How do you solve \frac { x } { x - 1} + 1= \frac { 1} { x ^ { 2} - x }?

1 Answer
Mrs. R
Dec 1, 2016

x = -1/2

Explanation:

Begin by getting a common denominator for all terms. Factor any denominator that can be factored first.
x/(x-1) + 1 = 1/((x)(x-1))
The common denominator is (x)(x-1). Multiply each fraction by the appropriate factors to make the denominators the same.
(x/(x-1))(x/x) +((x)(x-1))/((x)(x-1)) = 1/((x)(x-1))

Multiplying each term by the common denominator (x)(x-1) on both sides of the equation removes the denominator.

Now expand the numerator and rearrange:
(x)(x) +(x)(x-1) = 1
x^2 + x^2 - x = 1
2x^2 - x - 1 = 0

Factoring the quadratic,
(2x + 1)(x - 1) = 0

Applying the zero product rule, each factor equals 0.
2x + 1 = 0 OR x - 1 = 0
2x =-1 , x = 1
x = -1/2 , x = 1

Now we need to consider the restrictions.
In the original equation, each denominator cannot equal zero.
This means that x cannot equal zero or 1. So the solution x=1 is inadmissable.
The only solution then is x = -1/2.

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