{"id":1048,"date":"2023-07-16T20:38:34","date_gmt":"2023-07-16T20:38:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=1048"},"modified":"2023-09-07T16:44:59","modified_gmt":"2023-09-07T16:44:59","slug":"solving-inequalities-using-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/solving-inequalities-using-graphs-of-functions\/","title":{"raw":"\u25aa   Solving Inequalities Using Graphs of Functions","rendered":"\u25aa   Solving Inequalities Using Graphs of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve inequalities with one variable using graphs of functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Solving Inequalities with One Variable using One Function<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>Recall DISTINCT PARTS OF A GRAPH of a Function AND THEIR [latex]y[\/latex]\u00a0VALUES<\/h3>\r\n(a) When a graph is below the [latex]x[\/latex]-axis, [latex]y[\/latex] values\u00a0are negative. So, [latex]f(x)&lt;0[\/latex].\r\n\r\n(b) When a graph is on the [latex]x[\/latex]-axis, [latex]y[\/latex] values\u00a0are zero. So,\u00a0[latex]f(x)=0[\/latex].\r\n\r\n(c) When a graph is above the [latex]x[\/latex]-axis, [latex]y[\/latex] values\u00a0are positive. So,\u00a0[latex]f(x)&gt;0[\/latex].\r\n\r\n<\/div>\r\nAs we have seen, we can solve an equation [latex]f(x)=0[\/latex] by finding the [latex]x[\/latex]-intercepts of its function [latex]y=f(x)[\/latex]. That means we can solve an equation\u00a0[latex]f(x)=0[\/latex] by finding the [latex]x[\/latex] values when the graph of [latex]y=f(x)[\/latex] is on the [latex]x[\/latex]-axis. We can apply the idea to solve inequalities with one variable.\r\n\r\nIn the previous section, we talked about\u00a0[latex]-\\frac{3}{4}x+3=0[\/latex]. Now let's consider the inequalities [latex]-\\frac{3}{4}x+3&gt;0[\/latex] and\u00a0[latex]-\\frac{3}{4}x+3&lt;0[\/latex]. According to the relationship above, to solve\u00a0[latex]-\\frac{3}{4}x+3&gt;0[\/latex], we can find the [latex]x[\/latex] values when the graph of\u00a0[latex]f(x)=-\\frac{3}{4}x+3[\/latex] is above the [latex]x[\/latex]-axis and below the [latex]x[\/latex]-axis, respectively.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.0845%; text-align: center;\"><strong style=\"text-align: center;\">Inequality<\/strong><\/td>\r\n<td style=\"width: 12.2691%; text-align: center;\"><strong>Position<\/strong><\/td>\r\n<td style=\"width: 25.221%; text-align: center;\"><strong style=\"text-align: center;\">Graph<\/strong><\/td>\r\n<td style=\"width: 23.5141%;\"><\/td>\r\n<td style=\"width: 22.9116%; text-align: center;\"><strong style=\"text-align: center;\">Solution<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0845%; text-align: center;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex]-\\frac{3}{4}+3&gt;0[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.2691%; text-align: center; vertical-align: middle;\">above\r\nthe [latex]x[\/latex]-axis<\/td>\r\n<td style=\"width: 25.221%; text-align: center; vertical-align: middle;\"><img class=\"wp-image-1060 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt2.png\" alt=\"-3\/4x+3 is solid line, -3\/4x+3&gt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&lt;4 is solid line\" width=\"161\" height=\"148\" \/><\/td>\r\n<td style=\"width: 23.5141%;\"><img class=\"wp-image-1059 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt_eq2.png\" alt=\"Key: -3\/4x+3 is solid line, -3\/4x+3&gt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&lt;4 is solid line\" width=\"177\" height=\"143\" \/><\/td>\r\n<td style=\"width: 22.9116%;\">When the graph is above the [latex]x[\/latex]-axis, which is the orange part,\r\n<p style=\"text-align: left;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex](-\\infty, 4)[\/latex] or [latex]\\{x|x&lt;4\\}[\/latex]<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0845%; text-align: center;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex]-\\frac{3}{4}+3&lt;0[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.2691%; text-align: