Answer
$\left( -\dfrac{4}{3},\dfrac{2}{3} \right)$
Work Step by Step
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a,$ then the solution to the given equation, $
\left| 3x+1 \right|-1\lt2
,$ is
\begin{array}{l}\require{cancel}
\left| 3x+1 \right|\lt2+1
\\\\
\left| 3x+1 \right|\lt3
\\\\
-3\lt 3x+1\lt3
\\\\
-3-1\lt 3x+1-1\lt3-1
\\\\
-4\lt 3x\lt2
\\\\
-\dfrac{4}{3}\lt \dfrac{3x}{3}\lt\dfrac{2}{3}
\\\\
-\dfrac{4}{3}\lt x\lt\dfrac{2}{3}
.\end{array}
Hence, the solution set is the interval $
\left( -\dfrac{4}{3},\dfrac{2}{3} \right)
.$