Solve x+9/x-9+x-9/x+9/frac{x+9{x-9}-x-9/x+9}

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

Evaluate

$$\frac{x}{18}+\frac{9}{2x}$$

Short Solution Steps

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

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

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

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

Do the multiplications in \left(x+9\right)\left(x+9\right)+\left(x-9\right)\left(x-9\right).

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

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{x+9}{x-9}-\frac{x-9}{x+9}}$$

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{\left(x+9\right)\left(x+9\right)}{\left(x-9\right)\left(x+9\right)}-\frac{\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+9\right)}}$$

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{\left(x+9\right)\left(x+9\right)-\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+9\right)}}$$

Do the multiplications in \left(x+9\right)\left(x+9\right)-\left(x-9\right)\left(x-9\right).

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{x^{2}+9x+9x+81-x^{2}+9x+9x-81}{\left(x-9\right)\left(x+9\right)}}$$

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{36x}{\left(x-9\right)\left(x+9\right)}}$$

Divide \frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)} by \frac{36x}{\left(x-9\right)\left(x+9\right)} by multiplying \frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)} by the reciprocal of \frac{36x}{\left(x-9\right)\left(x+9\right)}.

$$\frac{\left(2x^{2}+162\right)\left(x-9\right)\left(x+9\right)}{\left(x-9\right)\left(x+9\right)\times 36x}$$

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

$$\frac{2x^{2}+162}{36x}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x^{2}+81\right)}{36x}$$

Cancel out 2 in both numerator and denominator.

$$\frac{x^{2}+81}{18x}$$

Expand

$$\frac{x}{18}+\frac{9}{2x}$$

Short Solution Steps

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

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

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

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

Do the multiplications in \left(x+9\right)\left(x+9\right)+\left(x-9\right)\left(x-9\right).

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

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{x+9}{x-9}-\frac{x-9}{x+9}}$$

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{\left(x+9\right)\left(x+9\right)}{\left(x-9\right)\left(x+9\right)}-\frac{\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+9\right)}}$$

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{\left(x+9\right)\left(x+9\right)-\left(x-9\right)\left(x-9\right)}{\left(x-9\right)\left(x+9\right)}}$$

Do the multiplications in \left(x+9\right)\left(x+9\right)-\left(x-9\right)\left(x-9\right).

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{x^{2}+9x+9x+81-x^{2}+9x+9x-81}{\left(x-9\right)\left(x+9\right)}}$$

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

$$\frac{\frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)}}{\frac{36x}{\left(x-9\right)\left(x+9\right)}}$$

Divide \frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)} by \frac{36x}{\left(x-9\right)\left(x+9\right)} by multiplying \frac{2x^{2}+162}{\left(x-9\right)\left(x+9\right)} by the reciprocal of \frac{36x}{\left(x-9\right)\left(x+9\right)}.

$$\frac{\left(2x^{2}+162\right)\left(x-9\right)\left(x+9\right)}{\left(x-9\right)\left(x+9\right)\times 36x}$$

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

$$\frac{2x^{2}+162}{36x}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x^{2}+81\right)}{36x}$$

Cancel out 2 in both numerator and denominator.

$$\frac{x^{2}+81}{18x}$$