center;\">below\r\nthe [latex]x[\/latex]-axis<\/td>\r\n<td style=\"width: 25.221%; text-align: center; vertical-align: middle;\"><img class=\"aligncenter wp-image-1062\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt2.png\" alt=\"Graphs: -3\/4x+3 is solid line, -3\/4x+3&lt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&gt;4 is solid line\" width=\"159\" height=\"146\" \/><\/td>\r\n<td style=\"width: 23.5141%;\"><img class=\"wp-image-1061 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt_eq2.png\" alt=\"Key: -3\/4x+3 is solid line, -3\/4x+3&lt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&gt;4 is solid line\" width=\"166\" height=\"136\" \/><\/td>\r\n<td style=\"width: 22.9116%; text-align: left;\">When the graph is below the [latex]x[\/latex]-axis, which is the green part,\r\n\r\n<span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex](4, \\infty)[\/latex] or [latex]\\{x|x&gt;4\\}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving [latex]f(x) \\geq 0[\/latex] Graphically<\/h3>\r\nSolve the inequality graphically. Use a graphing tool.\r\n\r\n[latex]x^3-x-4x^2+4 \\geq 0[\/latex]\r\n\r\n[reveal-answer q=\"18651\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"18651\"]\r\n\r\nLet [latex]f(x)=x^3-x-4x^2+4[\/latex].\u00a0Then find the parts that are <strong>above <\/strong>or <strong>on<\/strong> the [latex]x[\/latex]-axis because [latex]f(x)[\/latex] is \"greater than ([latex]&gt;[\/latex]),\" which is above the [latex]x[\/latex]-axis, or \"equal to ([latex]=[\/latex]),\" which is on the [latex]x[\/latex]-axis. From the graph, we can see that the graph of [latex]f(x)=x^3-x-4x^2+4[\/latex] is\u00a0above the [latex]x[\/latex]-axis when [latex]-1&lt;x&lt;1[\/latex] or [latex]x&gt;4[\/latex] and on\u00a0the [latex]x[\/latex]-axis when [latex]x=-1, 1, 4[\/latex]. So, the solution of the inequality\u00a0[latex]x^3-x-4x^2+4 \\leq 0[\/latex] is [latex][-1, 1] \\cup [4, \\infty)[\/latex] or [latex]\\{x|-1 \\leq x \\leq 1[\/latex] or [latex]x \\geq 4\\}[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-1065\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-2-Poly-Ineq-w-4-terms.png\" alt=\"graphs of f(x)=-2x^3-x^2+6x with above x-axis is shaded\" width=\"150\" height=\"160\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality graphically. Use a graphing tool.\r\n\r\n[latex](x-1)^4-3(x-1)^2-4 \\leq 0[\/latex]\r\n\r\n[reveal-answer q=\"544904\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"544904\"]\r\n\r\nLet [latex]f(x)=(x-1)^4-3(x-1)^2-4[\/latex].\u00a0Then find the parts that are <strong>below\u00a0<\/strong>or <strong>on\u00a0<\/strong>the [latex]x[\/latex]-axis. From the graph, we can see that the graph of [latex]f(x)=(x-1)^4-3(x-1)^2-4[\/latex] is\u00a0below the [latex]x[\/latex]-axis when [latex]-1&lt;x&lt;3[\/latex] and on the [latex]x[\/latex]-axis when [latex]x=-1, 3[\/latex]. So, the solution of the inequality\u00a0[latex](x-1)^4-3(x-1)^2-4&lt;0[\/latex] is [latex][-1, 3][\/latex] or [latex]\\{x|-1 \\leq x \\leq 3\\}[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-1066\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-3-Ineq-in-Quad-Form.png\" alt=\"graph of f(x)=(x-1)^4-3(x-1)^2-4 with below x-axis is shaded\" width=\"161\" height=\"158\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality graphically. Use a graphing tool.