Expert: Sherry Wallin - 4/23/2009

Question
PLEASE SHOW HOW TO SOLVE THE FOLLOWING RATIONAL EQUATION:
(6x)/(x+4)+(4)=(2x+2)/(x-1)

Answer
Hi Rebecca~
    The first thing you need to pay attention to is the domain of the rational expressions. On the left x cannot equal -4 and on the right x cannot equal 1. You may ask why and that is because in order to solve the rational equation you will need to multiply by a number that could possibly make one of the denominators 0 and that is not legal.

Next: find the LCD which in this case is (x+4)(x-1) so multiply all terms by these factors. On the right side notice that 2x+2 has a 2 in it so it is 2(x+1)

[6x/(x+4)](x+4)(x-1) + 4(x+4)(x-1)= [2(x+1)/(x-1)](x+4)(x-1)
the x-4 cancels leaving 6x(x-1)+4(x^2+3x-4)= 2(x+1)(x+4)
on the right side it is equal to 2x^2 + 10x + 8. So we have
6x^2 - 6x +4x^2 +12x -16 = 2x^2 + 10x + 8
gather like terms
10x^2 + 6x  -16 = 2x^2 + 10x + 8
move all to the left
8x^2 - 4x -24 = 0
factor getting 4(2x^2 -x-6) = (2x+3)(x-2) = 0
so x = -3/2 or x = 2 and both are solutions since none of them makes the denominator zero.


Math Prof