{"id":160454,"date":"2024-11-02T21:32:52","date_gmt":"2024-11-02T21:32:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=160454"},"modified":"2024-11-02T21:32:55","modified_gmt":"2024-11-02T21:32:55","slug":"if-the-replacement-set-is-the-set-of-integers-find-the-solution-set-for-the-inequality","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/02\/if-the-replacement-set-is-the-set-of-integers-find-the-solution-set-for-the-inequality\/","title":{"rendered":"If the replacement set is the set of integers, find the solution set for the inequality"},"content":{"rendered":"\n

If the replacement set is the set of integers, find the solution set for the inequality:
x\u22129<\u221215<\/p>\n\n\n\n

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

To solve the inequality ( x – 9 < -15 ) for ( x ) with the replacement set being the set of integers, we can follow these steps:<\/p>\n\n\n\n

    \n
  1. Isolate ( x ):<\/strong>
    We want to isolate ( x ) on one side of the inequality. To do this, add 9 to both sides:
    [
    x – 9 + 9 < -15 + 9
    ]
    Simplifying this gives:
    [
    x < -6
    ]<\/li>\n\n\n\n
  2. Identify the Solution Set with Integers:<\/strong>
    Now, we have the inequality ( x < -6 ). Since the replacement set is the set of integers, we are only interested in integer values of ( x ) that satisfy this inequality. The integer values less than -6 are:
    [
    -7, -8, -9, -10, \\ldots
    ] Therefore, the solution set is:
    [
    { x \\in \\mathbb{Z} \\,|\\, x < -6 } = {-7, -8, -9, -10, \\ldots}
    ]<\/li>\n<\/ol>\n\n\n\n

    Explanation of Solution<\/h3>\n\n\n\n

    This problem involves solving a basic linear inequality and finding the integer values that satisfy it. An inequality, unlike an equation, represents a range of possible solutions rather than a single solution. The goal here is to isolate the variable ( x ) on one side, allowing us to interpret the range of values it can take.<\/p>\n\n\n\n

    Starting with ( x – 9 < -15 ), we add 9 to both sides to isolate ( x ). Adding or subtracting the same value on both sides of an inequality preserves the inequality\u2019s direction. After this step, we get ( x < -6 ), which tells us that ( x ) must be any value smaller than -6.<\/p>\n\n\n\n

    Since we are limited to the set of integers, we ignore any decimal or fractional values. The integers less than -6 begin at -7 and continue indefinitely in the negative direction, so our solution set includes all integers less than -6, which we can write as ( {-7, -8, -9, -10, \\ldots} ).<\/p>\n\n\n\n

    This type of problem is foundational in algebra and helps build skills for understanding inequalities and solution sets, especially when restrictions are given, such as working within the integers.<\/p>\n","protected":false},"excerpt":{"rendered":"

    If the replacement set is the set of integers, find the solution set for the inequality:x\u22129<\u221215 The Correct Answer and Explanation is: To solve the inequality ( x – 9 < -15 ) for ( x ) with the replacement set being the set of integers, we can follow these steps: Explanation of Solution This […]<\/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-160454","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/160454","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=160454"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/160454\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=160454"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=160454"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=160454"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}