\r\n\r\n[latex]\\sqrt[3]{(x+1)^2}-4&gt;0[\/latex]\r\n\r\n[reveal-answer q=\"213567\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"213567\"]\r\n\r\nLet\u00a0[latex]f(x)=\\sqrt[3]{(x+1)^2}-4[\/latex]. Then find the parts that are <strong>above<\/strong> the [latex]x[\/latex]-axis. From the graph, we can see that the graph of\u00a0[latex]f(x)=\\sqrt[3]{(x+1)^2}-4[\/latex] is above the [latex]x[\/latex]-axis when [latex]x&lt;-9[\/latex] or [latex]x&gt;7[\/latex]. So, the solution of the inequality\u00a0[latex]\\sqrt[3]{(x+1)^2}&gt;4[\/latex] is [latex](-\\infty, -9)\\cup(7, \\infty)[\/latex] or [latex]\\{x|x&lt;-9[\/latex] or [latex]x&gt;7\\}[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-1064\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq.png\" alt=\"graph of f(x)=(x+1)^(2\/3)-4 with above x-axis is shaded\" width=\"243\" height=\"117\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solving Equations with One Variable using Multiple Functions<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving [latex]f(x)&gt;g(x)[\/latex] Graphically<\/h3>\r\nSolve the inequality graphically. Use a graphing tool.\r\n\r\n[latex]2x^2-3&gt;-2x+1[\/latex]\r\n\r\n[reveal-answer q=\"799147\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"799147\"]\r\n\r\n<span style=\"text-decoration: underline;\"><strong>Method 1<\/strong><\/span>\r\n\r\nMove all terms to the left: [latex]2x^2-3+2x-1&gt;0[\/latex]*\r\n<p style=\"padding-left: 30px;\">* If you are using a graphing tool, you don't need to simplify the inequality. Just let\u00a0[latex]f(x)=2x^2-3-(-2x+1)[\/latex] or\u00a0 [latex]f(x)=2x^2-3+2x-1[\/latex].<\/p>\r\nSo, [latex]2x^2+2x-4&gt;0[\/latex].\r\n\r\nLet\u00a0[latex]f(x)=2x^2+2x-4[\/latex].\u00a0Then find the parts that are <strong>above<\/strong><strong>\u00a0<\/strong>the [latex]x[\/latex]-axis.\r\n\r\n<img class=\"aligncenter wp-image-1068\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function.png\" alt=\"graph of f(x)=2x^2+2x-4 with above the x-axis is shaded\" width=\"172\" height=\"160\" \/>\r\n\r\nSince the graph of [latex]f(x)=2x^2+2x-4[\/latex]\u00a0<span style=\"font-size: 1rem; text-align: initial;\">is above the [latex]x[\/latex]-axis when [latex]x&lt;-2[\/latex] or [latex]x&gt;1[\/latex], the solution of the inequality\u00a0[latex]2x^2-3&gt;-2x+1[\/latex] is [latex](-\\infty, -2)\\cup(1, \\infty)[\/latex]\u00a0 or [latex]\\{x|x&lt;-2[\/latex] or [latex]x&gt;1\\}[\/latex].<\/span>\r\n\r\n<span style=\"text-decoration: underline;\"><strong>Method 2<\/strong><\/span>\r\n\r\nLet [latex]f(x)=2x^2-3[\/latex] and [latex]g(x)=-2x+1[\/latex]. Then find the parts where [latex]f(x)&gt;g(x)[\/latex], which means where the graph of [latex]f(x)[\/latex] is <strong>above<\/strong> the graph of [latex]g(x)[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-1070\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions.png\" alt=\"Graphs of f(x)=2x^2-3 and g(x)=-2x+1 with shaded when f(x) is higher than g(x)\" width=\"189\" height=\"164\" \/>\r\n\r\nSince the graph of [latex]f(x)=2x^2-3[\/latex]\u00a0<span style=\"font-size: 1rem; text-align: initial;\">is above the graph of [latex]g(x)=-2x+1[\/latex] when [latex]x&lt;-2[\/latex] or [latex]x&gt;1[\/latex], the solution of the inequality\u00a0[latex]2x^2-3&gt;-2x+1[\/latex] is [latex](-\\infty, -2)\\cup(1, \\infty)[\/latex] or [latex]\\{x|x&lt;-2[\/latex] or [latex]x&gt;1\\}[\/latex]<\/span>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality graphically. Use a graphing tool.\r\n\r\n[latex]-2x^3+6x-3 \\leq x^2-3[\/latex]\r\n\r\n[reveal-answer q=\"618302\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"618302\"]\r\n\r\nMove all terms to the lefthand side: [latex]-2x^3+6x-3-x^2+3 \\leq 0[\/latex]. Then let [latex]f(x)=-2x^3-x^2+6x[\/latex]. Since the graph of[latex]f(x)=-2x^3-x^2+6x[\/latex] is <strong>below<\/strong> the [latex]x[\/latex]-axis when [latex]-2&lt;x&lt;0[\/latex] or [latex]x&gt;\\frac{3}{2}[\/latex] and <strong>on<\/strong> the [latex]x[\/latex]-axis when [latex]x=-2, 0, \\frac{3}{2}[\/latex],\u00a0the solution of the inequality\u00a0[latex]-2x^3+6x-3 \\leq x^2-3[\/latex] is [latex][-2, 0] \\cup [\\frac{3}{2}, \\infty)[\/latex] or [latex]\\{x|-2 \\leq x \\leq 0[\/latex] or [latex]x \\geq \\frac{3}{2}\\}[\/latex].