{"id":2650,"date":"2025-08-13T18:21:20","date_gmt":"2025-08-13T18:21:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2650"},"modified":"2026-03-22T18:03:16","modified_gmt":"2026-03-22T18:03:16","slug":"systems-of-equations-and-inequalities-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-equations-and-inequalities-background-youll-need-3\/","title":{"raw":"Systems of Equations and Inequalities: Background You'll Need 3","rendered":"Systems of Equations and Inequalities: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph a linear inequality from standard and slope-intercept form<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>graphs of linear inequalities<\/h3>\r\nInequalities shade a half-plane.\r\n<ul>\r\n \t<li>Use a solid boundary line for [latex]\\le[\/latex] or [latex]\\ge[\/latex]\r\n<img class=\"alignnone wp-image-4041\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM.png\" alt=\"\" width=\"289\" height=\"275\" \/><\/li>\r\n \t<li>Use a dashed for [latex]&lt;[\/latex] or [latex]&gt;[\/latex].\r\n<img class=\"alignnone wp-image-4042\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM.png\" alt=\"\" width=\"292\" height=\"284\" \/><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Graphing Linear Inequalities in Slope-Intercept Form<\/h3>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]y\\le -\\tfrac{1}{2}x+3[\/latex].[reveal-answer q=\"241512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"241512\"]\r\n<ul>\r\n \t<li><strong><strong>Step 1: Draw the boundary line<\/strong><\/strong>Graph the line [latex]y=-\\frac{1}{2}x+3[\/latex]Because the inequality is [latex]\\leq[\/latex] the boundary is solid.<img class=\"alignnone wp-image-4043\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM.png\" alt=\"\" width=\"371\" height=\"370\" \/><\/li>\r\n \t<li><strong>Step 2: Test a point not on the line.<\/strong>Test [latex](0,0)[\/latex]Is the inequality true? [latex]\\begin{align}\r\n0 &amp;\\le -\\frac{1}{2}(0)+3 \\\\\r\n0 &amp;\\le 0+3 &amp;&amp; \\text{since } -\\frac{1}{2}\\cdot 0=0 \\\\\r\n0 &amp;\\le 3 &amp;&amp; \\text{simplify}\r\n\\end{align}[\/latex]This is true. Shade the half-plane that contains [latex](0,0)[\/latex]<img class=\"alignnone wp-image-4044\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM.png\" alt=\"\" width=\"403\" height=\"403\" \/>[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312225[\/ohm_question]<\/section>\r\n<h3>Graphing Linear Inequalities in Slope-Intercept Form<\/h3>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"4021\" data-end=\"4052\">Graph [latex]2x+3y&gt;6[\/latex].<\/p>\r\n<p data-start=\"4021\" data-end=\"4052\">[reveal-answer q=\"958824\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"958824\"]<\/p>\r\n\r\n<ul data-start=\"4053\" data-end=\"4226\">\r\n \t<li data-start=\"4053\" data-end=\"4121\">\r\n<p data-start=\"4055\" data-end=\"4121\">Convert [latex]3y&gt;6-2x[\/latex] to slope-intercept form:\r\n\\begin{align}\r\n3y &amp;&gt; 6 - 2x \\\\\r\ny &amp;&gt; \\frac{6 - 2x}{3} &amp;&amp; \\text{divide both sides by }3\\\\\r\ny &amp;&gt; \\frac{6}{3} - \\frac{2x}{3} &amp;&amp; \\text{split the fraction} \\\\\r\ny &amp;&gt; 2 - \\frac{2}{3}x &amp;&amp; \\text{simplify} \\\\\r\ny &amp;&gt; -\\frac{2}{3}x + 2 &amp;&amp; \\text{reorder to slope\u2013intercept form}\r\n\\end{align}<\/p>\r\n<\/li>\r\n \t<li data-start=\"4122\" data-end=\"4158\">\r\n<p data-start=\"4124\" data-end=\"4158\">Graph the boundary line using a dashed line (because [latex]&gt;[\/latex]).\r\n<img class=\"alignnone wp-image-4046\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM.png\" alt=\"\" width=\"310\" height=\"309\" \/><\/p>\r\n<\/li>\r\n \t<li data-start=\"4159\" data-end=\"4226\">\r\n<p data-start=\"4161\" data-end=\"4226\">Test [latex](0,0)[\/latex]:\r\n[latex]\r\n\\begin{align}\r\n0 &amp;&gt; -\\frac{2}{3}(0) + 2 \\\\\r\n0 &amp;&gt; 0 + 2 &amp;&amp; \\text{since } -\\tfrac{2}{3}\\cdot 0=0 \\\\\r\n0 &amp;&gt; 2 &amp;&amp; \\text{simplify}\r\n\\end{align}\r\n[\/latex]<\/p>\r\nThis is false so [latex](0,0)[\/latex] is\u00a0<strong>not<\/strong> in the solution. Shade the other side.\r\n<img class=\"alignnone wp-image-4047\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM.png\" alt=\"\" width=\"308\" height=\"308\" \/>[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">If you multiply or divide an inequality by a negative number, the inequality symbol will flip.