Answer
$(-\infty,-\frac{5}{2})\cup(-2,-1)$
Work Step by Step
Step 1. Rewrite the inequality as
$\frac{1}{x+2}-\frac{3}{x+1}\gt0$, $\frac{x+1-3x-6}{(x+2)(x+1)}\gt0$, $\frac{-2x-5}{(x+2)(x+1)}\gt0$
or
$\frac{2x+5}{(x+2)(x+1)}\lt0$
Step 2. Identify the boundary points as
$x=-\frac{5}{2}, -2, -1$
Step 3. Put the above boundary points on a number line to get four intervals.
Step 4. Use one test point for each interval $x=-3, -2.2, -1.5, 0$ and evaluate the signs of the left expression of the inequality.
We have signs $-,+,-,+$ for intervals
$(-\infty,-\frac{5}{2}),(-\frac{5}{2},-2),(-2,-1),(-1,\infty)$
Step 5. Choose the intervals giving negative signs for the solutions as
$(-\infty,-\frac{5}{2})\cup(-2,-1)$
