Solve (4/5-x+9/x-5)div(2/x+3/x-5)

$$( \frac { 4 } { 5 - x } + \frac { 9 } { x - 5 } ) \div ( \frac { 2 } { x } + \frac { 3 } { x - 5 } )$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5-x and x-5 is x-5. Multiply \frac{4}{5-x} times \frac{-1}{-1}.

$$\frac{\frac{4\left(-1\right)}{x-5}+\frac{9}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

Since \frac{4\left(-1\right)}{x-5} and \frac{9}{x-5} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\left(-1\right)+9}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

Do the multiplications in 4\left(-1\right)+9.

$$\frac{\frac{-4+9}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

Do the calculations in -4+9.

$$\frac{\frac{5}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x-5 is x\left(x-5\right). Multiply \frac{2}{x} times \frac{x-5}{x-5}. Multiply \frac{3}{x-5} times \frac{x}{x}.

$$\frac{\frac{5}{x-5}}{\frac{2\left(x-5\right)}{x\left(x-5\right)}+\frac{3x}{x\left(x-5\right)}}$$

Since \frac{2\left(x-5\right)}{x\left(x-5\right)} and \frac{3x}{x\left(x-5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5}{x-5}}{\frac{2\left(x-5\right)+3x}{x\left(x-5\right)}}$$

Do the multiplications in 2\left(x-5\right)+3x.

$$\frac{\frac{5}{x-5}}{\frac{2x-10+3x}{x\left(x-5\right)}}$$

Combine like terms in 2x-10+3x.

$$\frac{\frac{5}{x-5}}{\frac{5x-10}{x\left(x-5\right)}}$$

Divide \frac{5}{x-5} by \frac{5x-10}{x\left(x-5\right)} by multiplying \frac{5}{x-5} by the reciprocal of \frac{5x-10}{x\left(x-5\right)}.

$$\frac{5x\left(x-5\right)}{\left(x-5\right)\left(5x-10\right)}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{5x}{5x-10}$$

Factor the expressions that are not already factored.

$$\frac{5x}{5\left(x-2\right)}$$

Cancel out 5 in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5-x and x-5 is x-5. Multiply \frac{4}{5-x} times \frac{-1}{-1}.

$$\frac{\frac{4\left(-1\right)}{x-5}+\frac{9}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

Since \frac{4\left(-1\right)}{x-5} and \frac{9}{x-5} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\left(-1\right)+9}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

Do the multiplications in 4\left(-1\right)+9.

$$\frac{\frac{-4+9}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

Do the calculations in -4+9.

$$\frac{\frac{5}{x-5}}{\frac{2}{x}+\frac{3}{x-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x-5 is x\left(x-5\right). Multiply \frac{2}{x} times \frac{x-5}{x-5}. Multiply \frac{3}{x-5} times \frac{x}{x}.

$$\frac{\frac{5}{x-5}}{\frac{2\left(x-5\right)}{x\left(x-5\right)}+\frac{3x}{x\left(x-5\right)}}$$

Since \frac{2\left(x-5\right)}{x\left(x-5\right)} and \frac{3x}{x\left(x-5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5}{x-5}}{\frac{2\left(x-5\right)+3x}{x\left(x-5\right)}}$$

Do the multiplications in 2\left(x-5\right)+3x.

$$\frac{\frac{5}{x-5}}{\frac{2x-10+3x}{x\left(x-5\right)}}$$

Combine like terms in 2x-10+3x.

$$\frac{\frac{5}{x-5}}{\frac{5x-10}{x\left(x-5\right)}}$$

Divide \frac{5}{x-5} by \frac{5x-10}{x\left(x-5\right)} by multiplying \frac{5}{x-5} by the reciprocal of \frac{5x-10}{x\left(x-5\right)}.

$$\frac{5x\left(x-5\right)}{\left(x-5\right)\left(5x-10\right)}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{5x}{5x-10}$$

Factor the expressions that are not already factored.

$$\frac{5x}{5\left(x-2\right)}$$

Cancel out 5 in both numerator and denominator.

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