Solve (x-8/x^2+5x-6-2/x+6+2/x-1):1/x^2-1

$$( \frac { x - 8 } { x ^ { 2 } + 5 x - 6 } - \frac { 2 } { x + 6 } + \frac { 2 } { x - 1 } ) : \frac { 1 } { x ^ { 2 } - 1 }$$

Evaluate

$$x+1$$

Short Solution Steps

Factor x^{2}+5x-6.

$$\frac{\frac{x-8}{\left(x-1\right)\left(x+6\right)}-\frac{2}{x+6}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+6\right) and x+6 is \left(x-1\right)\left(x+6\right). Multiply \frac{2}{x+6} times \frac{x-1}{x-1}.

$$\frac{\frac{x-8}{\left(x-1\right)\left(x+6\right)}-\frac{2\left(x-1\right)}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Since \frac{x-8}{\left(x-1\right)\left(x+6\right)} and \frac{2\left(x-1\right)}{\left(x-1\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-8-2\left(x-1\right)}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Do the multiplications in x-8-2\left(x-1\right).

$$\frac{\frac{x-8-2x+2}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Combine like terms in x-8-2x+2.

$$\frac{\frac{-x-6}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Extract the negative sign in -x-6.

$$\frac{\frac{-\left(x+6\right)}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Cancel out x+6 in both numerator and denominator.

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

Since \frac{-1}{x-1} and \frac{2}{x-1} have the same denominator, add them by adding their numerators. Add -1 and 2 to get 1.

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

Divide \frac{1}{x-1} by \frac{1}{x^{2}-1} by multiplying \frac{1}{x-1} by the reciprocal of \frac{1}{x^{2}-1}.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

$$x+1$$

Expand

$$x+1$$

Short Solution Steps

Factor x^{2}+5x-6.

$$\frac{\frac{x-8}{\left(x-1\right)\left(x+6\right)}-\frac{2}{x+6}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+6\right) and x+6 is \left(x-1\right)\left(x+6\right). Multiply \frac{2}{x+6} times \frac{x-1}{x-1}.

$$\frac{\frac{x-8}{\left(x-1\right)\left(x+6\right)}-\frac{2\left(x-1\right)}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Since \frac{x-8}{\left(x-1\right)\left(x+6\right)} and \frac{2\left(x-1\right)}{\left(x-1\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-8-2\left(x-1\right)}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Do the multiplications in x-8-2\left(x-1\right).

$$\frac{\frac{x-8-2x+2}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Combine like terms in x-8-2x+2.

$$\frac{\frac{-x-6}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Extract the negative sign in -x-6.

$$\frac{\frac{-\left(x+6\right)}{\left(x-1\right)\left(x+6\right)}+\frac{2}{x-1}}{\frac{1}{x^{2}-1}}$$

Cancel out x+6 in both numerator and denominator.

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

Since \frac{-1}{x-1} and \frac{2}{x-1} have the same denominator, add them by adding their numerators. Add -1 and 2 to get 1.

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

Divide \frac{1}{x-1} by \frac{1}{x^{2}-1} by multiplying \frac{1}{x-1} by the reciprocal of \frac{1}{x^{2}-1}.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

$$x+1$$