Theory of Equations 1 Question 48
49. Find the set of all $x$ for which
$$ \frac{2 x}{2 x^{2}+5 x+2}>\frac{1}{x+1} $$
(1987, 3M)
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Solution:
- Given,
$$ \frac{2 x}{2 x^{2}+5 x+2}>\frac{1}{x+1} $$
$\Rightarrow \quad \frac{2 x}{(2 x+1)(x+2)}-\frac{1}{(x+1)}>0$
$\Rightarrow \frac{2 x(x+1)-(2 x+1)(x+2)}{(2 x+1)(x+2)(x+1)}>0$ $\Rightarrow \quad \frac{-(3 x+2)}{(2 x+1)(x+1)(x+2)}>0$; using number line rule
$$ \therefore \quad x \in(-2,-1) \cup-\frac{2}{3},-\frac{1}{2} $$
