# How do you solve 3/(x+6)=5/(x-2) and find any extraneous solutions?

Dec 6, 2017

$x = - 18$

#### Explanation:

$\text{multiply both sides by } \left(x + 6\right) \left(x - 2\right)$

$\cancel{\left(x + 6\right)} \left(x - 2\right) \times \frac{3}{\cancel{\left(x + 6\right)}} = \cancel{\left(x - 2\right)} \left(x + 6\right) \times \frac{5}{\cancel{\left(x - 2\right)}}$

$\Rightarrow 3 \left(x - 2\right) = 5 \left(x + 6\right) \leftarrow \textcolor{b l u e}{\text{no fractions}}$

$\text{distribute brackets on both sides}$

$3 x - 6 = 5 x + 30$

$\text{subtract 5x from both sides}$

$3 x - 5 x - 6 = \cancel{5 x} \cancel{- 5 x} + 30$

$\Rightarrow - 2 x - 6 = 30$

$\text{add 6 to both sides}$

$- 2 x \cancel{- 6} \cancel{+ 6} = 30 + 6$

$\Rightarrow - 2 x = 36$

$\text{divide both sides by } - 2$

$\frac{\cancel{- 2} x}{\cancel{- 2}} = \frac{36}{- 2}$

$\Rightarrow x = - 18$

$\textcolor{b l u e}{\text{As a check}}$

Substitute this value into the equation and if both sides are equal then it is the solution.

$\text{left } = \frac{3}{- 12} = - \frac{1}{4}$

$\text{right } = \frac{5}{- 20} = - \frac{1}{4}$

$\Rightarrow x = - 18 \text{ is the only solution}$