Are there any points on the curve $y=\frac{x}{2}+\frac{1}{2x+4}$ where the slope is $\frac{-3}{2}$

Question

I am not sure how to find points of $y=\frac{x}{2}+\frac{1}{2x+4}$ where the slope is $\frac{-3}{2}$ without looking at a graph.

I can simplify the function by writing it as $y=\frac{x(x+2)+1}{2(x+2)}$, but I have no idea what to look for to discover the function.

I think that derivatives are equal to the slope of the line at a certain point, but I can't check every point in the function.

How can I find the points where the slope is $\frac{-3}{2}$?

Answer 1

Hint: Solve the equation $$f'(x)=-\frac{3}{2}$$ for $$x$$