\r\n<p style=\"text-align: center;\">OR<\/p>\r\nLet [latex]g(x)=-2x^3+6x-3[\/latex] and [latex]h(x)=x^2-3[\/latex]. Since the graph of\u00a0[latex]g(x)=-2x^3+6x-3[\/latex] is <strong>below<\/strong> the graph of\u00a0[latex]h(x)=x^2-3[\/latex] when [latex]-2&lt;x&lt;0[\/latex] or [latex]x&gt;\\frac{3}{2}[\/latex] and <strong>intersects<\/strong> the graph of\u00a0[latex]h(x)=x^2-3[\/latex] when [latex]x=-2, 0, \\frac{3}{2}[\/latex], the solution of the inequality\u00a0[latex]-2x^3+6x-3 \\leq x^2-3[\/latex] is [latex][-2, 0] \\cup [\\frac{3}{2}, \\infty)[\/latex] or [latex]\\{x|-2 \\leq x \\leq 0[\/latex] or [latex]x \\geq \\frac{3}{2}\\}[\/latex].\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-1072\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function.png\" alt=\"graphs of f(x)=-2x^3-x^2+6x with the below the x-axis is shaded\" width=\"203\" height=\"162\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-1073\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function.png\" alt=\"graphs of g(x)=-2x^3+6x-3, h(x)=x^2-3 with shaded when g(x) is below h(x)\" width=\"215\" height=\"168\" \/><\/td>\r\n<td style=\"width: 33.3333%; vertical-align: middle;\"><img class=\"aligncenter wp-image-1074\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all.png\" alt=\"graphs of f(x)=-2x^3-x^2+6x, g(x)=-2x^3+6x-3, h(x)=x^2-3 with shaded when g(x) is below h(x)\" width=\"218\" height=\"168\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve inequalities with one variable using graphs of functions.<\/li>\n<\/ul>\n<\/div>\n<h2>Solving Inequalities with One Variable using One Function<\/h2>\n<div class=\"textbox examples\">\n<h3>Recall DISTINCT PARTS OF A GRAPH of a Function AND THEIR [latex]y[\/latex]\u00a0VALUES<\/h3>\n<p>(a) When a graph is below the [latex]x[\/latex]-axis, [latex]y[\/latex] values\u00a0are negative. So, [latex]f(x)<0[\/latex].\n\n(b) When a graph is on the [latex]x[\/latex]-axis, [latex]y[\/latex] values\u00a0are zero. So,\u00a0[latex]f(x)=0[\/latex].\n\n(c) When a graph is above the [latex]x[\/latex]-axis, [latex]y[\/latex] values\u00a0are positive. So,\u00a0[latex]f(x)>0[\/latex].<\/p>\n<\/div>\n<p>As we have seen, we can solve an equation [latex]f(x)=0[\/latex] by finding the [latex]x[\/latex]-intercepts of its function [latex]y=f(x)[\/latex]. That means we can solve an equation\u00a0[latex]f(x)=0[\/latex] by finding the [latex]x[\/latex] values when the graph of [latex]y=f(x)[\/latex] is on the [latex]x[\/latex]-axis. We can apply the idea to solve inequalities with one variable.<\/p>\n<p>In the previous section, we talked about\u00a0[latex]-\\frac{3}{4}x+3=0[\/latex]. Now let&#8217;s consider the inequalities [latex]-\\frac{3}{4}x+3>0[\/latex] and\u00a0[latex]-\\frac{3}{4}x+3<0[\/latex]. According to the relationship above, to solve\u00a0[latex]-\\frac{3}{4}x+3>0[\/latex], we can find the [latex]x[\/latex] values when the graph of\u00a0[latex]f(x)=-\\frac{3}{4}x+3[\/latex] is above the [latex]x[\/latex]-axis and below the [latex]x[\/latex]-axis, respectively.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 16.0845%; text-align: center;\"><strong style=\"text-align: center;\">Inequality<\/strong><\/td>\n<td style=\"width: 12.2691%; text-align: center;\"><strong>Position<\/strong><\/td>\n<td style=\"width: 25.221%; text-align: center;\"><strong style=\"text-align: center;\">Graph<\/strong><\/td>\n<td style=\"width: 23.5141%;\"><\/td>\n<td style=\"width: 22.9116%; text-align: center;\"><strong style=\"text-align: center;\">Solution<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.0845%; text-align: center;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex]-\\frac{3}{4}+3>0[\/latex]<\/span><\/td>\n<td style=\"width: 12.2691%; text-align: center; vertical-align: middle;\">above<br \/>\nthe [latex]x[\/latex]-axis<\/td>\n<td style=\"width: 25.221%; text-align: center; vertical-align: middle;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1060 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt2.png\" alt=\"-3\/4x+3 is solid line, -3\/4x+3&gt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&lt;4 is solid line\" width=\"161\" height=\"148\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt2.png 262w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt2-65x60.