{"id":157882,"date":"2024-10-23T18:26:28","date_gmt":"2024-10-23T18:26:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=157882"},"modified":"2024-10-23T18:26:31","modified_gmt":"2024-10-23T18:26:31","slug":"determine-the-solution-to-the-inequality","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/23\/determine-the-solution-to-the-inequality\/","title":{"rendered":"Determine the solution to the inequality."},"content":{"rendered":"\n<p>Determine the solution to the inequality. |4x \u2212 4| \u2265 8 x \u2264 \u22121 or x \u2265 3 x \u2264 \u22122 or x \u2265 3 x \u2264 \u22123 or x \u2265 4 x \u2264 \u22124 or x \u2265 4<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve the inequality (|4x &#8211; 4| \\geq 8), we will break it down into two separate cases, because the absolute value inequality can be rewritten without the absolute value as two separate inequalities:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Case 1:<\/strong> (4x &#8211; 4 \\geq 8)<\/li>\n\n\n\n<li><strong>Case 2:<\/strong> (4x &#8211; 4 \\leq -8)<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Solving Case 1: (4x &#8211; 4 \\geq 8)<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Start by isolating (4x):<br>[<br>4x &#8211; 4 \\geq 8<br>]<br>[<br>4x \\geq 12<br>]<\/li>\n\n\n\n<li>Divide both sides by 4:<br>[<br>x \\geq 3<br>]<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Solving Case 2: (4x &#8211; 4 \\leq -8)<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Isolate (4x):<br>[<br>4x &#8211; 4 \\leq -8<br>]<br>[<br>4x \\leq -4<br>]<\/li>\n\n\n\n<li>Divide both sides by 4:<br>[<br>x \\leq -1<br>]<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Combining the Results<\/h3>\n\n\n\n<p>From Case 1, we found (x \\geq 3) and from Case 2, we found (x \\leq -1). Therefore, the complete solution to the inequality (|4x &#8211; 4| \\geq 8) can be expressed in interval notation as:<\/p>\n\n\n\n<p>[<br>x \\leq -1 \\quad \\text{or} \\quad x \\geq 3<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>Thus, the solution is (x \\leq -1) or (x \\geq 3).<\/p>\n\n\n\n<p>This means that any value of (x) less than or equal to (-1) or greater than or equal to (3) satisfies the original inequality. The critical points that define these intervals are (-1) and (3). Graphically, on the number line, the solution would cover all numbers to the left of (-1) (inclusive) and all numbers to the right of (3) (inclusive).<\/p>\n\n\n\n<p>The provided multiple-choice options were:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(x \\leq -1) or (x \\geq 3)<\/li>\n\n\n\n<li>(x \\leq -2) or (x \\geq 3)<\/li>\n\n\n\n<li>(x \\leq -3) or (x \\geq 4)<\/li>\n\n\n\n<li>(x \\leq -4) or (x \\geq 4)<\/li>\n<\/ul>\n\n\n\n<p>The correct answer is:<br><strong>(x \\leq -1) or (x \\geq 3)<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Determine the solution to the inequality. |4x \u2212 4| \u2265 8 x \u2264 \u22121 or x \u2265 3 x \u2264 \u22122 or x \u2265 3 x \u2264 \u22123 or x \u2265 4 x \u2264 \u22124 or x \u2265 4 The Correct Answer and Explanation is: To solve the inequality (|4x &#8211; 4| \\geq 8), we [&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-157882","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157882","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=157882"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157882\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=157882"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=157882"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=157882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}