#### Answer

The solution is $\Big(-\dfrac{4}{3},\dfrac{2}{3}\Big)$

#### Work Step by Step

$|3x+1|-1\lt2$
Take $1$ to the right side of the inequality:
$|3x+1|\lt2+1$
$|3x+1|\lt3$
Solving this absolute value inequality is equivalent to solving the following inequality:
$-3\lt3x+1\lt3$
$\textbf{Solve the inequality shown above:}$
$-3\lt3x+1\lt3$
Subtract $1$ from all three parts of the inequality:
$-3-1\lt3x+1-1\lt3-1$
$-4\lt3x\lt2$
Divide all three parts of the inequality by $3$:
$-\dfrac{4}{3}\lt\dfrac{3x}{3}\lt\dfrac{2}{3}$
$-\dfrac{4}{3}\lt x\lt\dfrac{2}{3}$
Expressing the solution in interval notation:
$\Big(-\dfrac{4}{3},\dfrac{2}{3}\Big)$