Answer
$$x=-4\quad \mathrm{or}\quad \:x=3$$
Work Step by Step
Since the value inside the absolute value can be positive or negative and get the same number, we must consider both options. Doing this, we find:
$$ \left|2x+1\right|=7\\ 2x+1=-7\quad \mathrm{or}\quad \:2x+1=7\\ x=-4\quad \mathrm{or}\quad \:x=3$$
