{"id":1082,"date":"2023-07-18T02:39:04","date_gmt":"2023-07-18T02:39:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=1082"},"modified":"2023-09-07T16:45:34","modified_gmt":"2023-09-07T16:45:34","slug":"solving-equations-and-inequalities-using-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/solving-equations-and-inequalities-using-graphs-of-functions\/","title":{"raw":"\u25aa   Solving Equations and Inequalities Using Graphs of Functions","rendered":"\u25aa   Solving Equations and 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 equations and inequalities using graphs of functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example Solving Equations and Inequalities using Given Graphs<\/h3>\r\nSolve the equations and inequalities using the given graphs.\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 46.2525%;\"><img class=\"aligncenter size-full wp-image-1090\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1.png\" alt=\"Graphs of f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange, h(x)=-1 in green\" width=\"573\" height=\"359\" \/><\/td>\r\n<td style=\"width: 24.6378%;\"><img class=\"aligncenter wp-image-1089\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-2-1.png\" alt=\"Key: f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange, h(x)=-1 in green\" width=\"167\" height=\"135\" \/><\/td>\r\n<td style=\"width: 29.1096%;\"><img class=\"aligncenter wp-image-1088\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-3.png\" alt=\"Key: f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange, h(x)=-1 in green\" width=\"221\" height=\"130\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>Sove the equation [latex]f(x)=g(x)[\/latex].\r\n[reveal-answer q=\"84492\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"84492\"]\r\n[latex]f(x)=g(x)[\/latex] means that their [latex]y[\/latex] values are equal. So, we need to find the intersecting points of those two functions. From the graphs, we can find that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are intersecting at [latex](-3, 1)[\/latex] and [latex](5, 5)[\/latex]. Therefore, the solutions of the equation\u00a0[latex]f(x)=g(x)[\/latex] are [latex]x=-3, 5[\/latex].<img class=\"aligncenter wp-image-1092\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4.png\" alt=\"Graphs of f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange with intersecting points (-3, 1), (5, 5)\" width=\"260\" height=\"179\" \/>[\/hidden-answer]<\/li>\r\n \t<li>Solve the inequality [latex]f(x) \\geq h(x)[\/latex].\r\n[reveal-answer q=\"674183\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"674183\"]\r\n[latex]f(x) \\geq h(x)[\/latex] means that the graph of [latex]f(x)[\/latex] is above or intersecting the graph of [latex]h(x)[\/latex]. From the graphs, we can find\u00a0the graph of [latex]f(x)[\/latex] is above the graph of [latex]h(x)[\/latex] when [latex]x&gt;-7[\/latex] and is intersecting the graph of [latex]h(x)[\/latex] when [latex]x=-7[\/latex]. So, the solution of the inequality\u00a0[latex]f(x) \\geq h(x)[\/latex] is [latex][-7, \\infty)[\/latex] or [latex]\\{x|x \\geq -7\\}[\/latex].\r\n<img class=\"aligncenter wp-image-1093\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5.png\" alt=\"Graphs of f(x)=1\/2x+5\/2 in blue, h(x)=-1 in green with shaded when x&gt;-7\" width=\"235\" height=\"154\" \/>\r\n[\/hidden-answer]<\/li>\r\n \t<li>Solve the inequality [latex]g(x)&lt;h(x)[\/latex].\r\n[reveal-answer q=\"800838\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"800838\"]\r\n[latex]g(x)&lt;h(x)[\/latex] means that the graph of [latex]g(x)[\/latex] is below the graph of [latex]h(x)[\/latex]. From the graphs, we can find the graph of [latex]g(x)[\/latex] is below the graph of [latex]h(x)[\/latex] when [latex]-2&lt;x&lt;2[\/latex]. So, the solution of the inequality\u00a0[latex]g(x)&lt;h(x)[\/latex] is [latex][-2, 2][\/latex] or [latex]\\{x|-2&lt;x&lt;2\\}[\/latex].\r\n<img class=\"aligncenter wp-image-1094\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6.png\" alt=\"Graphs of g(x)=2|x|-5 in orange, h(x)=-1 in green with shaded when -2&lt;x&lt;2\" width=\"263\" height=\"146\" \/>\r\n[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve the equations and inequalities using the given graphs.\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 46.2525%;\"><img class=\"aligncenter wp-image-1102\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1.png\" alt=\"Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange, h(x)=2x-3\" width=\"325\" height=\"223\" \/><\/td>\r\n<td style=\"width: 24.6378%;\"><img class=\"aligncenter wp-image-1104\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-3.png\" alt=\"Key: Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange, h(x)=2x-3\" width=\"166\" height=\"126\" \/><\/td>\r\n<td style=\"width: 29.1096%;\"><img class=\"aligncenter size-full wp-image-1103\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-2.png\" alt=\"Key: Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange, h(x)=2x-3\" width=\"265\" height=\"159\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>Sove the equation [latex]f(x)=g(x)[\/latex].