To solve for x, combine the fractions on the left-hand side by finding a common denominator:
(x(x+1))/((x-2)(x+1)) + ((x-1)(x-2))/((x-2)(x+1)) = -1
Simplify by distributing the numerators:
(x^2 + x + x^2 - 3x + 2)/((x-2)(x+1)) = -1
Combine like terms in the numerator:
(2x^2 - 2x + 2)/((x-2)(x+1)) = -1
Multiply both sides by the denominator:
2x^2 - 2x + 2 = -((x-2)(x+1))
Expand the right-hand side:
2x^2 - 2x + 2 = -x^2 + x + 2
Move all terms to one side:
3x^2 - 3x = 0
Factor out 3x:
3x(x-1) = 0
Solve for x:
x = 0 or x = 1
However, we need to check for extraneous solutions by plugging them back into the original equation to make sure they are valid.
When x = 0:
(0/(0-2)) + ((-1)/(0+1)) = -1
0 - 1 = -1
This is true, so x = 0 is a valid solution.
When x = 1:
(1/(1-2)) + ((1-1)/(1+1)) = -1
-1/1 + 0/2 = -1
-1 + 0 = -1
This is also true, so x = 1 is a valid solution.
Therefore, the solutions to the equation are x = 0 and x = 1.
(x/(x-2))+(x-1)/(x+1)=-1
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