- #1

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## Homework Statement

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

## Homework Equations

## The Attempt at a Solution

I'm getting two different sets of answers with two different methods:

Method 1-Wrong

##\dfrac{2x}{2x^2 + 5x + 2} > \dfrac{1}{x + 1}##

##\dfrac{2x^2 + 5x + 2}{2x} < x + 1##

##\dfrac{2x^2 + 5x + 2}{2x} - (x+1) < 0##

##\dfrac{2x^2 + 5x + 2 - 2x(x+1)}{2x} < 0##

##\dfrac{3x + 2}{2x} < 0##

## x \in \left( \dfrac{-2}{3}, 0 \right)##

Method 2, the correct one

##\dfrac{2x}{2x^2 + 5x + 2} > \dfrac{1}{x + 1}##

##\dfrac{2x}{2x^2 + 5x + 2} - \dfrac{1}{x + 1} > 0##

##\dfrac{2x(x+1) - (2x^2 + 5x + 2)}{(2x^2 + 5x + 2)(x + 1)} > 0##

##\dfrac{3x + 2}{(2x + 1)(x + 1)(x + 2)} < 0 ##

##x \in (-2,-1) \cup \left(\dfrac{-2}{3} , \dfrac{-1}{2}\right)##