209_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cverify solutions to an inequality in two variables.\u201d, \u201crecognize the relation between the solutions of an inequality and its graph\u201d, and \u201cgraph linear inequalities\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\" \/><\/span><\/p>\n<p id=\"fs-id1167835328724\"><span class=\"token\">\u24d1<\/span> On a scale of 1\u201310, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/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-id1167834061630\">\n<dt>boundary line<\/dt>\n<dd id=\"fs-id1167834061636\">The line with equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0520a31036e9a951aea74693c8b23cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> is the boundary line that separates the region where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23b6e4813cdddb52957346c2cd079311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#62;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> from the region where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-760db367ad2bb780c3331d9cefc0ff43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#60;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1167835310131\">\n<dt>linear inequality<\/dt>\n<dd id=\"fs-id1167826779362\">A linear inequality is an inequality that can be written in one of the following forms: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f75e4c1456f35ac5689082d40649a26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#62;&#67;&#44;&#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;&#65;&#120;&#43;&#66;&#121;&#92;&#103;&#101;&#32;&#67;&#44;&#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;&#65;&#120;&#43;&#66;&#121;&#60;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"350\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a946ede12b2ae886b56e39570422dc0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#92;&#108;&#101;&#32;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834328459\">\n<dt>solution to a linear inequality<\/dt>\n<dd id=\"fs-id1167834328465\">An ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> is a solution to a linear inequality if the inequality is true when we substitute the values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1946","chapter","type-chapter","status-publish","hentry"],"part":1643,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1946","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":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1946\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/1643"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1946\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=1946"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=1946"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=1946"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=1946"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}