Answer
$\left( -\dfrac{8}{5},\dfrac{2}{5} \right)$
Work Step by Step
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x \lt a$ then the solution to the given inequality, $
\left| \dfrac{3}{5}+x \right|\lt 1
,$ is
\begin{array}{l}\require{cancel}
-1\lt \dfrac{3}{5}+x \lt 1
\\\\
-1-\dfrac{3}{5}\lt -\dfrac{3}{5}+\dfrac{3}{5}+x \lt 1-\dfrac{3}{5}
\\\\
-\dfrac{8}{5}\lt x \lt \dfrac{2}{5}
.\end{array}
Hence, the solution set is the interval $
\left( -\dfrac{8}{5},\dfrac{2}{5} \right)
.$