{"id":258334,"date":"2025-07-18T06:17:00","date_gmt":"2025-07-18T06:17:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=258334"},"modified":"2025-07-18T06:17:03","modified_gmt":"2025-07-18T06:17:03","slug":"graph-the-solution-to-the-following-inequality-on-the-number-line","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/18\/graph-the-solution-to-the-following-inequality-on-the-number-line\/","title":{"rendered":"Graph the solution to the following inequality on the number line."},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">Graph the solution to the following inequality on the number line.<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-518.png\" alt=\"\" class=\"wp-image-258335\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Here is the graph of the solution to the inequality.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>To find the solution for the quadratic inequality x\u00b2 + 10x &lt; -24, we first need to rearrange it so that one side is zero. We can accomplish this by adding 24 to both sides of the inequality, which gives us:<\/p>\n\n\n\n<p>x\u00b2 + 10x + 24 &lt; 0<\/p>\n\n\n\n<p>The next step is to find the critical points of the inequality. We do this by solving the corresponding equation, x\u00b2 + 10x + 24 = 0. This quadratic equation can be solved by factoring. We need to find two numbers that multiply to 24 and add up to 10. The numbers that fit this criteria are 4 and 6. This allows us to factor the equation as:<\/p>\n\n\n\n<p>(x + 4)(x + 6) = 0<\/p>\n\n\n\n<p>The solutions, or roots, of this equation are x = -4 and x = -6. These are the critical points that divide the number line into three distinct intervals: (-\u221e, -6), (-6, -4), and (-4, \u221e).<\/p>\n\n\n\n<p>Now, we test a value from each interval in the inequality x\u00b2 + 10x + 24 &lt; 0 to see which interval contains the solutions.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Test a value less than -6 (e.g., x = -7):<\/strong><br>(-7)\u00b2 + 10(-7) + 24 = 49 &#8211; 70 + 24 = 3. Since 3 is not less than 0, this interval is not a solution.<\/li>\n\n\n\n<li><strong>Test a value between -6 and -4 (e.g., x = -5):<\/strong><br>(-5)\u00b2 + 10(-5) + 24 = 25 &#8211; 50 + 24 = -1. Since -1 is less than 0, this interval is the solution.<\/li>\n\n\n\n<li><strong>Test a value greater than -4 (e.g., x = 0):<\/strong><br>(0)\u00b2 + 10(0) + 24 = 24. Since 24 is not less than 0, this interval is not a solution.<\/li>\n<\/ol>\n\n\n\n<p>The solution is the interval (-6, -4), which means -6 &lt; x &lt; -4. To graph this, we place open circles at -6 and -4 to show that these endpoints are not included in the solution, as indicated by the strict inequality symbol (&lt;). Then, we shade the line segment between these two points to represent all the numbers that satisfy the inequality.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-902.jpeg\" alt=\"\" class=\"wp-image-258337\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Graph the solution to the following inequality on the number line. The Correct Answer and Explanation is: Here is the graph of the solution to the inequality. Explanation To find the solution for the quadratic inequality x\u00b2 + 10x &lt; -24, we first need to rearrange it so that one side is zero. We can [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-258334","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258334","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=258334"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258334\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=258334"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=258334"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=258334"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}