There are different ways to solve the equation, but one possible method is to first find common denominators, combine like terms, and then isolate the variable. Here are the steps:
x/(x-2) + (x-1)/(x+1) = -1 original equation
To find common denominators, we need to multiply the first fraction by (x+1)/(x+1) and the second fraction by (x-2)/(x-2):
x(x+1)/[(x-2)(x+1)] + (x-1)(x-2)/[(x+1)(x-2)] = -1
Now we can combine the numerators over the common denominator (which is (x-2)(x+1)(x-2)):
x(x+1)(x-2) + (x-1)(x-2)(x+1) = -1(x-2)(x+1)(x-2)
Simplifying the left side:
x(x^2-x-2) + (x-1)(x^2-x-2) = -(x^3-3x^2+2x+4)
Expanding and collecting like terms:
x^3 - x^2 - 2x + x^3 - 3x^2 + 2x + 4x - 4 = -x^3 + 3x^2 - 2x - 4
Reordering:
2x^3 - 7x^2 + 4x = 0
Factoring out x:
x(2x^2 - 7x + 4) = 0
Now we can solve for x by setting each factor to zero:
x = 0 or 2x^2 - 7x + 4 = 0
To solve the quadratic equation, we can use the quadratic formula:
x = [ -(-7) ± sqrt((-7)^2 - 4(2)(4)) ] / (2(2))
x = [ 7 ± sqrt(25) ] / 4
x = [ 7 ± 5 ] / 4
x = 3 or 1/2
Therefore, the solutions to the equation are x = 0, x = 3, and x = 1/2. We can check them by plugging them back into the original equation and verifying that both sides are equal.