Question

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. Solve the following equation and. Show all your work x/x-2+x-1/x+1=-1

Answer 1

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

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

The common denominator is (x-2)(x+1):

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

Now we can combine the fractions:

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

Simplifying the left side:

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

Simplifying the right side:

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

Bringing all the terms to one side:

2x^2 - 2x - 4 = 0

Dividing both sides by 2:

x^2 - x - 2 = 0

This is a quadratic equation, which we can factor:

(x-2)(x+1) = 0

Therefore, the solutions are x = 2 and x = -1.

However, we need to check if these solutions are valid, because we might have introduced extraneous solutions when we found the common denominator.

Checking x = 2:

2/0 + 1/3 = -1
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The left side is undefined when x = 2, so this solution is extraneous and should be discarded.

Checking x = -1:

-1/-3 + 0/0 = -1
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The left side is still undefined when x = -1, so this solution is also extraneous and should be discarded.

Therefore, the equation has no valid solutions.

Answer 2

I will assume you meant:
x/(x-2) + (x-1)/(x+1) = -1

multiply each term by (x-2)(x+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 x = 1

I think the bot messed up the expansion of the right side of
x(x+1) + (x-1)(x-2) = -(x-2)(x+1)

Answer 3

I apologize for my mistake in the expansion of the right side of the equation. Thank you for providing the correct expansion and solution. You are correct that the solutions are x = 0 and x = 1, and we should check them for validity, which can be done by plugging in the values into the original equation.