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt2-225x207.png 225w\" sizes=\"auto, (max-width: 161px) 100vw, 161px\" \/><\/td>\n<td style=\"width: 23.5141%;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1059 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt_eq2.png\" alt=\"Key: -3\/4x+3 is solid line, -3\/4x+3&gt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&lt;4 is solid line\" width=\"177\" height=\"143\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt_eq2.png 282w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt_eq2-65x52.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_gt_eq2-225x181.png 225w\" sizes=\"auto, (max-width: 177px) 100vw, 177px\" \/><\/td>\n<td style=\"width: 22.9116%;\">When the graph is above the [latex]x[\/latex]-axis, which is the orange part,<\/p>\n<p style=\"text-align: left;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex](-\\infty, 4)[\/latex] or [latex]\\{x|x<4\\}[\/latex]<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.0845%; text-align: center;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex]-\\frac{3}{4}+3<0[\/latex]<\/span><\/td>\n<td style=\"width: 12.2691%; text-align: center;\">below<br \/>\nthe [latex]x[\/latex]-axis<\/td>\n<td style=\"width: 25.221%; text-align: center; vertical-align: middle;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1062\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt2.png\" alt=\"Graphs: -3\/4x+3 is solid line, -3\/4x+3&lt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&gt;4 is solid line\" width=\"159\" height=\"146\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt2.png 261w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt2-65x60.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt2-225x206.png 225w\" sizes=\"auto, (max-width: 159px) 100vw, 159px\" \/><\/td>\n<td style=\"width: 23.5141%;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1061 aligncenter\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt_eq2.png\" alt=\"Key: -3\/4x+3 is solid line, -3\/4x+3&lt;0 shaded with dotted line at x=4 and y=-3\/4x+3 when x&gt;4 is solid line\" width=\"166\" height=\"136\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt_eq2.png 273w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt_eq2-65x53.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-linear-inequality_lt_eq2-225x185.png 225w\" sizes=\"auto, (max-width: 166px) 100vw, 166px\" \/><\/td>\n<td style=\"width: 22.9116%; text-align: left;\">When the graph is below the [latex]x[\/latex]-axis, which is the green part,<\/p>\n<p><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex](4, \\infty)[\/latex] or [latex]\\{x|x>4\\}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example: Solving [latex]f(x) \\geq 0[\/latex] Graphically<\/h3>\n<p>Solve the inequality graphically. Use a graphing tool.<\/p>\n<p>[latex]x^3-x-4x^2+4 \\geq 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q18651\">Show Answer<\/span><\/p>\n<div id=\"q18651\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]f(x)=x^3-x-4x^2+4[\/latex].\u00a0Then find the parts that are <strong>above <\/strong>or <strong>on<\/strong> the [latex]x[\/latex]-axis because [latex]f(x)[\/latex] is &#8220;greater than ([latex]>[\/latex]),&#8221; which is above the [latex]x[\/latex]-axis, or &#8220;equal to ([latex]=[\/latex]),&#8221; which is on the [latex]x[\/latex]-axis. From the graph, we can see that the graph of [latex]f(x)=x^3-x-4x^2+4[\/latex] is\u00a0above the [latex]x[\/latex]-axis when [latex]-1<x<1[\/latex] or [latex]x>4[\/latex] and on\u00a0the [latex]x[\/latex]-axis when [latex]x=-1, 1, 4[\/latex]. So, the solution of the inequality\u00a0[latex]x^3-x-4x^2+4 \\leq 0[\/latex] is [latex][-1, 1] \\cup [4, \\infty)[\/latex] or [latex]\\{x|-1 \\leq x \\leq 1[\/latex] or [latex]x \\geq 4\\}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1065\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-2-Poly-Ineq-w-4-terms.png\" alt=\"graphs of f(x)=-2x^3-x^2+6x with above x-axis is shaded\" width=\"150\" height=\"160\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-2-Poly-Ineq-w-4-terms.png 331w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-2-Poly-Ineq-w-4-terms-282x300.png 282w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-2-Poly-Ineq-w-4-terms-65x69.