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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{1}{x} and \frac{x}{x} have the same denominator, add them by adding their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Divide \frac{1+x}{x} by \frac{x-1}{x} by multiplying \frac{1+x}{x} by the reciprocal of \frac{x-1}{x}.

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

Cancel out x in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{1}{x} and \frac{x}{x} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Divide \frac{1-x}{x} by \frac{x+1}{x} by multiplying \frac{1-x}{x} by the reciprocal of \frac{x+1}{x}.

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

Cancel out x in both numerator and denominator.

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

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{x+1}{x-1} times \frac{x+1}{x+1}. Multiply \frac{-x+1}{x+1} times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in x^{2}+x+x+1+x^{2}-x+1-x.

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

Expand \left(x-1\right)\left(x+1\right).

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{1}{x} and \frac{x}{x} have the same denominator, add them by adding their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Divide \frac{1+x}{x} by \frac{x-1}{x} by multiplying \frac{1+x}{x} by the reciprocal of \frac{x-1}{x}.

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

Cancel out x in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{1}{x} and \frac{x}{x} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Divide \frac{1-x}{x} by \frac{x+1}{x} by multiplying \frac{1-x}{x} by the reciprocal of \frac{x+1}{x}.

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

Cancel out x in both numerator and denominator.

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

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{x+1}{x-1} times \frac{x+1}{x+1}. Multiply \frac{-x+1}{x+1} times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in x^{2}+x+x+1+x^{2}-x+1-x.

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

Expand \left(x-1\right)\left(x+1\right).

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