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

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

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

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.

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

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

To find the opposite of x^{2}+x, find the opposite of each term.

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

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

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

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

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

Subtract 2x from both sides.

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

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

$$-1-3x=-2$$

Add 1 to both sides.

$$-3x=-2+1$$

Add -2 and 1 to get -1.

$$-3x=-1$$

Divide both sides by -3.

$$x=\frac{-1}{-3}$$

Fraction \frac{-1}{-3} can be simplified to \frac{1}{3} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{1}{3}$$