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-2-Poly-Ineq-w-4-terms-225x239.png 225w\" sizes=\"auto, (max-width: 150px) 100vw, 150px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve the inequality graphically. Use a graphing tool.<\/p>\n<p>[latex](x-1)^4-3(x-1)^2-4 \\leq 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q544904\">Show Answer<\/span><\/p>\n<div id=\"q544904\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]f(x)=(x-1)^4-3(x-1)^2-4[\/latex].\u00a0Then find the parts that are <strong>below\u00a0<\/strong>or <strong>on\u00a0<\/strong>the [latex]x[\/latex]-axis. From the graph, we can see that the graph of [latex]f(x)=(x-1)^4-3(x-1)^2-4[\/latex] is\u00a0below the [latex]x[\/latex]-axis when [latex]-1<x<3[\/latex] and on the [latex]x[\/latex]-axis when [latex]x=-1, 3[\/latex]. So, the solution of the inequality\u00a0[latex](x-1)^4-3(x-1)^2-4<0[\/latex] is [latex][-1, 3][\/latex] or [latex]\\{x|-1 \\leq x \\leq 3\\}[\/latex].\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1066\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-3-Ineq-in-Quad-Form.png\" alt=\"graph of f(x)=(x-1)^4-3(x-1)^2-4 with below x-axis is shaded\" width=\"161\" height=\"158\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-3-Ineq-in-Quad-Form.png 330w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-3-Ineq-in-Quad-Form-300x295.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-3-Ineq-in-Quad-Form-65x64.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-3-Ineq-in-Quad-Form-225x221.png 225w\" sizes=\"auto, (max-width: 161px) 100vw, 161px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve the inequality graphically. Use a graphing tool.<\/p>\n<p>[latex]\\sqrt[3]{(x+1)^2}-4>0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q213567\">Show Answer<\/span><\/p>\n<div id=\"q213567\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u00a0[latex]f(x)=\\sqrt[3]{(x+1)^2}-4[\/latex]. Then find the parts that are <strong>above<\/strong> the [latex]x[\/latex]-axis. From the graph, we can see that the graph of\u00a0[latex]f(x)=\\sqrt[3]{(x+1)^2}-4[\/latex] is above the [latex]x[\/latex]-axis when [latex]x<-9[\/latex] or [latex]x>7[\/latex]. So, the solution of the inequality\u00a0[latex]\\sqrt[3]{(x+1)^2}>4[\/latex] is [latex](-\\infty, -9)\\cup(7, \\infty)[\/latex] or [latex]\\{x|x<-9[\/latex] or [latex]x>7\\}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1064\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq.png\" alt=\"graph of f(x)=(x+1)^(2\/3)-4 with above x-axis is shaded\" width=\"243\" height=\"117\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq.png 591w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq-300x144.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq-65x31.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq-225x108.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-1-Rational-Power-Ineq-350x168.png 350w\" sizes=\"auto, (max-width: 243px) 100vw, 243px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solving Equations with One Variable using Multiple Functions<\/h2>\n<div class=\"textbox exercises\">\n<h3>Example: Solving [latex]f(x)>g(x)[\/latex] Graphically<\/h3>\n<p>Solve the inequality graphically. Use a graphing tool.