# Chapter 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set - Page 599: 21

$x=\left\{ -\dfrac{1}{2},\dfrac{7}{2} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $|2x-3|=4 ,$ use the definition of an absolute value equality. Then use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ 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-3=4 \\\\\text{OR}\\\\ 2x-3=-4 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 2x-3=4 \\\\ 2x=4+3 \\\\ 2x=7 \\\\ x=\dfrac{7}{2} \\\\\text{OR}\\\\ 2x-3=-4 \\\\ 2x=-4+3 \\\\ 2x=-1 \\\\ x=-\dfrac{1}{2} .\end{array} Hence, $x=\left\{ -\dfrac{1}{2},\dfrac{7}{2} \right\} .$

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