Solve 1/x-1+1/x^2-1/x-frac{2{x+1}}

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

Evaluate

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

Short Solution Steps

Factor x^{2}-1.

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

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

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

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

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

Combine like terms in x+1+1.

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

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

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

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

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

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

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+2 in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Factor x^{2}-1.

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

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

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

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

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

Combine like terms in x+1+1.

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

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

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

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

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

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

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+2 in both numerator and denominator.

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

Expand the expression.

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