Answer
$(-2,4)\cup(16,\infty)$
Work Step by Step
Step 1. Rewrite the inequality as $\frac{3}{x+2}-\frac{2}{x-4}\gt0\longrightarrow \frac{3x-12-2x-4}{(x+2)(x-4)}\gt0\longrightarrow \frac{x-16}{(x+2)(x-4)}\gt0$
Step 2. Identify the boundary points $x=-2,4,16$ and separate the number line into intervals $(-\infty, -2)$, $(-2,4)$, $(4,16)$ and $(16,\infty)$
Step 3. Use test points $x=-3,0,5,17$ to get the signs of the left side of the inequality as $-,+,-,+$
Step 4. Based on the signs, we have the solution as $(-2,4)\cup(16,\infty)$
