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 - Page 599: 40


$x=\left\{ \dfrac{3}{2},\dfrac{7}{2} \right\}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ 3|2x-5|-7=-1 ,$ isolate first the absolute value expression. Then use the definition of an absolute value equality. Finally, use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the proeprties of equality to isolate the absolute value expression, the given is equivalent to \begin{array}{l}\require{cancel} 3|2x-5|-7=-1 \\\\ 3|2x-5|=-1+7 \\\\ 3|2x-5|=6 \\\\ |2x-5|=\dfrac{6}{3} \\\\ |2x-5|=2 .\end{array} Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation is equivalent to \begin{array}{l}\require{cancel} 2x-5=2 \\\\\text{OR}\\\\ 2x-5=-2 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 2x-5=2 \\\\ 2x=2+5 \\\\ 2x=7 \\\\ x=\dfrac{7}{2} \\\\\text{OR}\\\\ 2x-5=-2 \\\\ 2x=-2+5 \\\\ 2x=3 \\\\ x=\dfrac{3}{2} .\end{array} Hence, $ x=\left\{ \dfrac{3}{2},\dfrac{7}{2} \right\} .$
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