First, you can get each fraction over a common denominator, in this case x(x - 2), so the fractions can be added:
((x-2)/(x-2))(9/x) + (x/x)(9/(x-2)) = 12
(9(x-2) + 9x)/(x(x-2)) = 12
We can now expand and add the numerator:
(9x-18 + 9x)/(x(x-2)) = 12
(18x-18)/(x(x-2)) = 12
We can now eliminate the fraction through multiplication on each side of the equation to keep the equation balanced:
(18x-18)/(x^2-2x) = 12
(x^2 - 2x)(18x - 18)/(x^2-2x) = 12(x^2 - 2x)
cancel((x^2 - 2x))(18x - 18)/cancel((x^2-2x)) = 12x^2 - 24x
18x - 18 = 12x^2 - 24x
We can now make a single quadratic and factor:
18x - 18 - 18x + 18 = 12x^2 - 24x - 18x + 18
12x^2 - 42x + 18 = 0
(6x - 3)(2x - 6) = 0
We can now solve each factor for 0:
6x - 3 = 0
6x - 3 + 3 = 0 + 3
6x = 3
(6x)/6 = 3/6
x = 1/2
and
2x - 6 = 0
2x - 6 + 6 = 0 + 6
2x = 6
(2x)/2 = 6/2
x = 3
