# How do you solve 1 / (1+x) - 1 / (2+x) = 1/4?

Jun 20, 2016

$x = \frac{- 3 \pm \sqrt{17}}{2}$

#### Explanation:

first the denominator shouldn't be 0
so $x \ne - 1 \mathmr{and} - 2$
$\frac{1}{x + 1} - \frac{1}{x + 2} = \frac{1}{4}$

$\frac{\left(x + 2\right) - \left(x + 1\right)}{\left(x + 1\right) \left(x + 2\right)} = \frac{1}{4}$

$\left(x + 1\right) \left(x + 2\right) = 4$

${x}^{2} + 3 x = 2$

${x}^{2} + 3 x + \frac{9}{4} = 2 + \frac{9}{4}$

${\left(x + \frac{3}{2}\right)}^{2} = \frac{17}{4}$

$x + \frac{3}{2} = \pm \frac{\sqrt{17}}{2}$

$x = \frac{- 3 \pm \sqrt{17}}{2}$