Using Analytical method, find the solution set of the inequality $\displaystyle \frac{3}{x-1}+\frac{5}{(x-2)^2} > \frac{3}{x-2}$
Getting a common denominator gives and moving everything to the right hand side gives
$\displaystyle 0 > \frac{3(x-1)-3(x+2)-5(x-1)}{(x-1)(x-2)}=\frac{-5x-4}{(x-1)(x-2)}$
So the inequality will change signs at
$\displaystyle x=1 \quad x=2 \quad x=-\frac{4}{5}$
can you finish from here?