# How do you solve 2/(x+1) + 5/(x-2)=-2?

Jul 24, 2016

$x = \frac{1}{2}$ or $x = - 3$

#### Explanation:

$\frac{2}{x + 1} + \frac{5}{x - 2} = - 2$

$\Leftrightarrow \frac{2 \left(x - 2\right) + 5 \left(x + 1\right)}{\left(x + 1\right) \left(x - 2\right)} = - 2$

$\Leftrightarrow 2 \left(x - 2\right) + 5 \left(x + 1\right) = - 2 \left(x + 1\right) \left(x - 2\right)$

$\Leftrightarrow 2 x - 4 + 5 x + 5 = - 2 \left({x}^{2} - 2 x + x - 2\right)$

$\Leftrightarrow 7 x + 1 = - 2 {x}^{2} + 2 x + 4$

$\Leftrightarrow 2 {x}^{2} + 5 x - 3 = 0$

$\Leftrightarrow 2 {x}^{2} + 6 x - x - 3 = 0$

$\Leftrightarrow 2 x \left(x + 3\right) - 1 \left(x + 3\right) = 0$

$\Leftrightarrow \left(2 x - 1\right) \left(x + 3\right) = 0$

Hence either $2 x - 1 = 0$ i.e. $x = \frac{1}{2}$

or $x + 3 = 0$ i.e. $x = - 3$