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

$$\frac { - x } { x + 1 } = - 1 + \frac { 2 } { x - 2 }$$

Solve for x

$$x=-4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+1\right), the least common multiple of x+1,x-2.

$$\left(x-2\right)\left(-x\right)=\left(x-2\right)\left(x+1\right)\left(-1\right)+\left(x+1\right)\times 2$$

Use the distributive property to multiply x-2 by -x.

$$x\left(-x\right)-2\left(-x\right)=\left(x-2\right)\left(x+1\right)\left(-1\right)+\left(x+1\right)\times 2$$

Multiply -2 and -1 to get 2.

$$x\left(-x\right)+2x=\left(x-2\right)\left(x+1\right)\left(-1\right)+\left(x+1\right)\times 2$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$x\left(-x\right)+2x=\left(x^{2}-x-2\right)\left(-1\right)+\left(x+1\right)\times 2$$

Use the distributive property to multiply x^{2}-x-2 by -1.

$$x\left(-x\right)+2x=-x^{2}+x+2+\left(x+1\right)\times 2$$

Use the distributive property to multiply x+1 by 2.

$$x\left(-x\right)+2x=-x^{2}+x+2+2x+2$$

Combine x and 2x to get 3x.

$$x\left(-x\right)+2x=-x^{2}+3x+2+2$$

Add 2 and 2 to get 4.

$$x\left(-x\right)+2x=-x^{2}+3x+4$$

Add x^{2} to both sides.

$$x\left(-x\right)+2x+x^{2}=3x+4$$

Subtract 3x from both sides.

$$x\left(-x\right)+2x+x^{2}-3x=4$$

Combine 2x and -3x to get -x.

$$x\left(-x\right)-x+x^{2}=4$$

Multiply x and x to get x^{2}.

$$x^{2}\left(-1\right)-x+x^{2}=4$$

Combine x^{2}\left(-1\right) and x^{2} to get 0.

$$-x=4$$

Multiply both sides by -1.

$$x=-4$$