First, we should put each fraction over a common denominator so we can add and subtract them problem. In this case a common denominator for each fraction would be #36x#. We need to multiply each fraction by the correct form of #1# to have them each over a common denominator:
#(9x)/(9x) * (x/4) + (4x)/(4x) * (2x)/9 = 4/4*((9x - 1))/(9x)#
#(9x^2)/(36x) + (8x^2)/(36x) = (4(9x - 1))/(36x)#
#(17x^2)/(36x) = (36x - 4)/(36x)#
We can now multiply each side of the equation by #36x# to eliminate the fractions while keeping the equation balanced:
#(36x) * (17x^2)/(36x) = (36x) * (36x - 4)/(36x)#
#cancel((36x)) * (17x^2)/cancel((36x)) = cancel((36x)) * (36x - 4)/cancel((36x))#
#17x^2 = 36x - 4#
We can now manage the equation to be a quadratic in standard for while keeping the equation balanced:
#17x^2 - 36x + 4 = 36x - 4 - 36x + 4#
#17x^2 - 36x + 4 = 0#
Factoring the quadratic gives:
#(17x - 2)(x - 2) = 0#
Solving each term for #0# gives:
#17x - 2 = 0#
#17x - 2 + 2 = 0 + 2#
#17x = 2#
#(17x)/17 = 2/17#
#x = 2/17#
and
#x - 2 = 0#
#x - 2 + 2 = 0 + 2#
#x = 2#
