Answer
$ (-7,-1)\cup(3,\infty)$.
Work Step by Step
Step 1. For $\frac{5}{x-3}-\frac{3}{x+1}\gt0\Longrightarrow \frac{2x+14}{(x-3)(x+1)}\gt0$, identify boundary points as $x=-7,-1,3$.
Step 2. Form intervals $(-\infty,-7),(-7,-1),(-1,3),(3,\infty)$.
Step 3. Choose test values for each interval $x=-8,-2,0,4$.
Step 4. Test the inequality to get results: $False,\ True,\ False,\ True $
Step 5. We have the solution $ (-7,-1)\cup(3,\infty)$.
