Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 2 - Equations, Inequalities, and Problem Solving - 2.6 Solving Inequalities - 2.6 Exercise Set - Page 136: 108


$\left( -\dfrac{27}{19},\infty \right)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the order of operations and the properties of inequality to solve the given inequality, $ 6[4-2(6+3t)]\gt5[3(7-t)-4(8+2t)]-20 .$ $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to \begin{array}{l}\require{cancel} 6[4-2(6+3t)]\gt5[3(7-t)-4(8+2t)]-20 \\\\ 6[4-2(6)-2(3t)]\gt5[3(7)+3(-t)-4(8)-4(2t)]-20 \\\\ 6[4-12-6t]\gt5[21-3t-32-8t]-20 \\\\ 6[-8-6t]\gt5[-11-11t]-20 \\\\ 6[-8]+6[-6t]\gt5[-11]+5[-11t]-20 \\\\ -48-36t\gt-55-55t-20 \\\\ -48-36t\gt-75-55t .\end{array} Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} -48-36t\gt-75-55t \\\\ -36t+55t\gt-75+48 \\\\ 19t\gt-27 \\\\ t\gt-\dfrac{27}{19} .\end{array} Hence, the solution set is $ \left( -\dfrac{27}{19},\infty \right) .$
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