Solving $\frac{1}{x} + \frac{1}{x+1} < \frac{2}{x+5}$ $\frac{1}{x} + \frac{1}{x+1} < \frac{2}{x+5}$
So, getting rid of the denominators I got to:
$(x+1)(x+5) + x(x+5) < 2x(x+1)$
$\rightarrow x < \frac{-5}{9}$
And also keeping in mind we can't have $-1,-5$ in the solution set, I got that the solution set was:
$\left(-\infty ,-5\right)\cup \left(-5,-1\right)\cup \left(-1,-\frac{5}{9}\right)$
But this is not correct. Help appreciated!

When multiply we need to take into account the sign; in this case your step are not correct since you are assuming positive terms.
As an alternative to avoid multiplication, we have
$$\frac{1}{x} + \frac{1}{x+1} < \frac{2}{x+5} \iff\frac{1}{x} + \frac{1}{x+1} - \frac{2}{x+5}<0 \iff \frac{9x+5}{x(x+1)(x+5)}<0$$
then study the sign of each term.
