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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.

AAAaannndd the bot gets it wrong yet again!

x/(x-2) + (x-1)/(x+1) = -1
(x(x+1)+(x-1)(x-2)) = -(x-2)(x+1)
x^2+x+x^2-3x+2 = -x^2+x+2
3x^2-3x = 0
3x(x-1) = 0
x = 0 or 1
check:
x=0: 0 + -1/1 = -1 ✅
x=1: 1/-1 + 0 = -1 ✅

I apologize for the mistake in my previous response. Thank you for providing the correct solution and for pointing out my error.

To solve the equation (x/x-2) + (x-1/x+1) = -1, we need to find the value of x that satisfies this equation. Here are the step-by-step solutions:

Step 1: Start by multiplying both sides of the equation by the least common denominator (LCD) to eliminate the fractions. The LCD in this case is (x - 2)(x + 1):

(x)(x + 1) + (x - 1)(x - 2) = -1(x - 2)(x + 1)

Step 2: Simplify both sides of the equation. Expand and combine like terms:

x^2 + x + (x^2 - 3x + 2) = -x^2 + x + 2

2x^2 - 3x + 2 = -x^2 + x + 2

Step 3: Move all the terms to one side of the equation:

2x^2 - 3x + x^2 - x = 0

3x^2 - 4x = 0

Step 4: Factor out the common terms:

x(3x - 4) = 0

Step 5: Set each factor equal to zero and solve for x:

x = 0
3x - 4 = 0
3x = 4
x = 4/3

So the solutions to the equation are x = 0 and x = 4/3.

To solve the equation, we first need to simplify the expression on the left-hand side by finding a common denominator.

The given equation is: x/(x - 2) + (x - 1)/(x + 1) = -1

Step 1: Find the common denominator for (x - 2) and (x + 1).
To do this, multiply (x - 2) by (x + 1) and (x + 1) by (x - 2):
[(x - 2) * (x + 1)] + [(x - 1) * (x - 2)] = -1

Step 2: Simplify the expression with the common denominator:
[(x^2 - x - 2) + (x^2 - 3x + 2)] / [(x - 2) * (x + 1)] = -1

Step 3: Combine like terms:
(x^2 - x - 2 + x^2 - 3x + 2) / [(x - 2) * (x + 1)] = -1

Simplifying further:
(2x^2 - 4x) / [(x - 2) * (x + 1)] = -1

Step 4: Multiply both sides of the equation by [(x - 2) * (x + 1)] to eliminate the denominator:
(2x^2 - 4x) = -1 * [(x - 2) * (x + 1)]

Step 5: Distribute -1 on the right-hand side:
2x^2 - 4x = -1 * (x^2 - x - 2)

Simplifying further:
2x^2 - 4x = -x^2 + x + 2

Step 6: Rearrange the equation to bring all the terms to one side:
2x^2 - 4x + x^2 - x - 2 = 0

Combine like terms:
3x^2 - 5x - 2 = 0

The equation is now quadratic. We can solve it by factoring, completing the square, or using the quadratic formula.

Step 7: Factor the quadratic expression:
(3x + 1)(x - 2) = 0

Setting each factor equal to zero:
3x + 1 = 0 or x - 2 = 0

Step 8: Solve for x:
For 3x + 1 = 0:
3x = -1
x = -1/3

For x - 2 = 0:
x = 2

So, the two solutions to the given equation are x = -1/3 and x = 2.

To summarize:
The solutions to the equation x/(x - 2) + (x - 1)/(x + 1) = -1 are x = -1/3 and x = 2.