Answer
$-2 \lt x \lt \dfrac{8}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
.$
