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

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

Evaluate

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

Short Solution Steps

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

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

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, add them by adding their numerators.

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

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

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

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

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

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

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

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)}.

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

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 1-x and 1+x is \left(x+1\right)\left(-x+1\right). Multiply \frac{1}{1-x} times \frac{x+1}{x+1}. Multiply \frac{1}{1+x} times \frac{-x+1}{-x+1}.

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

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, add them by adding their numerators.

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

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

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

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

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

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

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

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)}.

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

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

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