x/x-2 + x-1/x+1 =-1
How do I solve this?
Use a common denominator, just as you do when you figure 1/3 + 1/4
Here, that will be (x-2)(x+1). So, multiply through by that and you get
x(x+1) + (x-1)(x-2) = -1(x+1)(x-2)
expand and collect terms and you wind up with
3x^2 - 3x = 0
3x(x-1) = 0
x = 0,1
To solve the equation (x/x-2) + (x-1/x+1) = -1, we need to find a common denominator for the fractions and combine them into a single fraction.
Step 1: Find a common denominator
To find a common denominator, we multiply the denominators of the fractions together:
Denominator = (x-2)(x+1)
Step 2: Convert each fraction to have the common denominator
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
Step 3: Combine the fractions
Combine the numerators over the common denominator:
(x(x+1) + (x-1)(x-2))/(x-2)(x+1) = -1
Step 4: Simplify the numerator
Expand and simplify the numerator:
(x^2 + x + x^2 - 3x + 2)/(x-2)(x+1) = -1
(2x^2 - 2x + 2)/(x-2)(x+1) = -1
Step 5: Eliminate the denominator
Multiply both sides by (x-2)(x+1) to eliminate the denominator:
2x^2 - 2x + 2 = -1(x-2)(x+1)
Step 6: Simplify and solve for x
Expand the right side:
2x^2 - 2x + 2 = -x^2 + x + 2x - 2
Combine like terms:
3x^2 - 3x + 2 = 0
This is a quadratic equation. To solve it, apply the quadratic formula or factor the equation if possible.