# How do you solve  1/ (x-2) + (2x)/ ((x-2)(x-8)) = x/ (2(x-8)) ?

Apr 22, 2017

Multiply both sides by the least common multiple of the numerators.
Check.

#### Explanation:

Given: 1/ (x-2) + (2x)/ ((x-2)(x-8)) = x/ (2(x-8));x!=2,x!=8

Multiply both sides by $2 \left(x - 2\right) \left(x - 8\right)$

2(x-8) + 2(2x) = x(x-2);x!=2,x!=8

2x-16 + 4x = x^2-2x;x!=2,x!=8

Combine like terms:

0 = x^2-8x+16;x!=2,x!=8

Neither 2 nor 8 are roots, therefore, we drop the restrictions:

$0 = {x}^{2} - 8 x + 16$

The above factors into a perfect square:

${\left(x - 4\right)}^{2} = 0$

$x = 4$

check:

$\frac{1}{4 - 2} + \frac{2 \left(4\right)}{\left(4 - 2\right) \left(4 - 8\right)} = \frac{4}{2 \left(4 - 8\right)}$
$\frac{1}{2} + \frac{8}{\left(2\right) \left(- 4\right)} = \frac{2}{- 4}$
$- \frac{1}{2} = - \frac{1}{2}$

This checks.