What would be the fastest way of solving the following inequality $\frac{(x+1)}{(x-1)(x+2)}>\frac{(x)}{(x-1)(x-3)}$ What would be the fastest way of solving the following inequality:
$\frac{(x+1)}{(x-1)(x+2)}>\frac{(x)}{(x-1)(x-3)}$

Clearly
$\frac{x+1}{\left(x-1\right)\left(x+2\right)}-\frac{x}{\left(x-1\right)\left(x-3\right)}>0$
$\Rightarrow \frac{-4x-3}{\left(x+2\right)\left(x-1\right)\left(x-3\right)}>0$
Then draw a straight line and mark $x$ values that make the fractors zero. That is $-2 ,-\frac{3}{4} ,1,3$.
Then check the validity for some $x$ value less than $-2$. 
Say $x=-3$.
For $x=-3$ we have that inequality does not hold.
So LHS is negative for $x=-3$
Then clearly $x \in \:\left(-2,-\frac{3}{4}\right)\cup \left(1,\:3\right)$
