{"id":258326,"date":"2025-07-18T06:14:14","date_gmt":"2025-07-18T06:14:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=258326"},"modified":"2025-07-18T06:14:16","modified_gmt":"2025-07-18T06:14:16","slug":"solve-the-following-inequalities-starting-the-solution-in-words","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/18\/solve-the-following-inequalities-starting-the-solution-in-words\/","title":{"rendered":"Solve the following inequalities, starting the solution in words."},"content":{"rendered":"\n

Solve the following inequalities, starting the solution in words. Graph the inequality on the number line. (a) x – 25 > 7 (b) 5 < 2x + 15 (c) 25\/x < 10 (d) x\/2 + x\/2 + x\/3 \u00e2\u2030\u00a5 17\/6 (e) 3x – 29 \u00e2\u2030\u00a4 7x + 11<\/p>\n\n\n\n

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

Let’s go through each inequality step by step and explain the solutions.<\/p>\n\n\n\n

(a) x\u221225>7x – 25 > 7x\u221225>7<\/h3>\n\n\n\n
    \n
  1. Start by isolating xxx<\/strong>. To do this, add 25 to both sides: x>7+25x > 7 + 25x>7+25 x>32x > 32x>32 Solution<\/strong>: x>32x > 32x>32. This means that xxx must be greater than 32.<\/li>\n\n\n\n
  2. Graphing<\/strong>: On the number line, represent an open circle at 32, and shade to the right of 32 to show all values greater than 32.<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    (b) 5<2x+155 < 2x + 155<2x+15<\/h3>\n\n\n\n
      \n
    1. Isolate the xxx-term<\/strong>. Subtract 15 from both sides: 5\u221215<2×5 – 15 < 2×5\u221215<2x \u221210<2x-10 < 2x\u221210<2x<\/li>\n\n\n\n
    2. Solve for xxx<\/strong> by dividing both sides by 2: \u2212102<x\\frac{-10}{2} < x2\u221210\u200b<x \u22125<x-5 < x\u22125<x Solution<\/strong>: x>\u22125x > -5x>\u22125. This means that xxx is greater than -5.<\/li>\n\n\n\n
    3. Graphing<\/strong>: On the number line, represent an open circle at -5, and shade to the right of -5.<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      (c) 25x<10\\frac{25}{x} < 10×25\u200b<10<\/h3>\n\n\n\n
        \n
      1. Isolate xxx<\/strong>. Multiply both sides by xxx (but remember, this changes the direction of the inequality if xxx is negative): 25<10×25 < 10×25<10x<\/li>\n\n\n\n
      2. Solve for xxx<\/strong> by dividing both sides by 10: 2510<x\\frac{25}{10} < x1025\u200b<x 2.5<x2.5 < x2.5<x Solution<\/strong>: x>2.5x > 2.5x>2.5. This means that xxx must be greater than 2.5.<\/li>\n\n\n\n
      3. Graphing<\/strong>: On the number line, represent an open circle at 2.5, and shade to the right of 2.5.<\/li>\n<\/ol>\n\n\n\n
        \n\n\n\n

        (d) x2+x2+x3\u2265176\\frac{x}{2} + \\frac{x}{2} + \\frac{x}{3} \\geq \\frac{17}{6}2x\u200b+2x\u200b+3x\u200b\u2265617\u200b<\/h3>\n\n\n\n
          \n
        1. Combine like terms<\/strong>. The first two terms can be added together: x2+x2=2×2=x\\frac{x}{2} + \\frac{x}{2} = \\frac{2x}{2} = x2x\u200b+2x\u200b=22x\u200b=x Now the inequality becomes: x+x3\u2265176x + \\frac{x}{3} \\geq \\frac{17}{6}x+3x\u200b\u2265617\u200b<\/li>\n\n\n\n
        2. Get a common denominator<\/strong> for the terms on the left-hand side. The common denominator between 1 and 3 is 3: 3×3+x3=4×3\\frac{3x}{3} + \\frac{x}{3} = \\frac{4x}{3}33x\u200b+3x\u200b=34x\u200b So the inequality becomes: 4×3\u2265176\\frac{4x}{3} \\geq \\frac{17}{6}34x\u200b\u2265617\u200b<\/li>\n\n\n\n
        3. Clear the fraction<\/strong> by multiplying both sides by 6 (the least common denominator of 3 and 6): 6\u00d74×3\u22656\u00d71766 \\times \\frac{4x}{3} \\geq 6 \\times \\frac{17}{6}6\u00d734x\u200b\u22656\u00d7617\u200b 8x\u2265178x \\geq 178x\u226517<\/li>\n\n\n\n
        4. Solve for xxx<\/strong> by dividing both sides by 8: x\u2265178x \\geq \\frac{17}{8}x\u2265817\u200b x\u22652.125x \\geq 2.125x\u22652.125 Solution<\/strong>: x\u22652.125x \\geq 2.125x\u22652.125. This means xxx is greater than or equal to 2.125.<\/li>\n\n\n\n
        5. Graphing<\/strong>: On the number line, represent a closed circle at 2.125, and shade to the right of it.<\/li>\n<\/ol>\n\n\n\n
          \n\n\n\n

          (e) 3x\u221229\u22647x+113x – 29 \\leq 7x + 113x\u221229\u22647x+11<\/h3>\n\n\n\n
            \n
          1. Move all terms involving xxx to one side<\/strong>. Subtract 3x3x3x from both sides: \u221229\u22644x+11-29 \\leq 4x + 11\u221229\u22644x+11<\/li>\n\n\n\n
          2. Move the constants to the other side<\/strong> by subtracting 11 from both sides: \u221229\u221211\u22644x-29 – 11 \\leq 4x\u221229\u221211\u22644x \u221240\u22644x-40 \\leq 4x\u221240\u22644x<\/li>\n\n\n\n
          3. Solve for xxx<\/strong> by dividing both sides by 4: \u2212404\u2264x\\frac{-40}{4} \\leq x4\u221240\u200b\u2264x \u221210\u2264x-10 \\leq x\u221210\u2264x Solution<\/strong>: x\u2265\u221210x \\geq -10x\u2265\u221210. This means xxx is greater than or equal to -10.<\/li>\n\n\n\n
          4. Graphing<\/strong>: On the number line, represent a closed circle at -10, and shade to the right of it.<\/li>\n<\/ol>\n\n\n\n
            \n\n\n\n

            Summary of Solutions:<\/h3>\n\n\n\n
              \n
            • (a) x>32x > 32x>32<\/li>\n\n\n\n
            • (b) x>\u22125x > -5x>\u22125<\/li>\n\n\n\n
            • (c) x>2.5x > 2.5x>2.5<\/li>\n\n\n\n
            • (d) x\u22652.125x \\geq 2.125x\u22652.125<\/li>\n\n\n\n
            • (e) x\u2265\u221210x \\geq -10x\u2265\u221210<\/li>\n<\/ul>\n\n\n\n
              \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

              Solve the following inequalities, starting the solution in words. Graph the inequality on the number line. (a) x – 25 > 7 (b) 5 < 2x + 15 (c) 25\/x < 10 (d) x\/2 + x\/2 + x\/3 \u00e2\u2030\u00a5 17\/6 (e) 3x – 29 \u00e2\u2030\u00a4 7x + 11 The Correct Answer and Explanation is: Let’s […]<\/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-258326","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258326","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=258326"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258326\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=258326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=258326"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=258326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}