Question

If $\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}$, then

• A. $-2>x>-1$
• B. $-2\ge x\ge 1$
• C. $-2<x<-1$
• D. $-2<x\le 1$

Answer 1

The correct option is C $-2<x<-1$

$\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}$

$\Rightarrow \frac{2x}{2x^2+5x+2}-\frac{1}{x+1}>0$

$\Rightarrow \frac{-3x-2}{(2x^2+4x+x+2)(x+1)}>0$

$\Rightarrow \frac{-3(x+\frac{2}{3})}{2x(x+2)+1(x+2)(x+1)}>0$

$\Rightarrow \frac{x+\frac{2}{3}}{(x+2)2(x+\frac{1}{2})(x+1)}<0$

$\Rightarrow \frac{x+\frac{2}{3}}{(x+2)(x+\frac{1}{2})(x+1)}<0$

$\Rightarrow x\in (-2-1)\cup (-\frac{2}{3},-\frac{-1}{2})$

$\Rightarrow -2<x<-1$ or $-\frac{-2}{3}<x<-\frac{1}{2}$