Solve 1/x+1+1/1-x/frac{1{x+1}+1/x-1}

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

Evaluate

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

Short Solution Steps

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

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

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.

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

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

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

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}.

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

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.

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

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

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

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

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

Extract the negative sign in x-1.

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

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

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

Expand

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

Short Solution Steps

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

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

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.

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

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

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

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}.

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

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.

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

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

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

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

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

Extract the negative sign in x-1.

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

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

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