Answer
$\left( -\infty,-\dfrac{3}{8}\right)\cup\left( -\dfrac{3}{8},\infty\right)$
Work Step by Step
The given inequality, $
|8x+3|\gt0
,$ is equivalent to
\begin{array}{l}\require{cancel}
8x+3\gt0
\\\\
8x\gt-3
\\\\
x\gt-\dfrac{3}{8}
,\\\\\text{OR}\\\\
8x+3\lt0
\\\\
8x\lt-3
\\\\
x\lt-\dfrac{3}{8}
.\end{array}
Hence, the solution set is $
\left( -\infty,-\dfrac{3}{8}\right)\cup\left( -\dfrac{3}{8},\infty\right)
.$
See graph below.