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  4. Find the set of all x for which (2 x/2 x2+5 x+2) > (1/x+1)

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

IIT JEEIIT JEE 1987Complex Numbers and Quadratic Equations

Solution:

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