# How do you solve \frac{3}{2x-4}-\frac{2}{x^{2}-x-2}=\frac{9}{x+1}?

Apr 19, 2018

$x = \frac{7}{3}$

#### Explanation:

First we factor the denominators, then find a common denominator so we can subtract, then cross multiply and solve the resulting polynomial equation.

$\frac{3}{2 x - 4} - \frac{2}{{x}^{2} - x - 2} = \frac{9}{x + 1}$

$\frac{3}{2 \left(x - 2\right)} - \frac{2}{\left(x + 1\right) \left(x - 2\right)} = \frac{9}{x + 1}$

$\frac{3 \left(x + 1\right) - 2 \left(2\right)}{2 \left(x + 1\right) \left(x - 2\right)} = \frac{9}{x + 1}$

$\frac{3 x - 1}{2 \left(x + 1\right) \left(x - 2\right)} = \frac{9}{x + 1}$

$\left(3 x - 1\right) \left(x + 1\right) = 18 \left(x + 1\right) \left(x - 2\right)$

$\left(x + 1\right) \left(\left(3 x - 1\right) - 18 \left(x - 2\right)\right) = 0$

$\left(x + 1\right) \left(- 15 x + 35\right) = 0$

$x = - 1$ or $x = \frac{35}{15} = \frac{7}{3}$

Apr 19, 2018

$x = \frac{7}{3}$

#### Explanation:

$\frac{3}{2 x - 4} - \frac{2}{{x}^{2} - x - 2} = \frac{9}{x + 1}$

Factorising

$2 x - 4 = 2 \left(x - 2\right)$

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

$= x \left(x - 2\right) + \left(x - 2\right)$

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

$\frac{3}{2 x - 4} - \frac{2}{{x}^{2} - x - 2} = \frac{3}{2 \left(x - 2\right)} - \frac{2}{\left(x + 1\right) \left(x - 2\right)}$

$\frac{3}{2 \left(x - 2\right)} - \frac{2}{\left(x + 1\right) \left(x - 2\right)} = \frac{9}{x + 1}$

$\frac{3}{2 \left(x - 2\right)} - \frac{9}{x + 1} = \frac{2}{\left(x + 1\right) \left(x - 2\right)}$

Multiplying throughout by $\left(x + 1\right) \left(x - 2\right)$

$\frac{3}{2} \left(x + 1\right) - 9 \left(x - 2\right) = 2$

$\frac{3}{2} x + \frac{3}{2} - 9 x + 18 = 2$

$\left(\frac{3}{2} - 9\right) x + \left(\frac{3}{2} + 18 - 2\right) = 0$

$- \frac{15}{2} x + \frac{35}{2} = 0$

$- 15 x + 35 = 0$

$15 x = 35$

$x = \frac{35}{15}$

$x = \frac{7}{3}$

Check:

lhs=
$\frac{3}{2 x - 4} - \frac{2}{{x}^{2} - x - 2} = \frac{3}{2 \times \frac{7}{3} - 4} - \frac{2}{{\left(\frac{7}{3}\right)}^{2} - \frac{7}{3} - 2}$

$= \frac{3}{\frac{14}{3} - 4} - \frac{2}{\frac{49}{9} - \frac{7}{3} - 2} = \frac{9}{14 - 12} - \frac{18}{49 - 21 - 18}$

$= \frac{9}{2} - \frac{18}{10} = \frac{45}{10} - \frac{18}{10} = \frac{45 - 18}{10} = \frac{27}{10}$

$r h s = \frac{9}{x + 1} = \frac{9}{\frac{7}{3} + 1} = \frac{27}{7 + 3} = \frac{27}{10}$

$l h s = r h s$