Question

Find the set of all $x$ for which $\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}$

Answer 1

Given, $\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}$
$\Rightarrow \frac{2x}{(2x+1)(x+2)}-\frac{1}{(x+1)}>0$
$\Rightarrow \frac{2x(x+1)-(2x+1)(x+2)}{(2x+1)(x+2)(x+1)}>0$
$\Rightarrow \frac{-(3x+2)}{(2x+1)(x+2)(x+1)}>0$; using number line rule

$\therefore x\in (-2,-1)\cup \left(-\frac{2}{3},-\frac{1}{2}\right)$