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 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set: 78

Answer

$-\dfrac{7}{2} \le a \le 6$
1515743433

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $ 7+|4a-5|\le26 ,$ isolate first the absolute value expression. Then use the definition of a less than absolute value inequality. Use the properties of equality to isolate the variable. Finally, graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Using the properties of inequality to isolate the absolute value expression, the given is equivalent to \begin{array}{l}\require{cancel} 7+|4a-5|\le26 \\\\ |4a-5|\le26-7 \\\\ |4a-5|\le19 .\end{array} Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} -19 \le 4a-5 \le19 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -19 \le 4a-5 \le19 \\\\ -19+5 \le 4a-5+5 \le19+5 \\\\ -14 \le 4a \le24 \\\\ -\dfrac{14}{4} \le \dfrac{4a}{4} \le\dfrac{24}{4} \\\\ -\dfrac{7}{2} \le a \le 6 .\end{array} Hence, the solution set $ -\dfrac{7}{2} \le a \le 6 .$
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