Question

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

Answer 1

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

Explanation:

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

Multiply both sides by `2(x-2)(x-8)`

`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^2-8x+16`

The above factors into a perfect square:

`(x - 4)^2 = 0`

`x = 4`

check:

`1/ (4-2) + (2(4))/ ((4-2)(4-8)) = 4/ (2(4-8))`
`1/2 + 8/ ((2)(-4)) = 2/(-4)`
`-1/2 = -1/2`

This checks.