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.
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
3 answers
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 ✅
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.
2 answers