Is my process correct?
Solve: $\dfrac{x-3}{x+5}\leq3$
$$\dfrac{x-3}{x+5}-\dfrac{3(x+5)}{x+5}\leq0$$
$$\dfrac{x-3-3x-15}{x+5}\leq0$$
$$\dfrac{-2(x+9)}{x+5}\leq0$$
$$\dfrac{x+9}{x+5}\geq0$$
Points on number line are $-9$ with a closed interval, $-5$ with an open interval.
Upon plotting, I have concluded the solution to be:
$$\boxed{(-\infty,-9]\cup(-5,\infty)}$$
Going from line 3 to line 4, you've made a tiny mistake.
Remember, that when we multiply (or divide) an inequality by a negative number, we reverse the direction of the inequality, so it should be
Other than that, your final solution is, indeed, correct.
Remember: if in doubt, ask Wolfram Alpha!
http://www.wolframalpha.com/input/?i=solve+%28x-3%29%2F%28x%2B5%29+%3C%3D3