{"id":256493,"date":"2025-07-16T09:47:20","date_gmt":"2025-07-16T09:47:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=256493"},"modified":"2025-07-16T09:47:28","modified_gmt":"2025-07-16T09:47:28","slug":"solve-the-absolute-value-equation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/16\/solve-the-absolute-value-equation\/","title":{"rendered":"Solve the absolute value equation"},"content":{"rendered":"\n<p>Solve the absolute value equation. 4- 10x, =5 Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is (Use a comma to separate answers as needed.) B. The solution set is all real numbers. C. The solution is the empty set.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>We are given the absolute value equation:<br><strong>|4 &#8211; 10x| = 5<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand the definition of absolute value<\/h3>\n\n\n\n<p>The absolute value of a number is its distance from 0 on the number line, so:<br><strong>|A| = B<\/strong> means <strong>A = B<\/strong> or <strong>A = -B<\/strong>, as long as B is positive.<\/p>\n\n\n\n<p>So, from <strong>|4 &#8211; 10x| = 5<\/strong>, we write two equations:<\/p>\n\n\n\n<p><strong>1) 4 &#8211; 10x = 5<\/strong><br><strong>2) 4 &#8211; 10x = -5<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Solve each equation<\/h3>\n\n\n\n<p><strong>Equation 1: 4 &#8211; 10x = 5<\/strong><br>Subtract 4 from both sides:<br><strong>-10x = 1<\/strong><br>Divide both sides by -10:<br><strong>x = -1\/10<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Equation 2: 4 &#8211; 10x = -5<\/strong><br>Subtract 4 from both sides:<br><strong>-10x = -9<\/strong><br>Divide both sides by -10:<br><strong>x = 9\/10<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: State the solution set<\/h3>\n\n\n\n<p>We found two values that make the original equation true:<br><strong>x = -1\/10<\/strong> and <strong>x = 9\/10<\/strong><\/p>\n\n\n\n<p>So the solution set is:<br><strong>A. The solution set is -1\/10, 9\/10<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation <\/h3>\n\n\n\n<p>This problem involves solving an absolute value equation. The absolute value expression <strong>|4 &#8211; 10x|<\/strong> represents how far the quantity <strong>4 &#8211; 10x<\/strong> is from zero, regardless of direction. The equation says this distance equals 5, which means the inside expression could be either <strong>5<\/strong> or <strong>-5<\/strong>, because both are 5 units from zero.<\/p>\n\n\n\n<p>To solve the equation, we break it into two simpler linear equations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>One assumes the quantity inside the absolute value is positive: <strong>4 &#8211; 10x = 5<\/strong><\/li>\n\n\n\n<li>The other assumes it&#8217;s negative: <strong>4 &#8211; 10x = -5<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Solving both equations gives us two possible x-values. For the first, subtract 4 from both sides to get <strong>-10x = 1<\/strong>, then divide by -10 to find <strong>x = -1\/10<\/strong>. For the second, subtract 4 to get <strong>-10x = -9<\/strong>, then divide by -10 to find <strong>x = 9\/10<\/strong>.<\/p>\n\n\n\n<p>Since both solutions are valid, the solution set consists of these two values. Therefore, the correct answer is:<br><strong>A. The solution set is -1\/10, 9\/10<\/strong><\/p>\n\n\n\n<p>This process demonstrates a basic principle in algebra: absolute value equations usually lead to two separate equations, which must be solved individually. Always check for extraneous solutions in more complex cases, but here, both solutions work.<\/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-banner6-634.jpeg\" alt=\"\" class=\"wp-image-256494\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Solve the absolute value equation. 4- 10x, =5 Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is (Use a comma to separate answers as needed.) B. The solution set is all real numbers. C. The solution is the empty set. The Correct Answer and Explanation [&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-256493","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/256493","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=256493"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/256493\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=256493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=256493"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=256493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}