Intermediate Algebra (12th Edition)

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