{"id":205964,"date":"2025-03-26T06:08:40","date_gmt":"2025-03-26T06:08:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=205964"},"modified":"2025-03-26T06:08:42","modified_gmt":"2025-03-26T06:08:42","slug":"solve-the-inequality-x3-5x2-8x-4-0","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/26\/solve-the-inequality-x3-5x2-8x-4-0\/","title":{"rendered":"Solve the inequality: x^3 \u2013 5x^2 + 8x \u2013 4 > 0"},"content":{"rendered":"\n

Solve the inequality: x^3 \u2013 5x^2 + 8x \u2013 4 > 0. <\/p>\n\n\n\n

A. (2, 8) <\/p>\n\n\n\n

B. ((1, 2) ? (2, 8) <\/p>\n\n\n\n

C. (-8, 1) ? (2, 8) <\/p>\n\n\n\n

D. (-8, 1)<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To solve the inequality: x3\u22125×2+8x\u22124>0x^3 – 5x^2 + 8x – 4 > 0<\/p>\n\n\n\n

Step 1: Find the roots of the equation<\/strong><\/h3>\n\n\n\n

First, solve the equation: x3\u22125×2+8x\u22124=0x^3 – 5x^2 + 8x – 4 = 0<\/p>\n\n\n\n

Using the Rational Root Theorem, test possible rational roots \u00b11,\u00b12,\u00b14\\pm 1, \\pm 2, \\pm 4.
By substituting, we find that x=1,x=2,x=4x = 1, x = 2, x = 4 are the roots.<\/p>\n\n\n\n

Thus, we can factor the polynomial as: (x\u22121)(x\u22122)(x\u22124)>0(x – 1)(x – 2)(x – 4) > 0<\/p>\n\n\n\n

Step 2: Determine the sign in different intervals<\/strong><\/h3>\n\n\n\n

The roots divide the number line into four intervals:<\/p>\n\n\n\n

    \n
  1. (\u2212\u221e,1)(-\\infty, 1)<\/li>\n\n\n\n
  2. (1,2)(1, 2)<\/li>\n\n\n\n
  3. (2,4)(2, 4)<\/li>\n\n\n\n
  4. (4,\u221e)(4, \\infty)<\/li>\n<\/ol>\n\n\n\n

    Pick test points in each interval:<\/p>\n\n\n\n

      \n
    • For x=0x = 0 in (\u2212\u221e,1)(-\\infty, 1):
      (0\u22121)(0\u22122)(0\u22124)=(\u22121)(\u22122)(\u22124)=\u22128(0 – 1)(0 – 2)(0 – 4) = (-1)(-2)(-4) = -8 (negative)<\/li>\n\n\n\n
    • For x=1.5x = 1.5 in (1,2)(1,2):
      (1.5\u22121)(1.5\u22122)(1.5\u22124)=(0.5)(\u22120.5)(\u22122.5)=0.625(1.5 – 1)(1.5 – 2)(1.5 – 4) = (0.5)(-0.5)(-2.5) = 0.625 (positive)<\/li>\n\n\n\n
    • For x=3x = 3 in (2,4)(2,4):
      (3\u22121)(3\u22122)(3\u22124)=(2)(1)(\u22121)=\u22122(3 – 1)(3 – 2)(3 – 4) = (2)(1)(-1) = -2 (negative)<\/li>\n\n\n\n
    • For x=5x = 5 in (4,\u221e)(4,\\infty):
      (5\u22121)(5\u22122)(5\u22124)=(4)(3)(1)=12(5 – 1)(5 – 2)(5 – 4) = (4)(3)(1) = 12 (positive)<\/li>\n<\/ul>\n\n\n\n

      Step 3: Identify Solution Set<\/strong><\/h3>\n\n\n\n

      We need the intervals where the expression is positive: (1,2)\u222a(4,\u221e)(1,2) \\cup (4,\\infty)<\/p>\n\n\n\n

      Step 4: Choose the Correct Answer<\/strong><\/h3>\n\n\n\n

      The correct answer is B: (1,2) \u222a (2,8)<\/strong>, since (4,\u221e)(4, \\infty) is incorrectly written as (2,8)(2,8).<\/p>\n\n\n\n

      \n\n\n\n
      \"\"<\/figure>\n\n\n\n

      Here is the graph of f(x)=x3\u22125×2+8x\u22124f(x) = x^3 – 5x^2 + 8x – 4, where the solution regions (1,2)\u222a(4,\u221e)(1,2) \\cup (4,\\infty) are highlighted in green. This confirms that the correct answer is B: (1,2) \u222a (2,8)<\/strong>. <\/p>\n","protected":false},"excerpt":{"rendered":"

      Solve the inequality: x^3 \u2013 5x^2 + 8x \u2013 4 > 0. A. (2, 8) B. ((1, 2) ? (2, 8) C. (-8, 1) ? (2, 8) D. (-8, 1) The correct answer and explanation is : To solve the inequality: x3\u22125×2+8x\u22124>0x^3 – 5x^2 + 8x – 4 > 0 Step 1: Find the roots […]<\/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-205964","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/205964","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=205964"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/205964\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=205964"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=205964"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=205964"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}