Solve x-1/9-x+2x^2-16x+82/x^2-8x-9-x-9/x+1

$$\frac { x - 1 } { 9 - x } + \frac { 2 x ^ { 2 } - 16 x + 82 } { x ^ { 2 } - 8 x - 9 } - \frac { x - 9 } { x + 1 }$$

Evaluate

$$\frac{2}{x-9}$$

Short Solution Steps

Factor x^{2}-8x-9.

$$\frac{x-1}{9-x}+\frac{2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9-x and \left(x-9\right)\left(x+1\right) is \left(x-9\right)\left(x+1\right). Multiply \frac{x-1}{9-x} times \frac{-\left(x+1\right)}{-\left(x+1\right)}.

$$\frac{\left(x-1\right)\left(-1\right)\left(x+1\right)}{\left(x-9\right)\left(x+1\right)}+\frac{2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

Since \frac{\left(x-1\right)\left(-1\right)\left(x+1\right)}{\left(x-9\right)\left(x+1\right)} and \frac{2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-1\right)\left(-1\right)\left(x+1\right)+2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

Do the multiplications in \left(x-1\right)\left(-1\right)\left(x+1\right)+2x^{2}-16x+82.

$$\frac{-x^{2}-x+x+1+2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

Combine like terms in -x^{2}-x+x+1+2x^{2}-16x+82.

$$\frac{x^{2}-16x+83}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-9\right)\left(x+1\right) and x+1 is \left(x-9\right)\left(x+1\right). Multiply \frac{x-9}{x+1} times \frac{x-9}{x-9}.

$$\frac{x^{2}-16x+83}{\left(x-9\right)\left(x+1\right)}-\frac{\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+1\right)}$$

Since \frac{x^{2}-16x+83}{\left(x-9\right)\left(x+1\right)} and \frac{\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-16x+83-\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+1\right)}$$

Do the multiplications in x^{2}-16x+83-\left(x-9\right)\left(x-9\right).

$$\frac{x^{2}-16x+83-x^{2}+9x+9x-81}{\left(x-9\right)\left(x+1\right)}$$

Combine like terms in x^{2}-16x+83-x^{2}+9x+9x-81.

$$\frac{2x+2}{\left(x-9\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{2x+2}{\left(x-9\right)\left(x+1\right)}.

$$\frac{2\left(x+1\right)}{\left(x-9\right)\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{2}{x-9}$$

Expand

$$\frac{2}{x-9}$$

Short Solution Steps

Factor x^{2}-8x-9.

$$\frac{x-1}{9-x}+\frac{2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9-x and \left(x-9\right)\left(x+1\right) is \left(x-9\right)\left(x+1\right). Multiply \frac{x-1}{9-x} times \frac{-\left(x+1\right)}{-\left(x+1\right)}.

$$\frac{\left(x-1\right)\left(-1\right)\left(x+1\right)}{\left(x-9\right)\left(x+1\right)}+\frac{2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

Since \frac{\left(x-1\right)\left(-1\right)\left(x+1\right)}{\left(x-9\right)\left(x+1\right)} and \frac{2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-1\right)\left(-1\right)\left(x+1\right)+2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

Do the multiplications in \left(x-1\right)\left(-1\right)\left(x+1\right)+2x^{2}-16x+82.

$$\frac{-x^{2}-x+x+1+2x^{2}-16x+82}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

Combine like terms in -x^{2}-x+x+1+2x^{2}-16x+82.

$$\frac{x^{2}-16x+83}{\left(x-9\right)\left(x+1\right)}-\frac{x-9}{x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-9\right)\left(x+1\right) and x+1 is \left(x-9\right)\left(x+1\right). Multiply \frac{x-9}{x+1} times \frac{x-9}{x-9}.

$$\frac{x^{2}-16x+83}{\left(x-9\right)\left(x+1\right)}-\frac{\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+1\right)}$$

Since \frac{x^{2}-16x+83}{\left(x-9\right)\left(x+1\right)} and \frac{\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-16x+83-\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+1\right)}$$

Do the multiplications in x^{2}-16x+83-\left(x-9\right)\left(x-9\right).

$$\frac{x^{2}-16x+83-x^{2}+9x+9x-81}{\left(x-9\right)\left(x+1\right)}$$

Combine like terms in x^{2}-16x+83-x^{2}+9x+9x-81.

$$\frac{2x+2}{\left(x-9\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{2x+2}{\left(x-9\right)\left(x+1\right)}.

$$\frac{2\left(x+1\right)}{\left(x-9\right)\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{2}{x-9}$$