#### Answer

$(-\infty,-3)\cup(2,\infty)$

#### Work Step by Step

Using the properties of inequality, the given statement, $
|2x+1|+3\gt8
,$ is equivalent to
\begin{array}{l}\require{cancel}
|2x+1|\gt8-3
\\\\
|2x+1|\gt5
.\end{array}
Since for any $a\gt0$, $|x|\gt a$ implies $x\gt a$ OR $x\lt-a$, then the equation above is equivalent to
\begin{array}{l}\require{cancel}
2x+1\gt5 \text{ OR } 2x+1\lt-5
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
2x+1\gt5
\\\\
2x\gt5-1
\\\\
2x\gt4
\\\\
x\gt\dfrac{4}{2}
\\\\
x\gt2
\\\\\text{ OR }\\\\
2x+1\lt-5
\\\\
2x\lt-5-1
\\\\
2x\lt-6
\\\\
x\lt-\dfrac{6}{2}
\\\\
x\lt-3
.\end{array}
Hence, the solution set is the interval $
(-\infty,-3)\cup(2,\infty)
.$