{"id":1096,"date":"2023-07-18T13:23:27","date_gmt":"2023-07-18T13:23:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=1096"},"modified":"2023-08-10T22:44:59","modified_gmt":"2023-08-10T22:44:59","slug":"summary-solving-equations-and-inequalities-using-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-solving-equations-and-inequalities-using-graphs-of-functions\/","title":{"raw":"Summary: Solving Equations and Inequalities using Graphs of Functions","rendered":"Summary: Solving Equations and Inequalities using Graphs of Functions"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>When a graph of [latex]y=f(x)[\/latex] is above the [latex]x[\/latex]-axis, [latex]f(x)&gt;0[\/latex].\r\nSo, the solution of the inequality\u00a0[latex]f(x)&gt;0[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is above the [latex]x[\/latex]-axis.<\/li>\r\n \t<li>When a graph of [latex]y=f(x)[\/latex] is on the [latex]x[\/latex]-axis, [latex]f(x)=0[\/latex].\r\nSo, the solution of the equation\u00a0[latex]f(x)=0[\/latex] is the [latex]x[\/latex] values of its [latex]x[\/latex]-intercepts.<\/li>\r\n \t<li>When a graph of [latex]y=f(x)[\/latex] is below the [latex]x[\/latex]-axis, [latex]f(x)&lt;0[\/latex].\r\nSo, the solution of the inequality\u00a0[latex]f(x)&lt;0[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is below the [latex]x[\/latex]-axis.\r\nAlso, the solution of the inequality\u00a0[latex]f(x) \\geq g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is above or intersecting the graph of [latex]y=g(x)[\/latex].<\/li>\r\n \t<li>When a graph of [latex]y=f(x)[\/latex] is above the graph of [latex]y=g(x)[\/latex], [latex]f(x)&gt;g(x)[\/latex].\r\nSo, the solution of the inequality\u00a0[latex]f(x)&gt;g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is above the graph of [latex]y=g(x)[\/latex].<\/li>\r\n \t<li>When a graph of [latex]y=f(x)[\/latex] is intersecting the graph of [latex]y=g(x)[\/latex], [latex]f(x)=g(x)[\/latex].\r\nSo, the solution of the equation\u00a0[latex]f(x)=g(x)[\/latex] is the [latex]x[\/latex] values of the intersecting points of [latex]y=f(x)[\/latex] and [latex]y=g(x)[\/latex].<\/li>\r\n \t<li>When a graph of [latex]y=f(x)[\/latex] is below the graph of [latex]y=g(x)[\/latex], [latex]f(x)&lt;g(x)[\/latex].\r\nSo, the solution of the inequality\u00a0[latex]f(x)&lt;g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is below the graph of [latex]y=g(x)[\/latex].\r\nAlso, the solution of the inequality\u00a0[latex]f(x) \\leq g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is below or intersecting the graph of [latex]y=g(x)[\/latex].<\/li>\r\n<\/ul>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>When a graph of [latex]y=f(x)[\/latex] is above the [latex]x[\/latex]-axis, [latex]f(x)>0[\/latex].<br \/>\nSo, the solution of the inequality\u00a0[latex]f(x)>0[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is above the [latex]x[\/latex]-axis.<\/li>\n<li>When a graph of [latex]y=f(x)[\/latex] is on the [latex]x[\/latex]-axis, [latex]f(x)=0[\/latex].<br \/>\nSo, the solution of the equation\u00a0[latex]f(x)=0[\/latex] is the [latex]x[\/latex] values of its [latex]x[\/latex]-intercepts.<\/li>\n<li>When a graph of [latex]y=f(x)[\/latex] is below the [latex]x[\/latex]-axis, [latex]f(x)<0[\/latex].\nSo, the solution of the inequality\u00a0[latex]f(x)<0[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is below the [latex]x[\/latex]-axis.\nAlso, the solution of the inequality\u00a0[latex]f(x) \\geq g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is above or intersecting the graph of [latex]y=g(x)[\/latex].<\/li>\n<li>When a graph of [latex]y=f(x)[\/latex] is above the graph of [latex]y=g(x)[\/latex], [latex]f(x)>g(x)[\/latex].<br \/>\nSo, the solution of the inequality\u00a0[latex]f(x)>g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is above the graph of [latex]y=g(x)[\/latex].<\/li>\n<li>When a graph of [latex]y=f(x)[\/latex] is intersecting the graph of [latex]y=g(x)[\/latex], [latex]f(x)=g(x)[\/latex].<br \/>\nSo, the solution of the equation\u00a0[latex]f(x)=g(x)[\/latex] is the [latex]x[\/latex] values of the intersecting points of [latex]y=f(x)[\/latex] and [latex]y=g(x)[\/latex].<\/li>\n<li>When a graph of [latex]y=f(x)[\/latex] is below the graph of [latex]y=g(x)[\/latex], [latex]f(x)<g(x)[\/latex].\nSo, the solution of the inequality\u00a0[latex]f(x)<g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is below the graph of [latex]y=g(x)[\/latex].\nAlso, the solution of the inequality\u00a0[latex]f(x) \\leq g(x)[\/latex] is the interval of [latex]x[\/latex] where the\u00a0graph of [latex]y=f(x)[\/latex] is below or intersecting the graph of [latex]y=g(x)[\/latex].<\/li>\n<\/ul>\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-1096\">\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>Summary: 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":34,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Summary: 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-1096","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":91,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1096","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":2,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1096\/revisions"}],"predecessor-version":[{"id":1099,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1096\/revisions\/1099"}],"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\/1096\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=1096"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1096"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1096"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=1096"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}