Solve x/x-10+x/x+9/frac{x{x-10}-13x/x+9}][139-12x]

$$\frac { \frac { x } { x - 10 } + \frac { x } { x + 9 } } { \frac { x } { x - 10 } - \frac { 13 x } { x + 9 } } ] [ 139 - 12 x ]$$

Evaluate

$$\frac{2x-1}{139-12x}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-10 and x+9 is \left(x-10\right)\left(x+9\right). Multiply \frac{x}{x-10} times \frac{x+9}{x+9}. Multiply \frac{x}{x+9} times \frac{x-10}{x-10}.

$$\frac{\frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)}+\frac{x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

Since \frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)} and \frac{x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x\left(x+9\right)+x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

Do the multiplications in x\left(x+9\right)+x\left(x-10\right).

$$\frac{\frac{x^{2}+9x+x^{2}-10x}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

Combine like terms in x^{2}+9x+x^{2}-10x.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-10 and x+9 is \left(x-10\right)\left(x+9\right). Multiply \frac{x}{x-10} times \frac{x+9}{x+9}. Multiply \frac{13x}{x+9} times \frac{x-10}{x-10}.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)}-\frac{13x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}$$

Since \frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)} and \frac{13x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x\left(x+9\right)-13x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}$$

Do the multiplications in x\left(x+9\right)-13x\left(x-10\right).

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x^{2}+9x-13x^{2}+130x}{\left(x-10\right)\left(x+9\right)}}$$

Combine like terms in x^{2}+9x-13x^{2}+130x.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{-12x^{2}+139x}{\left(x-10\right)\left(x+9\right)}}$$

Divide \frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)} by \frac{-12x^{2}+139x}{\left(x-10\right)\left(x+9\right)} by multiplying \frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)} by the reciprocal of \frac{-12x^{2}+139x}{\left(x-10\right)\left(x+9\right)}.

$$\frac{\left(2x^{2}-x\right)\left(x-10\right)\left(x+9\right)}{\left(x-10\right)\left(x+9\right)\left(-12x^{2}+139x\right)}$$

Cancel out \left(x-10\right)\left(x+9\right) in both numerator and denominator.

$$\frac{2x^{2}-x}{-12x^{2}+139x}$$

Factor the expressions that are not already factored.

$$\frac{x\left(2x-1\right)}{x\left(-12x+139\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{2x-1}{-12x+139}$$

Expand

$$\frac{2x-1}{139-12x}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-10 and x+9 is \left(x-10\right)\left(x+9\right). Multiply \frac{x}{x-10} times \frac{x+9}{x+9}. Multiply \frac{x}{x+9} times \frac{x-10}{x-10}.

$$\frac{\frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)}+\frac{x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

Since \frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)} and \frac{x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x\left(x+9\right)+x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

Do the multiplications in x\left(x+9\right)+x\left(x-10\right).

$$\frac{\frac{x^{2}+9x+x^{2}-10x}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

Combine like terms in x^{2}+9x+x^{2}-10x.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x}{x-10}-\frac{13x}{x+9}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-10 and x+9 is \left(x-10\right)\left(x+9\right). Multiply \frac{x}{x-10} times \frac{x+9}{x+9}. Multiply \frac{13x}{x+9} times \frac{x-10}{x-10}.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)}-\frac{13x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}$$

Since \frac{x\left(x+9\right)}{\left(x-10\right)\left(x+9\right)} and \frac{13x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x\left(x+9\right)-13x\left(x-10\right)}{\left(x-10\right)\left(x+9\right)}}$$

Do the multiplications in x\left(x+9\right)-13x\left(x-10\right).

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{x^{2}+9x-13x^{2}+130x}{\left(x-10\right)\left(x+9\right)}}$$

Combine like terms in x^{2}+9x-13x^{2}+130x.

$$\frac{\frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)}}{\frac{-12x^{2}+139x}{\left(x-10\right)\left(x+9\right)}}$$

Divide \frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)} by \frac{-12x^{2}+139x}{\left(x-10\right)\left(x+9\right)} by multiplying \frac{2x^{2}-x}{\left(x-10\right)\left(x+9\right)} by the reciprocal of \frac{-12x^{2}+139x}{\left(x-10\right)\left(x+9\right)}.

$$\frac{\left(2x^{2}-x\right)\left(x-10\right)\left(x+9\right)}{\left(x-10\right)\left(x+9\right)\left(-12x^{2}+139x\right)}$$

Cancel out \left(x-10\right)\left(x+9\right) in both numerator and denominator.

$$\frac{2x^{2}-x}{-12x^{2}+139x}$$

Factor the expressions that are not already factored.

$$\frac{x\left(2x-1\right)}{x\left(-12x+139\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{2x-1}{-12x+139}$$