Solve the equation. Show all your work.

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

Multiply both sides by (x-2) (x+1) to clea fractions

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

distribute

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

add like terms

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

add x^2 to both sdes

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

minus x from both sides

3x^2-3x+2=2

minus 2 both sides

3x^2-3x=0

factor

(3x) (x-1) = 0

set to zero

3x=0

x=0

x-1=0

x=1

x=0 or 1