NEED HELP ASAP!
x/x-2 + x-1/x+1 = - 1
Please show me how you got your answer as well. :)
multiply by [(x-2)(x+1)] ... common denominator
[x(x+1)] + [(x-1)(x-2)] = (x-2)(x+1)
multiply and collect like terms
you'll be solving a quadratic equation
you'll get a better answer if you don't forget that "-1" on the right:
[x(x+1)] + [(x-1)(x-2)] = -(x-2)(x+1)
To solve the given equation, we can start by finding a common denominator for the fractions on the left-hand side.
Step 1: Find the common denominator
The denominators are (x - 2) and (x + 1). Their common denominator will be (x - 2)(x + 1).
Step 2: Adjust the fractions
To get the fractions to have a common denominator, we need to multiply the numerator and denominator of each fraction by the missing factor(s) of the other fraction. For the first fraction, we need to multiply by (x + 1), and for the second fraction, we need to multiply by (x - 2).
(x / (x - 2)) * ((x + 1) / (x + 1)) + ((x - 1) / (x + 1)) * ((x - 2) / (x - 2))
= (x(x + 1) / [(x - 2)(x + 1)]) + ((x - 1)(x - 2) / [(x + 1)(x - 2)])
Step 3: Combine the fractions
Now that we have a common denominator, we can combine the two fractions over that common denominator.
[(x(x + 1) + (x - 1)(x - 2)) / [(x - 2)(x + 1)]]
= [x^2 + x + x^2 - 3x + 2) / [(x - 2)(x + 1)]
= [2x^2 - 2x + 2) / [(x - 2)(x + 1)]
Step 4: Simplify the equation
Now, we can simplify the equation by setting the numerator equal to the product of the denominator and -1.
2x^2 - 2x + 2 = -1 * (x - 2)(x + 1)
2x^2 - 2x + 2 = -x^2 + x + 2
3x^2 - 3x = 0
Step 5: Solve for x
Factoring out x, we get:
3x(x - 1) = 0
Setting each factor equal to zero, we have two solutions:
3x = 0 -> x = 0
x - 1 = 0 -> x = 1
Therefore, the solutions to the equation x/x-2 + x-1/x+1 = -1 are x = 0 and x = 1.
Please note that it's essential to verify these solutions by substituting them back into the original equation.