\(\frac{x}{x-2} + \frac{x-1}{x+1} =1\)
Least common denominator is (x-2)(x+1)
Multiply this times each term.
\((x-2)(x+1)( \frac{x}{x-2} )+(x-2)(x+1)( \frac{x-1}{x+1} )=(x-2)(x+1)(-1)\)
After cancelling the like factors from numerator and denominator, you get
\((x+1)x+(x-2)(x-1)=(x-2)(x+1)(-1)\)
Simplify the equation:
\(x^2+x+x^2-3x+2=-x^2+x+2\)
Add the opposite of all the terms on the right side to each side of the equation:
\(3x^2-3x=0\)
Factor out the GCF:
\(3x(x-1)=0\)
Set each factor equal to zero and solve:
\(3x = 0\) or \(x-1=0\)
x = 0 or x = 1
After checking each answer in the original equation you will see both numbers work. It's important to check because sometimes the answers can be extraneous (they don't work).
