Question

Enter your answer and show all the steps that you use to solve this problem in the space provided.

Solve the following equation. Show all your work.

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

Answer 1

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 ✅

Answer 2

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.

Answer 3

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