[reveal-answer q=\"576161\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"576161\"]\r\nSince [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are intersecting at [latex](-3, 3)[\/latex] and [latex](3, 3)[\/latex], the solutions of the equation\u00a0[latex]f(x)=g(x)[\/latex] are [latex]x=-3, 3[\/latex].\r\n<img class=\"aligncenter wp-image-1106\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4.png\" alt=\"Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange with intersecting points (-3, 3), (3, 3)\" width=\"246\" height=\"167\" \/>[\/hidden-answer]<\/li>\r\n \t<li>Solve the inequality [latex]f(x) &lt; g(x)[\/latex].[reveal-answer q=\"422763\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"422763\"]\r\nSince the graph of [latex]g(x)[\/latex] is below the graph of [latex]g(x)[\/latex] when [latex]x&lt;-3[\/latex] or [latex]x&gt;3[\/latex], the solution of the inequality\u00a0[latex]f(x)&lt;h(x)[\/latex] is [latex](-\\infty, -3) \\cup (3, \\infty)[\/latex] or [latex]\\{x|x&lt;-3[\/latex] or [latex]x&gt;3\\}[\/latex].\r\n<img class=\"aligncenter wp-image-1107\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5.png\" alt=\"Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange with shaded when x&lt;-3 and x&gt;3\" width=\"246\" height=\"167\" \/>\r\n[\/hidden-answer]<\/li>\r\n \t<li>Solve the inequality [latex]g(x) \\leq h(x)[\/latex].[reveal-answer q=\"65246\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"65246\"]\r\nSince the graph of [latex]g(x)[\/latex] is below the graph of [latex]h(x)[\/latex] when [latex]-1&lt;x&lt;3[\/latex] and is intersecting the graph of [latex]h(x)[\/latex] when [latex]x=-1, 3[\/latex], the solution of the inequality\u00a0[latex]g(x) \\leq h(x)[\/latex] is [latex][-1, 3][\/latex] or [latex]\\{x|-1 \\leq x \\leq 3\\}[\/latex].\r\n<img class=\"aligncenter wp-image-1108\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6.png\" alt=\"Graphs of g(x)=x^2-6 in orange, h(x)=2x-3 with shaded when -1&lt;x&lt;3\" width=\"246\" height=\"169\" \/>\r\n[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve equations and inequalities using graphs of functions.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example Solving Equations and Inequalities using Given Graphs<\/h3>\n<p>Solve the equations and inequalities using the given graphs.<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 46.2525%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1090\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1.png\" alt=\"Graphs of f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange, h(x)=-1 in green\" width=\"573\" height=\"359\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1.png 573w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1-300x188.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1-65x41.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1-225x141.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-1-1-350x219.png 350w\" sizes=\"auto, (max-width: 573px) 100vw, 573px\" \/><\/td>\n<td style=\"width: 24.6378%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1089\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-2-1.png\" alt=\"Key: f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange, h(x)=-1 in green\" width=\"167\" height=\"135\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-2-1.png 289w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-2-1-65x53.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-2-1-225x182.png 225w\" sizes=\"auto, (max-width: 167px) 100vw, 167px\" \/><\/td>\n<td style=\"width: 29.1096%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1088\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-3.png\" alt=\"Key: f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange, h(x)=-1 in green\" width=\"221\" height=\"130\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-3.png 333w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-3-300x177.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-3-65x38.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-3-225x132.png 225w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Sove the equation [latex]f(x)=g(x)[\/latex].