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: 41


$x=\left\{ -7,1 \right\}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Given that $ f(x)=|2x+6| ,$ to find $x$ for which $ f(x)=8 ,$ use substitution. Then use the definition of an absolute value equality. Finally, use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Replacing $f(x)$ with $ 8 ,$ then \begin{array}{l}\require{cancel} f(x)=|2x+6| \\\\ 8=|2x+6| \\\\ |2x+6|=8 .\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+6=8 \\\\\text{OR}\\\\ 2x+6=-8 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 2x+6=8 \\\\ 2x=8-6 \\\\ 2x=2 \\\\ x=\dfrac{2}{2} \\\\ x=1 \\\\\text{OR}\\\\ 2x+6=-8 \\\\ 2x=-8-6 \\\\ 2x=-14 \\\\ x=-\dfrac{14}{2} \\\\ x=-7 .\end{array} Hence, $ x=\left\{ -7,1 \right\} .$
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