Answer
$(-3,1)\cup[7,\infty)$
Work Step by Step
Step 1. Rewrite the inequality as $\frac{3}{x-1}-\frac{5}{x+3}\leq0\longrightarrow \frac{3x+9-5x+5}{(x-1)(x+3)}\leq0\longrightarrow \frac{-2x+14}{(x-1)(x+3)}\leq0\longrightarrow \frac{2x-14}{(x-1)(x+3)}\geq0$
Step 2. Identify the boundary points $x=-3,1,7$ and separate the number line into intervals $(-\infty, -3)$, $(-3,1)$, $(1,7)$ and $(7,\infty)$
Step 3. Use test points $x=-4,0,2,8$ to get the signs of the left side of the inequality as $-,+,-,+$
Step 4. Based on the signs and consider the boundary points for the equal sign, we have the solution as $(-3,1)\cup[7,\infty)$
