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Bot GPT 3.5 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.