## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$-2 \lt x \lt \dfrac{8}{3}$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|3x-1|\lt7 ,$ use the definition of a less than (less than or equal to) absolute value inequality. Then use the properties of inequality to isolate the variable. Graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} -7\lt 3x-1 \lt7 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -7\lt 3x-1 \lt7 \\\\ -7+1\lt 3x-1+1 \lt7+1 \\\\ -6 \lt 3x \lt8 \\\\ -\dfrac{6}{3} \lt \dfrac{3x}{3} \lt \dfrac{8}{3} \\\\ -2 \lt x \lt \dfrac{8}{3} .\end{array} Hence, the solution set $-2 \lt x \lt \dfrac{8}{3} .$