Complex Numbers And Quadratic Equations question 202
Question: If $ \frac{2x}{2x^{2}+5x+2} $ > $ \frac{1}{x+1} $ , then [IIT 1987]
Options:
A) $ -2>x>-1 $
B) $ -2\ge x\ge -1 $
C) $ -2<x<-1 $
D) $ -2<x\le -1 $
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Answer:
Correct Answer: C
Solution:
Given $ \frac{2x}{2x^{2}+5x+2}>\frac{1}{x+1} $
Þ $ \frac{2x}{(2x+1)(x+2)}>\frac{1}{(x+1)} $
Þ $ \frac{2x}{(2x+1)(x+2)}-\frac{1}{(x+1)}>0 $
Þ $ \frac{2x(x+1)-(2x+1)(x+2)}{(x+1)(2x+1)(x+2)}>0 $
Þ $ \frac{2x^{2}+2x-2x^{2}-4x-x-2}{(x+1)(x+2)(2x+1)}>0 $
Þ $ \frac{-3x-2}{(x+1)(x+2)(2x+1)}>0 $ Equating each factor equal to 0, we have $ x=-2,-1,-\frac{2}{3},-\frac{1}{2} $ . It is clear that $ -\frac{2}{3}<x<-\frac{1}{2} $ or $ -2<x<-1 $ .
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