Solve (1/x+1-1/x-1)div2/1-x

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

Evaluate

$$\frac{1}{x+1}$$

Short Solution Steps

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

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

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, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

Extract the negative sign in 1-x.

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

Cancel out 2\left(x-1\right) in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

$$\frac{1}{x+1}$$

Expand

$$\frac{1}{x+1}$$

Short Solution Steps

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

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

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, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

Extract the negative sign in 1-x.

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

Cancel out 2\left(x-1\right) in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

$$\frac{1}{x+1}$$