What is the solution to |x \u2212 6] \u2265 5?<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n The inequality given is ( |x – 6| \\geq 5 ).<\/p>\n\n\n\n The absolute value inequality ( |A| \\geq B ) means that the quantity inside the absolute value (denoted as ( A )) is either greater than or equal to ( B ), or less than or equal to ( -B ). In other words, if ( |A| \\geq B ), then: Solve for ( x ) by adding 6 to both sides: Solve for ( x ) by adding 6 to both sides: The solution to the inequality ( |x – 6| \\geq 5 ) is the union of the two inequalities: Thus, the solution to ( |x – 6| \\geq 5 ) is: The absolute value function measures the distance of a number from zero, regardless of direction. In this inequality, ( |x – 6| \\geq 5 ) indicates that the distance between ( x ) and 6 must be at least 5 units. This means ( x ) must either be 5 or more units to the right of 6 (i.e., ( x \\geq 11 )) or 5 or more units to the left of 6 (i.e., ( x \\leq 1 )). Therefore, the solution consists of all values of ( x ) in these two ranges, which are ( x \\leq 1 ) or ( x \\geq 11 ).<\/p>\n","protected":false},"excerpt":{"rendered":" What is the solution to |x \u2212 6] \u2265 5? The Correct Answer and Explanation is: The inequality given is ( |x – 6| \\geq 5 ). Step 1: Understanding the absolute value inequality The absolute value inequality ( |A| \\geq B ) means that the quantity inside the absolute value (denoted as ( A […]<\/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-167562","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/167562","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=167562"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/167562\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=167562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=167562"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=167562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Understanding the absolute value inequality<\/h3>\n\n\n\n
[
A \\geq B \\quad \\text{or} \\quad A \\leq -B
]
Thus, for our inequality ( |x – 6| \\geq 5 ), this translates to:
[
x – 6 \\geq 5 \\quad \\text{or} \\quad x – 6 \\leq -5
]<\/p>\n\n\n\nStep 2: Solve each inequality separately<\/h3>\n\n\n\n
Case 1: ( x – 6 \\geq 5 )<\/h4>\n\n\n\n
[
x \\geq 5 + 6
]
[
x \\geq 11
]<\/p>\n\n\n\nCase 2: ( x – 6 \\leq -5 )<\/h4>\n\n\n\n
[
x \\leq -5 + 6
]
[
x \\leq 1
]<\/p>\n\n\n\nStep 3: Combine the solutions<\/h3>\n\n\n\n
[
x \\geq 11 \\quad \\text{or} \\quad x \\leq 1
]<\/p>\n\n\n\nStep 4: Final solution<\/h3>\n\n\n\n
[
x \\leq 1 \\quad \\text{or} \\quad x \\geq 11
]
In interval notation, this is written as:
[
(-\\infty, 1] \\cup [11, \\infty)
]<\/p>\n\n\n\nExplanation<\/h3>\n\n\n\n