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

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

Solve for x

$$x=0$$

Steps for Solving Linear Equation

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

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

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

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

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

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

$$x^{2}-2x+1=2x^{2}+2x-x^{2}+1$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-2x+1=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$x^{2}-2x+1-x^{2}=2x+1$$

Combine x^{2} and -x^{2} to get 0.

$$-2x+1=2x+1$$

Subtract 2x from both sides.

$$-2x+1-2x=1$$

Combine -2x and -2x to get -4x.

$$-4x+1=1$$

Subtract 1 from both sides.

$$-4x=1-1$$

Subtract 1 from 1 to get 0.

$$-4x=0$$

Product of two numbers is equal to 0 if at least one of them is 0. Since -4 is not equal to 0, x must be equal to 0.

$$x=0$$