<\/p>\n<p>[latex]2x^2-3>-2x+1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q799147\">Show Answer<\/span><\/p>\n<div id=\"q799147\" class=\"hidden-answer\" style=\"display: none\">\n<p><span style=\"text-decoration: underline;\"><strong>Method 1<\/strong><\/span><\/p>\n<p>Move all terms to the left: [latex]2x^2-3+2x-1>0[\/latex]*<\/p>\n<p style=\"padding-left: 30px;\">* If you are using a graphing tool, you don&#8217;t need to simplify the inequality. Just let\u00a0[latex]f(x)=2x^2-3-(-2x+1)[\/latex] or\u00a0 [latex]f(x)=2x^2-3+2x-1[\/latex].<\/p>\n<p>So, [latex]2x^2+2x-4>0[\/latex].<\/p>\n<p>Let\u00a0[latex]f(x)=2x^2+2x-4[\/latex].\u00a0Then find the parts that are <strong>above<\/strong><strong>\u00a0<\/strong>the [latex]x[\/latex]-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1068\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function.png\" alt=\"graph of f(x)=2x^2+2x-4 with above the x-axis is shaded\" width=\"172\" height=\"160\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function.png 356w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function-300x279.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function-65x60.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function-225x209.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_One-function-350x325.png 350w\" sizes=\"auto, (max-width: 172px) 100vw, 172px\" \/><\/p>\n<p>Since the graph of [latex]f(x)=2x^2+2x-4[\/latex]\u00a0<span style=\"font-size: 1rem; text-align: initial;\">is above the [latex]x[\/latex]-axis when [latex]x<-2[\/latex] or [latex]x>1[\/latex], the solution of the inequality\u00a0[latex]2x^2-3>-2x+1[\/latex] is [latex](-\\infty, -2)\\cup(1, \\infty)[\/latex]\u00a0 or [latex]\\{x|x<-2[\/latex] or [latex]x>1\\}[\/latex].<\/span><\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Method 2<\/strong><\/span><\/p>\n<p>Let [latex]f(x)=2x^2-3[\/latex] and [latex]g(x)=-2x+1[\/latex]. Then find the parts where [latex]f(x)>g(x)[\/latex], which means where the graph of [latex]f(x)[\/latex] is <strong>above<\/strong> the graph of [latex]g(x)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1070\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions.png\" alt=\"Graphs of f(x)=2x^2-3 and g(x)=-2x+1 with shaded when f(x) is higher than g(x)\" width=\"189\" height=\"164\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions.png 351w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions-300x260.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions-65x56.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions-225x195.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-5-quad-Ineq_Two-functions-350x303.png 350w\" sizes=\"auto, (max-width: 189px) 100vw, 189px\" \/><\/p>\n<p>Since the graph of [latex]f(x)=2x^2-3[\/latex]\u00a0<span style=\"font-size: 1rem; text-align: initial;\">is above the graph of [latex]g(x)=-2x+1[\/latex] when [latex]x<-2[\/latex] or [latex]x>1[\/latex], the solution of the inequality\u00a0[latex]2x^2-3>-2x+1[\/latex] is [latex](-\\infty, -2)\\cup(1, \\infty)[\/latex] or [latex]\\{x|x<-2[\/latex] or [latex]x>1\\}[\/latex]<\/span>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve the inequality graphically. Use a graphing tool.<\/p>\n<p>[latex]-2x^3+6x-3 \\leq x^2-3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q618302\">Show Answer<\/span><\/p>\n<div id=\"q618302\" class=\"hidden-answer\" style=\"display: none\">\n<p>Move all terms to the lefthand side: [latex]-2x^3+6x-3-x^2+3 \\leq 0[\/latex]. Then let [latex]f(x)=-2x^3-x^2+6x[\/latex]. Since the graph of[latex]f(x)=-2x^3-x^2+6x[\/latex] is <strong>below<\/strong> the [latex]x[\/latex]-axis when [latex]-2<x<0[\/latex] or [latex]x>\\frac{3}{2}[\/latex] and <strong>on<\/strong> the [latex]x[\/latex]-axis when [latex]x=-2, 0, \\frac{3}{2}[\/latex],\u00a0the solution of the inequality\u00a0[latex]-2x^3+6x-3 \\leq x^2-3[\/latex] is [latex][-2, 0] \\cup [\\frac{3}{2}, \\infty)[\/latex] or [latex]\\{x|-2 \\leq x \\leq 0[\/latex] or [latex]x \\geq \\frac{3}{2}\\}[\/latex].