\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q84492\">Show Answer<\/span><\/p>\n<div id=\"q84492\" class=\"hidden-answer\" style=\"display: none\">\n[latex]f(x)=g(x)[\/latex] means that their [latex]y[\/latex] values are equal. So, we need to find the intersecting points of those two functions. From the graphs, we can find that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are intersecting at [latex](-3, 1)[\/latex] and [latex](5, 5)[\/latex]. Therefore, the solutions of the equation\u00a0[latex]f(x)=g(x)[\/latex] are [latex]x=-3, 5[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1092\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4.png\" alt=\"Graphs of f(x)=1\/2x+5\/2 in blue, g(x)=2|x|-5 in orange with intersecting points (-3, 1), (5, 5)\" width=\"260\" height=\"179\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4.png 389w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4-300x206.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4-65x45.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4-225x154.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-4-350x240.png 350w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/div>\n<\/div>\n<\/li>\n<li>Solve the inequality [latex]f(x) \\geq h(x)[\/latex].\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q674183\">Show Answer<\/span><\/p>\n<div id=\"q674183\" class=\"hidden-answer\" style=\"display: none\">\n[latex]f(x) \\geq h(x)[\/latex] means that the graph of [latex]f(x)[\/latex] is above or intersecting the graph of [latex]h(x)[\/latex]. From the graphs, we can find\u00a0the graph of [latex]f(x)[\/latex] is above the graph of [latex]h(x)[\/latex] when [latex]x>-7[\/latex] and is intersecting the graph of [latex]h(x)[\/latex] when [latex]x=-7[\/latex]. So, the solution of the inequality\u00a0[latex]f(x) \\geq h(x)[\/latex] is [latex][-7, \\infty)[\/latex] or [latex]\\{x|x \\geq -7\\}[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1093\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5.png\" alt=\"Graphs of f(x)=1\/2x+5\/2 in blue, h(x)=-1 in green with shaded when x&gt;-7\" width=\"235\" height=\"154\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5.png 353w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5-300x196.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5-65x43.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5-225x147.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-5-350x229.png 350w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/>\n<\/div>\n<\/div>\n<\/li>\n<li>Solve the inequality [latex]g(x)<h(x)[\/latex].\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q800838\">Show Answer<\/span>\n<div id=\"q800838\" class=\"hidden-answer\" style=\"display: none\">\n[latex]g(x)<h(x)[\/latex] means that the graph of [latex]g(x)[\/latex] is below the graph of [latex]h(x)[\/latex]. From the graphs, we can find the graph of [latex]g(x)[\/latex] is below the graph of [latex]h(x)[\/latex] when [latex]-2<x<2[\/latex]. So, the solution of the inequality\u00a0[latex]g(x)<h(x)[\/latex] is [latex][-2, 2][\/latex] or [latex]\\{x|-2<x<2\\}[\/latex].\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1094\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6.png\" alt=\"Graphs of g(x)=2|x|-5 in orange, h(x)=-1 in green with shaded when -2&lt;x&lt;2\" width=\"263\" height=\"146\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6.png 385w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6-300x167.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6-65x36.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6-225x125.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-7-Eq_Ineq_Graphs-6-350x195.png 350w\" sizes=\"auto, (max-width: 263px) 100vw, 263px\" \/>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve the equations and inequalities using the given graphs.<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 46.2525%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1102\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1.png\" alt=\"Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange, h(x)=2x-3\" width=\"325\" height=\"223\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1.png 389w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1-300x206.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1-65x45.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1-225x154.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-1-350x240.png 350w\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" \/><\/td>\n<td style=\"width: 24.6378%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1104\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-3.png\" alt=\"Key: Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange, h(x)=2x-3\" width=\"166\" height=\"126\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-3.png 221w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-3-65x49.png 65w\" sizes=\"auto, (max-width: 166px) 100vw, 166px\" \/><\/td>\n<td style=\"width: 29.1096%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1103\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-2.png\" alt=\"Key: Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange, h(x)=2x-3\" width=\"265\" height=\"159\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-2.png 265w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-2-65x39.