{"id":153909,"date":"2024-10-13T19:14:30","date_gmt":"2024-10-13T19:14:30","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=153909"},"modified":"2024-10-13T19:14:32","modified_gmt":"2024-10-13T19:14:32","slug":"predicate","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/13\/predicate\/","title":{"rendered":"Predicate"},"content":{"rendered":"\n<p>Predicate: 1 \u2264 x2 \u2264 9, domain: R Predicate: 1 \u2264 x2 \u2264 9, domain: Z<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To analyze the predicate ( 1 \\leq x^2 \\leq 9 ) under the domains of the real numbers ( \\mathbb{R} ) and the integers ( \\mathbb{Z} ), we need to consider the implications of the inequality for each domain.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Domain: ( \\mathbb{R} )<\/h3>\n\n\n\n<p>In the real number domain, the inequality ( 1 \\leq x^2 \\leq 9 ) can be separated into two parts:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>From ( 1 \\leq x^2 )<\/strong>: This implies that ( x^2 ) is at least 1. Therefore, ( x ) must be either greater than or equal to 1 or less than or equal to -1. In mathematical terms:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( x \\geq 1 ) or ( x \\leq -1 ).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>From ( x^2 \\leq 9 )<\/strong>: This implies that ( x^2 ) is at most 9. Hence, ( x ) must lie between -3 and 3. Formally, this can be expressed as:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( -3 \\leq x \\leq 3 ).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Combining the Results<\/h3>\n\n\n\n<p>Now, combining the results from the two inequalities:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For ( 1 \\leq x^2 ): ( x \\leq -1 ) or ( x \\geq 1 ).<\/li>\n\n\n\n<li>For ( x^2 \\leq 9 ): ( -3 \\leq x \\leq 3 ).<\/li>\n<\/ul>\n\n\n\n<p>Thus, the valid intervals for ( x ) are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( -3 \\leq x \\leq -1 )<\/li>\n\n\n\n<li>( 1 \\leq x \\leq 3 )<\/li>\n<\/ul>\n\n\n\n<p>This gives us the solution set ( x \\in [-3, -1] \\cup [1, 3] ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Domain: ( \\mathbb{Z} )<\/h3>\n\n\n\n<p>When considering the predicate in the integer domain ( \\mathbb{Z} ), we can evaluate the integers that satisfy the combined conditions:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>From ( -3 \\leq x \\leq -1 ), the integer solutions are ( -3, -2, -1 ).<\/li>\n\n\n\n<li>From ( 1 \\leq x \\leq 3 ), the integer solutions are ( 1, 2, 3 ).<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>In ( \\mathbb{R} ), the solution is the interval ( [-3, -1] \\cup [1, 3] ). In ( \\mathbb{Z} ), the valid integer solutions are ( {-3, -2, -1, 1, 2, 3} ).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Predicate: 1 \u2264 x2 \u2264 9, domain: R Predicate: 1 \u2264 x2 \u2264 9, domain: Z The Correct Answer and Explanation is : To analyze the predicate ( 1 \\leq x^2 \\leq 9 ) under the domains of the real numbers ( \\mathbb{R} ) and the integers ( \\mathbb{Z} ), we need to consider the [&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-153909","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/153909","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=153909"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/153909\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=153909"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=153909"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=153909"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}