{"id":168341,"date":"2024-11-17T09:24:15","date_gmt":"2024-11-17T09:24:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=168341"},"modified":"2024-11-17T09:24:17","modified_gmt":"2024-11-17T09:24:17","slug":"predicate1-%e2%89%a4-x2-%e2%89%a4-4","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/17\/predicate1-%e2%89%a4-x2-%e2%89%a4-4\/","title":{"rendered":"Predicate:1 \u2264 x^2 \u2264 4,"},"content":{"rendered":"\n<p>Predicate:<br>1 \u2264 x^2 \u2264 4,<br>domain: R<\/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>The given predicate is:<\/p>\n\n\n\n<p>[ 1 \\leq x^2 \\leq 4 ]<\/p>\n\n\n\n<p><strong>Domain: R (Real Numbers)<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Interpretation:<\/h3>\n\n\n\n<p>The inequality expresses a condition on the value of ( x^2 ). The domain is the set of real numbers ( \\mathbb{R} ), meaning that ( x ) can take any real value.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Breaking down the inequality:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>First part of the inequality:<\/strong> ( 1 \\leq x^2 )<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>This means that ( x^2 ) must be greater than or equal to 1.<\/li>\n\n\n\n<li>Since ( x^2 \\geq 0 ) for all real ( x ) (because squaring any real number results in a non-negative number), this condition restricts ( x ) to values such that the square of ( x ) is at least 1. This can happen in two cases:\n<ul class=\"wp-block-list\">\n<li>( x \\geq 1 ) or<\/li>\n\n\n\n<li>( x \\leq -1 )<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Second part of the inequality:<\/strong> ( x^2 \\leq 4 )<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>This means that ( x^2 ) must be less than or equal to 4.<\/li>\n\n\n\n<li>For the square of ( x ) to be less than or equal to 4, ( x ) must lie between ( -2 ) and ( 2 ), inclusive. That is:\n<ul class=\"wp-block-list\">\n<li>( -2 \\leq x \\leq 2 )<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Combining the two inequalities:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From the first condition (( 1 \\leq x^2 )), we know that ( x ) must be outside the range ( (-1, 1) ), i.e., ( x \\geq 1 ) or ( x \\leq -1 ).<\/li>\n\n\n\n<li>From the second condition (( x^2 \\leq 4 )), we know that ( x ) must lie within the range ( -2 \\leq x \\leq 2 ).<\/li>\n<\/ul>\n\n\n\n<p>Thus, combining both conditions, we conclude that ( x ) must satisfy:<\/p>\n\n\n\n<p>[ x \\in [-2, -1] \\cup [1, 2] ]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>The solution to the predicate ( 1 \\leq x^2 \\leq 4 ) in the domain of real numbers is:<\/p>\n\n\n\n<p>[ x \\in [-2, -1] \\cup [1, 2] ]<\/p>\n\n\n\n<p>This means that ( x ) can be any value in the intervals ( [-2, -1] ) or ( [1, 2] ).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Predicate:1 \u2264 x^2 \u2264 4,domain: R The Correct Answer and Explanation is: The given predicate is: [ 1 \\leq x^2 \\leq 4 ] Domain: R (Real Numbers) Interpretation: The inequality expresses a condition on the value of ( x^2 ). The domain is the set of real numbers ( \\mathbb{R} ), meaning that ( x [&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-168341","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168341","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=168341"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168341\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=168341"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=168341"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=168341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}