Solve x+6/x+8+x-7/x^2-64-x-7/x-8

$$\frac { x + 6 } { x + 8 } + \frac { x - 7 } { x ^ { 2 } - 64 } - \frac { x - 7 } { x - 8 }$$

Evaluate

$$\frac{1-2x}{x^{2}-64}$$

Short Solution Steps

Factor x^{2}-64.

$$\frac{x+6}{x+8}+\frac{x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+8 and \left(x-8\right)\left(x+8\right) is \left(x-8\right)\left(x+8\right). Multiply \frac{x+6}{x+8} times \frac{x-8}{x-8}.

$$\frac{\left(x+6\right)\left(x-8\right)}{\left(x-8\right)\left(x+8\right)}+\frac{x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

Since \frac{\left(x+6\right)\left(x-8\right)}{\left(x-8\right)\left(x+8\right)} and \frac{x-7}{\left(x-8\right)\left(x+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+6\right)\left(x-8\right)+x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

Do the multiplications in \left(x+6\right)\left(x-8\right)+x-7.

$$\frac{x^{2}-8x+6x-48+x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

Combine like terms in x^{2}-8x+6x-48+x-7.

$$\frac{x^{2}-x-55}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x+8\right) and x-8 is \left(x-8\right)\left(x+8\right). Multiply \frac{x-7}{x-8} times \frac{x+8}{x+8}.

$$\frac{x^{2}-x-55}{\left(x-8\right)\left(x+8\right)}-\frac{\left(x-7\right)\left(x+8\right)}{\left(x-8\right)\left(x+8\right)}$$

Since \frac{x^{2}-x-55}{\left(x-8\right)\left(x+8\right)} and \frac{\left(x-7\right)\left(x+8\right)}{\left(x-8\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-x-55-\left(x-7\right)\left(x+8\right)}{\left(x-8\right)\left(x+8\right)}$$

Do the multiplications in x^{2}-x-55-\left(x-7\right)\left(x+8\right).

$$\frac{x^{2}-x-55-x^{2}-8x+7x+56}{\left(x-8\right)\left(x+8\right)}$$

Combine like terms in x^{2}-x-55-x^{2}-8x+7x+56.

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

Expand \left(x-8\right)\left(x+8\right).

$$\frac{-2x+1}{x^{2}-64}$$

Expand

$$\frac{1-2x}{x^{2}-64}$$

Short Solution Steps

Factor x^{2}-64.

$$\frac{x+6}{x+8}+\frac{x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+8 and \left(x-8\right)\left(x+8\right) is \left(x-8\right)\left(x+8\right). Multiply \frac{x+6}{x+8} times \frac{x-8}{x-8}.

$$\frac{\left(x+6\right)\left(x-8\right)}{\left(x-8\right)\left(x+8\right)}+\frac{x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

Since \frac{\left(x+6\right)\left(x-8\right)}{\left(x-8\right)\left(x+8\right)} and \frac{x-7}{\left(x-8\right)\left(x+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+6\right)\left(x-8\right)+x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

Do the multiplications in \left(x+6\right)\left(x-8\right)+x-7.

$$\frac{x^{2}-8x+6x-48+x-7}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

Combine like terms in x^{2}-8x+6x-48+x-7.

$$\frac{x^{2}-x-55}{\left(x-8\right)\left(x+8\right)}-\frac{x-7}{x-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x+8\right) and x-8 is \left(x-8\right)\left(x+8\right). Multiply \frac{x-7}{x-8} times \frac{x+8}{x+8}.

$$\frac{x^{2}-x-55}{\left(x-8\right)\left(x+8\right)}-\frac{\left(x-7\right)\left(x+8\right)}{\left(x-8\right)\left(x+8\right)}$$

Since \frac{x^{2}-x-55}{\left(x-8\right)\left(x+8\right)} and \frac{\left(x-7\right)\left(x+8\right)}{\left(x-8\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-x-55-\left(x-7\right)\left(x+8\right)}{\left(x-8\right)\left(x+8\right)}$$

Do the multiplications in x^{2}-x-55-\left(x-7\right)\left(x+8\right).

$$\frac{x^{2}-x-55-x^{2}-8x+7x+56}{\left(x-8\right)\left(x+8\right)}$$

Combine like terms in x^{2}-x-55-x^{2}-8x+7x+56.

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

Expand \left(x-8\right)\left(x+8\right).

$$\frac{-2x+1}{x^{2}-64}$$