# Chapter 1 - Section 1.8 - Absolute Value Equations and Inequalities - 1.8 Exercises - Page 154: 43

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

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $|4x+3|-2=-1 ,$ use the properties of equality to isolate the absolute value expression. Then use the properties of absolute value equality. $\bf{\text{Solution Details:}}$ Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} |4x+3|=-1+2 \\\\ |4x+3|=1 .\end{array} Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to \begin{array}{l}\require{cancel} 4x+3=1 \\\\\text{OR}\\\\ 4x+3=-1 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 4x+3=1 \\\\ 4x=1-3 \\\\ 4x=-2 \\\\ x=-\dfrac{2}{4} \\\\ x=-\dfrac{1}{2} \\\\\text{OR}\\\\ 4x+3=-1 \\\\ 4x=-1-3 \\\\ 4x=-4 \\\\ x=-\dfrac{4}{4} \\\\ x=-1 .\end{array} Hence, $x=\left\{ -1,-\dfrac{1}{2} \right\} .$

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