solve the equation

Question

x/x-2 + x-1/x+1 = -1

Answer 1

To solve the equation x/(x-2) + (x-1)/(x+1) = -1, we first need to find a common denominator.

The common denominator for x-2 and x+1 is (x-2)(x+1).

Multiplying both terms by the appropriate factors to get the common denominator, we get:

x(x+1)/[(x-2)(x+1)] + (x-1)(x-2)/[(x-2)(x+1)] = -1

Simplifying, we get:

(x^2 + x)/[(x-2)(x+1)] + (x^2 - 3x + 2)/[(x-2)(x+1)] = -1

Combining the terms, we get:

(2x^2 - 2x + 2)/[(x-2)(x+1)] = -1

Multiplying both sides by [(x-2)(x+1)], we get:

2x^2 - 2x + 2 = -[(x-2)(x+1)]

Expanding the right side, we get:

2x^2 - 2x + 2 = -[x^2 - x - 2]

Simplifying further, we get:

2x^2 - 2x + 2 = -x^2 + x + 2

Rearranging the terms, we get:

3x^2 - 3x = 0

Factoring out a common factor of 3x, we get:

3x(x - 1) = 0

Setting each factor equal to zero, we get two solutions:

3x = 0 -> x=0

x - 1 = 0 -> x=1

Therefore, the solutions to the equation x/(x-2) + (x-1)/(x+1) = -1 are x=0 and x=1.