Answer
$(-7,-1)U(3,\infty)$
Work Step by Step
Step 1. Rewrite the inequality $\frac{5}{x-3}-\frac{3}{x+1}\gt0$ as $\frac{2x+14}{(x-3)(x+1)}\gt0$. Identify boundary points (zeros and asymptotes) as $x=-7,-1,3$ which divide the x-axis into four intervals $(-\infty,-7),(-7,-1),(-1,3),(3,\infty)$
Step 2. Choose test values in each interval $x=-8,-2,0,4$ and test the inequality to get $False,\ True,\ False,\ True$
Step 3. Thus the solution intervals are $(-7,-1)U(3,\infty)$
