x/x-2 +x-1/x+1 = -1
How do I find the solution to this equation?
To find the solution to the equation (x/x-2) + (x-1/x+1) = -1, you need to find the value of x that satisfies the equation. Here's how you can do it:
Step 1: Simplify the equation by finding a common denominator for both fractions. The common denominator is (x - 2)(x + 1). Multiply each fraction by the necessary factor to get the common denominator:
(x * (x + 1))/((x - 2)(x + 1)) + ((x - 1) * (x - 2))/((x + 1)(x - 2)) = -1
Simplifying this further:
(x^2 + x)/[(x - 2)(x + 1)] + (x^2 - 3x + 2)/[(x + 1)(x - 2)] = -1
Step 2: Combine the two fractions by adding their numerators, since they have the same common denominator:
(x^2 + x + x^2 - 3x + 2)/[(x - 2)(x + 1)] = -1
Simplifying the numerator:
(2x^2 - 2x + 2)/[(x - 2)(x + 1)] = -1
Step 3: Multiply both sides of the equation by the common denominator to eliminate the denominator:
(2x^2 - 2x + 2) = -1 * [(x - 2)(x + 1)]
Simplifying the right side:
(2x^2 - 2x + 2) = -1 * (x^2 - x - 2)
Step 4: Distribute the negative sign on the right side and simplify:
2x^2 - 2x + 2 = -x^2 + x + 2
Step 5: Combine like terms by moving all terms to one side of the equation:
2x^2 - 2x + x^2 - x + 2 - 2 = 0
3x^2 - 3x = 0
Step 6: Factor out the common factor, which is x:
3x(x - 1) = 0
Step 7: Set each factor equal to zero and solve for x:
3x = 0 => x = 0
x - 1 = 0 => x = 1
Therefore, the equation has two solutions: x = 0 and x = 1.