Solve (3/x^2-36-1/x+6)div(x-9/x-6)

$$( \frac { 3 } { x ^ { 2 } - 36 } - \frac { 1 } { x + 6 } ) \div ( \frac { x - 9 } { x - 6 } )$$

Evaluate

$$-\frac{1}{x+6}$$

Short Solution Steps

Factor x^{2}-36.

$$\frac{\frac{3}{\left(x-6\right)\left(x+6\right)}-\frac{1}{x+6}}{\frac{x-9}{x-6}}$$

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

$$\frac{\frac{3}{\left(x-6\right)\left(x+6\right)}-\frac{x-6}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Since \frac{3}{\left(x-6\right)\left(x+6\right)} and \frac{x-6}{\left(x-6\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3-\left(x-6\right)}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Do the multiplications in 3-\left(x-6\right).

$$\frac{\frac{3-x+6}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Combine like terms in 3-x+6.

$$\frac{\frac{9-x}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Divide \frac{9-x}{\left(x-6\right)\left(x+6\right)} by \frac{x-9}{x-6} by multiplying \frac{9-x}{\left(x-6\right)\left(x+6\right)} by the reciprocal of \frac{x-9}{x-6}.

$$\frac{\left(9-x\right)\left(x-6\right)}{\left(x-6\right)\left(x+6\right)\left(x-9\right)}$$

Extract the negative sign in 9-x.

$$\frac{-\left(x-9\right)\left(x-6\right)}{\left(x-9\right)\left(x-6\right)\left(x+6\right)}$$

Cancel out \left(x-9\right)\left(x-6\right) in both numerator and denominator.

$$\frac{-1}{x+6}$$

Expand

$$-\frac{1}{x+6}$$

Short Solution Steps

Factor x^{2}-36.

$$\frac{\frac{3}{\left(x-6\right)\left(x+6\right)}-\frac{1}{x+6}}{\frac{x-9}{x-6}}$$

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

$$\frac{\frac{3}{\left(x-6\right)\left(x+6\right)}-\frac{x-6}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Since \frac{3}{\left(x-6\right)\left(x+6\right)} and \frac{x-6}{\left(x-6\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3-\left(x-6\right)}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Do the multiplications in 3-\left(x-6\right).

$$\frac{\frac{3-x+6}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Combine like terms in 3-x+6.

$$\frac{\frac{9-x}{\left(x-6\right)\left(x+6\right)}}{\frac{x-9}{x-6}}$$

Divide \frac{9-x}{\left(x-6\right)\left(x+6\right)} by \frac{x-9}{x-6} by multiplying \frac{9-x}{\left(x-6\right)\left(x+6\right)} by the reciprocal of \frac{x-9}{x-6}.

$$\frac{\left(9-x\right)\left(x-6\right)}{\left(x-6\right)\left(x+6\right)\left(x-9\right)}$$

Extract the negative sign in 9-x.

$$\frac{-\left(x-9\right)\left(x-6\right)}{\left(x-9\right)\left(x-6\right)\left(x+6\right)}$$

Cancel out \left(x-9\right)\left(x-6\right) in both numerator and denominator.

$$\frac{-1}{x+6}$$