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312226[\/ohm_question]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<ul data-start=\"4507\" data-end=\"4749\">\r\n \t<li data-start=\"4507\" data-end=\"4605\">\r\n<p data-start=\"4509\" data-end=\"4605\">If the line already passes through (0,0), pick a different easy test point, like (0,1) or (1,0).<\/p>\r\n<\/li>\r\n \t<li data-start=\"4606\" data-end=\"4749\">\r\n<p data-start=\"4608\" data-end=\"4749\">Converting to [latex]y=mx+b[\/latex] makes shading direction clearer (shade <strong data-start=\"4683\" data-end=\"4692\">above<\/strong> for [latex]y&gt;[\/latex], <strong data-start=\"4716\" data-end=\"4725\">below<\/strong> for [latex]y&lt;[\/latex]).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph a linear inequality from standard and slope-intercept form<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>graphs of linear inequalities<\/h3>\n<p>Inequalities shade a half-plane.<\/p>\n<ul>\n<li>Use a solid boundary line for [latex]\\le[\/latex] or [latex]\\ge[\/latex]<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4041\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM.png\" alt=\"\" width=\"289\" height=\"275\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM.png 860w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM-300x285.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM-768x730.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM-65x62.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM-225x214.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194341\/Screenshot-2025-09-16-at-12.43.12%E2%80%AFPM-350x333.png 350w\" sizes=\"(max-width: 289px) 100vw, 289px\" \/><\/li>\n<li>Use a dashed for [latex]<[\/latex] or [latex]>[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4042\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM.png\" alt=\"\" width=\"292\" height=\"284\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM.png 874w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM-300x292.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM-768x747.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM-65x63.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM-225x219.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194342\/Screenshot-2025-09-16-at-12.43.27%E2%80%AFPM-350x340.png 350w\" sizes=\"(max-width: 292px) 100vw, 292px\" \/><\/li>\n<\/ul>\n<\/section>\n<h3>Graphing Linear Inequalities in Slope-Intercept Form<\/h3>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]y\\le -\\tfrac{1}{2}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q241512\">Show Solution<\/button><\/p>\n<div id=\"q241512\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li><strong><strong>Step 1: Draw the boundary line<\/strong><\/strong>Graph the line [latex]y=-\\frac{1}{2}x+3[\/latex]Because the inequality is [latex]\\leq[\/latex] the boundary is solid.<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4043\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM.png\" alt=\"\" width=\"371\" height=\"370\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM.png 824w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM-768x766.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM-225x224.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16194820\/Screenshot-2025-09-16-at-12.48.11%E2%80%AFPM-350x349.png 350w\" sizes=\"(max-width: 371px) 100vw, 371px\" \/><\/li>\n<li><strong>Step 2: Test a point not on the line.<\/strong>Test [latex](0,0)[\/latex]Is the inequality true? [latex]\\begin{align}  0 &\\le -\\frac{1}{2}(0)+3 \\\\  0 &\\le 0+3 && \\text{since } -\\frac{1}{2}\\cdot 0=0 \\\\  0 &\\le 3 && \\text{simplify}  \\end{align}[\/latex]This is true. Shade the half-plane that contains [latex](0,0)[\/latex]<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4044\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM.png\" alt=\"\" width=\"403\" height=\"403\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM.png 834w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM-768x768.