Answer
$$x=-8\quad \mathrm{or}\quad \:x=2$$
Work Step by Step
Since absolute values are positive no matter what, we have to consider both the case where the value inside the absolute value is positive and when it is negative. To check solutions, we would plug them into the original equation to see if they resulted in a true statement. Doing this, we find:
$$x+3=-5\quad \mathrm{or}\quad \:x+3=5 \\ x=-8\quad \mathrm{or}\quad \:x=2$$