<\/p>\n<p style=\"text-align: center;\">OR<\/p>\n<p>Let [latex]g(x)=-2x^3+6x-3[\/latex] and [latex]h(x)=x^2-3[\/latex]. Since the graph of\u00a0[latex]g(x)=-2x^3+6x-3[\/latex] is <strong>below<\/strong> the graph of\u00a0[latex]h(x)=x^2-3[\/latex] when [latex]-2<x<0[\/latex] or [latex]x>\\frac{3}{2}[\/latex] and <strong>intersects<\/strong> the graph of\u00a0[latex]h(x)=x^2-3[\/latex] when [latex]x=-2, 0, \\frac{3}{2}[\/latex], the solution of the inequality\u00a0[latex]-2x^3+6x-3 \\leq x^2-3[\/latex] is [latex][-2, 0] \\cup [\\frac{3}{2}, \\infty)[\/latex] or [latex]\\{x|-2 \\leq x \\leq 0[\/latex] or [latex]x \\geq \\frac{3}{2}\\}[\/latex].<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1072\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function.png\" alt=\"graphs of f(x)=-2x^3-x^2+6x with the below the x-axis is shaded\" width=\"203\" height=\"162\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function.png 376w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function-300x239.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function-65x52.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function-225x180.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_One-function-350x279.png 350w\" sizes=\"auto, (max-width: 203px) 100vw, 203px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1073\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function.png\" alt=\"graphs of g(x)=-2x^3+6x-3, h(x)=x^2-3 with shaded when g(x) is below h(x)\" width=\"215\" height=\"168\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function.png 451w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function-300x235.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function-65x51.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function-225x176.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_Two-function-350x274.png 350w\" sizes=\"auto, (max-width: 215px) 100vw, 215px\" \/><\/td>\n<td style=\"width: 33.3333%; vertical-align: middle;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1074\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all.png\" alt=\"graphs of f(x)=-2x^3-x^2+6x, g(x)=-2x^3+6x-3, h(x)=x^2-3 with shaded when g(x) is below h(x)\" width=\"218\" height=\"168\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all.png 449w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all-300x231.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all-65x50.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all-225x173.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-EX-6-poly-Ineq_all-350x270.png 350w\" sizes=\"auto, (max-width: 218px) 100vw, 218px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1048\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Solving Inequalities Using Graphs of Functions. <strong>Authored by<\/strong>: Michelle Eunhee Chung. <strong>Provided by<\/strong>: Georgia State University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":705214,"menu_order":32,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Solving Inequalities Using Graphs of Functions\",\"author\":\"Michelle Eunhee Chung\",\"organization\":\"Georgia State University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mchung12"],"pb_section_license":""},"chapter-type":[],"contributor":[62],"license":[],"class_list":["post-1048","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":91,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1048","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/705214"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1048\/revisions"}],"predecessor-version":[{"id":1404,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1048\/revisions\/1404"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/91"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1048\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=1048"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1048"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1048"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=1048"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}