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16195114\/Screenshot-2025-09-16-at-12.51.05%E2%80%AFPM-350x350.png 350w\" sizes=\"(max-width: 403px) 100vw, 403px\" \/><\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312225\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312225&theme=lumen&iframe_resize_id=ohm312225&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Graphing Linear Inequalities in Slope-Intercept Form<\/h3>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"4021\" data-end=\"4052\">Graph [latex]2x+3y>6[\/latex].<\/p>\n<p data-start=\"4021\" data-end=\"4052\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q958824\">Show Solution<\/button><\/p>\n<div id=\"q958824\" class=\"hidden-answer\" style=\"display: none\">\n<ul data-start=\"4053\" data-end=\"4226\">\n<li data-start=\"4053\" data-end=\"4121\">\n<p data-start=\"4055\" data-end=\"4121\">Convert [latex]3y>6-2x[\/latex] to slope-intercept form:<br \/>\n\\begin{align}<br \/>\n3y &amp;&gt; 6 &#8211; 2x \\\\<br \/>\ny &amp;&gt; \\frac{6 &#8211; 2x}{3} &amp;&amp; \\text{divide both sides by }3\\\\<br \/>\ny &amp;&gt; \\frac{6}{3} &#8211; \\frac{2x}{3} &amp;&amp; \\text{split the fraction} \\\\<br \/>\ny &amp;&gt; 2 &#8211; \\frac{2}{3}x &amp;&amp; \\text{simplify} \\\\<br \/>\ny &amp;&gt; -\\frac{2}{3}x + 2 &amp;&amp; \\text{reorder to slope\u2013intercept form}<br \/>\n\\end{align}<\/p>\n<\/li>\n<li data-start=\"4122\" data-end=\"4158\">\n<p data-start=\"4124\" data-end=\"4158\">Graph the boundary line using a dashed line (because [latex]>[\/latex]).<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4046\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM.png\" alt=\"\" width=\"310\" height=\"309\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM.png 824w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM-768x766.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM-225x224.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211447\/Screenshot-2025-09-16-at-2.14.32%E2%80%AFPM-350x349.png 350w\" sizes=\"(max-width: 310px) 100vw, 310px\" \/><\/p>\n<\/li>\n<li data-start=\"4159\" data-end=\"4226\">\n<p data-start=\"4161\" data-end=\"4226\">Test [latex](0,0)[\/latex]:<br \/>\n[latex]\\begin{align}  0 &> -\\frac{2}{3}(0) + 2 \\\\  0 &> 0 + 2 && \\text{since } -\\tfrac{2}{3}\\cdot 0=0 \\\\  0 &> 2 && \\text{simplify}  \\end{align}[\/latex]<\/p>\n<p>This is false so [latex](0,0)[\/latex] is\u00a0<strong>not<\/strong> in the solution. Shade the other side.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4047\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM.png\" alt=\"\" width=\"308\" height=\"308\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM.png 856w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM-768x768.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/16211526\/Screenshot-2025-09-16-at-2.15.15%E2%80%AFPM-350x350.png 350w\" sizes=\"(max-width: 308px) 100vw, 308px\" \/><\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">If you multiply or divide an inequality by a negative number, the inequality symbol will flip.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312226\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312226&theme=lumen&iframe_resize_id=ohm312226&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<ul data-start=\"4507\" data-end=\"4749\">\n<li data-start=\"4507\" data-end=\"4605\">\n<p data-start=\"4509\" data-end=\"4605\">If the line already passes through (0,0), pick a different easy test point, like (0,1) or (1,0).<\/p>\n<\/li>\n<li data-start=\"4606\" data-end=\"4749\">\n<p data-start=\"4608\" data-end=\"4749\">Converting to [latex]y=mx+b[\/latex] makes shading direction clearer (shade <strong data-start=\"4683\" data-end=\"4692\">above<\/strong> for [latex]y>[\/latex], <strong data-start=\"4716\" data-end=\"4725\">below<\/strong> for [latex]y<[\/latex]).<\/p>\n<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2650"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2650\/revisions"}],"predecessor-version":[{"id":5957,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2650\/revisions\/5957"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2650\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2650"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2650"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2650"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2650"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}