Solve (1/x^2-1-2/x^2+x-2/x^2-x):(1/1-x+frac{1}{x+1})

$$( \frac { 1 } { x ^ { 2 } - 1 } - \frac { 2 } { x ^ { 2 } + x } - \frac { 2 } { x ^ { 2 } - x } ) : ( \frac { 1 } { 1 - x } + \frac { 1 } { x + 1 } )$$

Evaluate

$$\frac{3}{2}=1.5$$

Short Solution Steps

Factor x^{2}-1. Factor x^{2}+x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x\left(x+1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{1}{\left(x-1\right)\left(x+1\right)} times \frac{x}{x}. Multiply \frac{2}{x\left(x+1\right)} times \frac{x-1}{x-1}.

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

Since \frac{x}{x\left(x-1\right)\left(x+1\right)} and \frac{2\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x-2\left(x-1\right).

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

Combine like terms in x-2x+2.

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

Factor x^{2}-x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-1\right)\left(x+1\right) and x\left(x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{2}{x\left(x-1\right)} times \frac{x+1}{x+1}.

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

Since \frac{-x+2}{x\left(x-1\right)\left(x+1\right)} and \frac{2\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -x+2-2\left(x+1\right).

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

Combine like terms in -x+2-2x-2.

$$\frac{\frac{-3x}{x\left(x-1\right)\left(x+1\right)}}{\frac{1}{1-x}+\frac{1}{x+1}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{-3}{\left(x-1\right)\left(x+1\right)}}{\frac{1}{1-x}+\frac{1}{x+1}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1-x and x+1 is \left(x+1\right)\left(-x+1\right). Multiply \frac{1}{1-x} times \frac{x+1}{x+1}. Multiply \frac{1}{x+1} times \frac{-x+1}{-x+1}.

$$\frac{\frac{-3}{\left(x-1\right)\left(x+1\right)}}{\frac{x+1}{\left(x+1\right)\left(-x+1\right)}+\frac{-x+1}{\left(x+1\right)\left(-x+1\right)}}$$

Since \frac{x+1}{\left(x+1\right)\left(-x+1\right)} and \frac{-x+1}{\left(x+1\right)\left(-x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-3}{\left(x-1\right)\left(x+1\right)}}{\frac{x+1-x+1}{\left(x+1\right)\left(-x+1\right)}}$$

Combine like terms in x+1-x+1.

$$\frac{\frac{-3}{\left(x-1\right)\left(x+1\right)}}{\frac{2}{\left(x+1\right)\left(-x+1\right)}}$$

Divide \frac{-3}{\left(x-1\right)\left(x+1\right)} by \frac{2}{\left(x+1\right)\left(-x+1\right)} by multiplying \frac{-3}{\left(x-1\right)\left(x+1\right)} by the reciprocal of \frac{2}{\left(x+1\right)\left(-x+1\right)}.

$$\frac{-3\left(x+1\right)\left(-x+1\right)}{\left(x-1\right)\left(x+1\right)\times 2}$$

Extract the negative sign in -x+1.

$$\frac{-3\left(-1\right)\left(x-1\right)\left(x+1\right)}{2\left(x-1\right)\left(x+1\right)}$$

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

$$\frac{-3\left(-1\right)}{2}$$

Multiply -3 and -1 to get 3.

$$\frac{3}{2}$$

Factor

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$