# How do you solve the rational equation (2x) / (x+2) - 2 = (x-8) / (x-2)?

Dec 24, 2016

The solution for the above rational equation is $x = 6 , - 4$

#### Explanation:

$\frac{2 x}{x + 2} - 2 = \frac{x - 8}{x - 2}$

or, $\frac{2 x - 2 \left(x + 2\right)}{x + 2} = \frac{x - 8}{x - 2} \rightarrow$Take LCM on LHS.

or, $\frac{2 x - 2 x - 4}{x + 2} = \frac{x - 8}{x - 2} \rightarrow$Simplify LHS.

or, $\frac{- 4}{x + 2} = \frac{x - 8}{x - 2}$

or, $\left(- 4\right) \cdot \left(x - 2\right) = \left(x + 2\right) \left(x - 8\right) \rightarrow$ Cross Multiply.

or, $- 4 x + 8 = x \left(x - 8\right) + 2 \left(x - 8\right) \rightarrow$Simplify.

or, $- 4 x + 8 = {x}^{2} - 8 x + 2 x - 16 \rightarrow$Simplify.

or, $- 4 x + 8 = {x}^{2} - 6 x - 16 \rightarrow$Simplify

or, $- 4 x + 4 x + 8 - 8 = {x}^{2} - 6 x + 4 x - 16 - 8 \rightarrow$Adding $4 x$

and subtracting $8$ from both sides.

or, ${x}^{2} - 2 x + 24 = 0 \rightarrow$By simplifying RHS.

or, ${x}^{2} + 4 x - 6 x - 24 = 0 \rightarrow$Factoring new LHS.

or, $x \left(x + 4\right) - 6 \left(x + 4\right) = 0$

or, $\left(x - 6\right) \left(x + 4\right) = 0 \rightarrow$Factored new LHS.

Either

$x - 6 = 0$

or, $x = 6$

Or,

$x + 4 = 0$

or, $x = - 4$

Hence, the solution for this equation is $x = 6 , - 4$