{"id":161155,"date":"2024-11-04T12:30:11","date_gmt":"2024-11-04T12:30:11","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=161155"},"modified":"2024-11-04T12:30:15","modified_gmt":"2024-11-04T12:30:15","slug":"what-is-the-solution-if-any-to-the-inequality-%e2%88%a33x%e2%88%a3%e2%89%a50","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/04\/what-is-the-solution-if-any-to-the-inequality-%e2%88%a33x%e2%88%a3%e2%89%a50\/","title":{"rendered":"What is the solution, if any, to the inequality \u22233x\u2223\u22650"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">What is the solution, if any, to the inequality \u22233x\u2223\u22650?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><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 class=\"wp-block-paragraph\">To solve the inequality (|3x| \\geq 0), we need to analyze the absolute value expression (|3x|).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The absolute value of a number is always non-negative, meaning it can either be zero or positive. Therefore, (|3x| \\geq 0) is always true for any real number (x). Let\u2019s break this down further:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Understanding Absolute Value<\/strong>: The expression (|3x|) represents the distance of (3x) from zero on the number line. Regardless of whether (3x) is positive, negative, or zero, the absolute value will either equal zero or a positive value.<\/li>\n\n\n\n<li><strong>Finding the Zero Point<\/strong>: The only time (|3x| = 0) is when (3x = 0). This occurs when (x = 0). Thus, (|3x|) is equal to zero at this point.<\/li>\n\n\n\n<li><strong>Positive Values<\/strong>: For all other values of (x) (whether positive or negative), (|3x|) is positive. For example:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If (x > 0), then (3x > 0) and (|3x| = 3x > 0).<\/li>\n\n\n\n<li>If (x &lt; 0), then (3x &lt; 0) and (|3x| = -3x > 0).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Conclusion<\/strong>: Since (|3x|) can only be zero or positive, it is always greater than or equal to zero for any real number (x). This leads us to conclude that the inequality (|3x| \\geq 0) holds true for all real numbers.<\/li>\n\n\n\n<li><strong>Final Solution<\/strong>: Therefore, the solution set to the inequality (|3x| \\geq 0) is all real numbers. In interval notation, this is expressed as ( (-\\infty, \\infty) ).<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">In summary, because the absolute value of any expression is either zero or positive, and since the expression (|3x|) satisfies this condition for all values of (x), we conclude that the inequality (|3x| \\geq 0) is valid for every real number (x).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is the solution, if any, to the inequality \u22233x\u2223\u22650? The Correct Answer and Explanation is: To solve the inequality (|3x| \\geq 0), we need to analyze the absolute value expression (|3x|). The absolute value of a number is always non-negative, meaning it can either be zero or positive. Therefore, (|3x| \\geq 0) is always [&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-161155","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/161155","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=161155"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/161155\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=161155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=161155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=161155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}