Intermediate Algebra (12th Edition)

Published by Pearson

Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises - Page 118: 26

Answer

$x=\left\{ -\dfrac{24}{5},\dfrac{32}{5} \right\}$

Work Step by Step

Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $\left|2-\dfrac{5}{2}x\right|=14 ,$ is equivalent to \begin{array}{l}\require{cancel} 2-\dfrac{5}{2}x=14 \text{ OR } 2-\dfrac{5}{2}x=-14 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 2-\dfrac{5}{2}x=14 \\\\ 2\left(2-\dfrac{5}{2}x\right)=(14)2 \\\\ 4-5x=28 \\\\ -5x=28-4 \\\\ -5x=24 \\\\ x=\dfrac{24}{-5} \\\\ x=-\dfrac{24}{5} \\\\\text{ OR }\\\\ 2-\dfrac{5}{2}x=-14 \\\\ 2\left(2-\dfrac{5}{2}x\right)=(-14)2 \\\\ 4-5x=-28 \\\\ -5x=-28-4 \\\\ -5x=-32 \\\\ x=\dfrac{-32}{-5} \\\\ x=\dfrac{32}{5} .\end{array} Hence, the solutions are $x=\left\{ -\dfrac{24}{5},\dfrac{32}{5} \right\} .$

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