{"id":159196,"date":"2024-10-31T13:52:34","date_gmt":"2024-10-31T13:52:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=159196"},"modified":"2024-10-31T13:52:37","modified_gmt":"2024-10-31T13:52:37","slug":"which-inequality-is-true-for-all-real-numbers","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/31\/which-inequality-is-true-for-all-real-numbers\/","title":{"rendered":"Which inequality is true for all real numbers"},"content":{"rendered":"\n

Which inequality is true for all real numbers? Check all that apply.
A. a\u2212b\u2265c\u2265b B. ac\u2265bc C. If a\u2264b, then a+c\u2264b+c D. If c>d, then a\u2212c\u2264b\u2212c E. If a\u2265b, then a\u2212c\u2265b\u2212c Note: The exact inequalities in the<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To determine which inequalities are true for all real numbers, let’s analyze each option one by one.<\/p>\n\n\n\n

A. ( a – b \\geq c \\geq b )<\/h3>\n\n\n\n

This statement can be broken down into two parts: ( a – b \\geq c ) and ( c \\geq b ).<\/p>\n\n\n\n

    \n
  1. For ( a – b \\geq c ) to hold, it is not universally true because if we choose values such as ( a = 1, b = 2, c = 3 ), then ( 1 – 2 \\geq 3 ) is false.<\/li>\n\n\n\n
  2. Similarly, for ( c \\geq b ), this does not hold for all real numbers. Thus, this inequality is not true for all real numbers<\/strong>.<\/li>\n<\/ol>\n\n\n\n

    B. ( ac \\geq bc )<\/h3>\n\n\n\n

    For this inequality to hold for all ( a, b, c ), we can consider the case where ( c = 0 ). In this case, both sides equal zero ((0 \\geq 0)). However, if ( c ) is positive and ( a < b ), the inequality becomes ( ac < bc ), which is not valid. If ( c ) is negative and ( a < b ), the inequality becomes ( ac > bc ). Thus, this inequality is not true for all real numbers<\/strong>.<\/p>\n\n\n\n

    C. If ( a \\leq b ), then ( a + c \\leq b + c )<\/h3>\n\n\n\n

    This statement is true. If we add the same number ( c ) to both sides of the inequality ( a \\leq b ), the direction of the inequality remains unchanged. Therefore, this inequality is true for all real numbers<\/strong>.<\/p>\n\n\n\n

    D. If ( c > d ), then ( a – c \\leq b – c )<\/h3>\n\n\n\n

    Rearranging gives ( a \\leq b ). This is not guaranteed to hold for all ( a ) and ( b ). For example, if ( a = 5, b = 3, c = 4, d = 2 ), the inequality ( 5 – 4 \\leq 3 – 4 ) translates to ( 1 \\leq -1 ), which is false. Thus, this inequality is not true for all real numbers<\/strong>.<\/p>\n\n\n\n

    E. If ( a \\geq b ), then ( a – c \\geq b – c )<\/h3>\n\n\n\n

    Similar to option C, if we subtract ( c ) from both sides of the inequality ( a \\geq b ), the inequality remains valid. Hence, this inequality is true for all real numbers<\/strong>.<\/p>\n\n\n\n

    Conclusion<\/h3>\n\n\n\n

    The inequalities that hold true for all real numbers are C<\/strong> and E<\/strong>. These properties are rooted in the basic properties of inequalities, ensuring that adding or subtracting the same value does not affect the relationship between the numbers involved.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Which inequality is true for all real numbers? Check all that apply.A. a\u2212b\u2265c\u2265b B. ac\u2265bc C. If a\u2264b, then a+c\u2264b+c D. If c>d, then a\u2212c\u2264b\u2212c E. If a\u2265b, then a\u2212c\u2265b\u2212c Note: The exact inequalities in the The Correct Answer and Explanation is : To determine which inequalities are true for all real numbers, let’s analyze […]<\/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-159196","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/159196","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=159196"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/159196\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=159196"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=159196"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=159196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}