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-2-225x135.png 225w\" sizes=\"auto, (max-width: 265px) 100vw, 265px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Sove the equation [latex]f(x)=g(x)[\/latex].\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q576161\">Show Answer<\/span><\/p>\n<div id=\"q576161\" class=\"hidden-answer\" style=\"display: none\">\nSince [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are intersecting at [latex](-3, 3)[\/latex] and [latex](3, 3)[\/latex], the solutions of the equation\u00a0[latex]f(x)=g(x)[\/latex] are [latex]x=-3, 3[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1106\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4.png\" alt=\"Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange with intersecting points (-3, 3), (3, 3)\" width=\"246\" height=\"167\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4.png 388w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4-300x204.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4-65x44.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4-225x153.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-4-350x238.png 350w\" sizes=\"auto, (max-width: 246px) 100vw, 246px\" \/><\/div>\n<\/div>\n<\/li>\n<li>Solve the inequality [latex]f(x) < g(x)[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q422763\">Show Answer<\/span>\n<div id=\"q422763\" class=\"hidden-answer\" style=\"display: none\">\nSince the graph of [latex]g(x)[\/latex] is below the graph of [latex]g(x)[\/latex] when [latex]x<-3[\/latex] or [latex]x>3[\/latex], the solution of the inequality\u00a0[latex]f(x)<h(x)[\/latex] is [latex](-\\infty, -3) \\cup (3, \\infty)[\/latex] or [latex]\\{x|x<-3[\/latex] or [latex]x>3\\}[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1107\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5.png\" alt=\"Graphs of f(x)=3 in blue, g(x)=x^2-6 in orange with shaded when x&lt;-3 and x&gt;3\" width=\"246\" height=\"167\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5.png 387w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5-300x205.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5-65x44.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5-225x153.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-5-350x239.png 350w\" sizes=\"auto, (max-width: 246px) 100vw, 246px\" \/>\n<\/div>\n<\/div>\n<\/li>\n<li>Solve the inequality [latex]g(x) \\leq h(x)[\/latex].\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q65246\">Show Answer<\/span><\/p>\n<div id=\"q65246\" class=\"hidden-answer\" style=\"display: none\">\nSince the graph of [latex]g(x)[\/latex] is below the graph of [latex]h(x)[\/latex] when [latex]-1<x<3[\/latex] and is intersecting the graph of [latex]h(x)[\/latex] when [latex]x=-1, 3[\/latex], the solution of the inequality\u00a0[latex]g(x) \\leq h(x)[\/latex] is [latex][-1, 3][\/latex] or [latex]\\{x|-1 \\leq x \\leq 3\\}[\/latex].\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1108\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6.png\" alt=\"Graphs of g(x)=x^2-6 in orange, h(x)=2x-3 with shaded when -1&lt;x&lt;3\" width=\"246\" height=\"169\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6.png 386w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6-300x205.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6-65x44.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6-225x154.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Ex-8-Eq_Ineq_Graphs-6-350x239.png 350w\" sizes=\"auto, (max-width: 246px) 100vw, 246px\" \/>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\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-1082\">\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 Equations and 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":33,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Solving Equations and 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-1082","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":91,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1082","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":9,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1082\/revisions"}],"predecessor-version":[{"id":1405,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1082\/revisions\/1405"}],"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\/1082\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=1082"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1082"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1082"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=1082"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}