Solve 2/6-x+3/x-6/frac{2{x}+4/x-6}

$$\frac { \frac { 2 } { 6 - x } + \frac { 3 } { x - 6 } } { \frac { 2 } { x } + \frac { 4 } { x - 6 } }$$

Evaluate

$$\frac{x}{6\left(x-2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6-x and x-6 is x-6. Multiply \frac{2}{6-x} times \frac{-1}{-1}.

$$\frac{\frac{2\left(-1\right)}{x-6}+\frac{3}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

Since \frac{2\left(-1\right)}{x-6} and \frac{3}{x-6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(-1\right)+3}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

Do the multiplications in 2\left(-1\right)+3.

$$\frac{\frac{-2+3}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

Do the calculations in -2+3.

$$\frac{\frac{1}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x-6 is x\left(x-6\right). Multiply \frac{2}{x} times \frac{x-6}{x-6}. Multiply \frac{4}{x-6} times \frac{x}{x}.

$$\frac{\frac{1}{x-6}}{\frac{2\left(x-6\right)}{x\left(x-6\right)}+\frac{4x}{x\left(x-6\right)}}$$

Since \frac{2\left(x-6\right)}{x\left(x-6\right)} and \frac{4x}{x\left(x-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1}{x-6}}{\frac{2\left(x-6\right)+4x}{x\left(x-6\right)}}$$

Do the multiplications in 2\left(x-6\right)+4x.

$$\frac{\frac{1}{x-6}}{\frac{2x-12+4x}{x\left(x-6\right)}}$$

Combine like terms in 2x-12+4x.

$$\frac{\frac{1}{x-6}}{\frac{6x-12}{x\left(x-6\right)}}$$

Divide \frac{1}{x-6} by \frac{6x-12}{x\left(x-6\right)} by multiplying \frac{1}{x-6} by the reciprocal of \frac{6x-12}{x\left(x-6\right)}.

$$\frac{x\left(x-6\right)}{\left(x-6\right)\left(6x-12\right)}$$

Cancel out x-6 in both numerator and denominator.

$$\frac{x}{6x-12}$$

Expand

$$\frac{x}{6\left(x-2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6-x and x-6 is x-6. Multiply \frac{2}{6-x} times \frac{-1}{-1}.

$$\frac{\frac{2\left(-1\right)}{x-6}+\frac{3}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

Since \frac{2\left(-1\right)}{x-6} and \frac{3}{x-6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(-1\right)+3}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

Do the multiplications in 2\left(-1\right)+3.

$$\frac{\frac{-2+3}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

Do the calculations in -2+3.

$$\frac{\frac{1}{x-6}}{\frac{2}{x}+\frac{4}{x-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x-6 is x\left(x-6\right). Multiply \frac{2}{x} times \frac{x-6}{x-6}. Multiply \frac{4}{x-6} times \frac{x}{x}.

$$\frac{\frac{1}{x-6}}{\frac{2\left(x-6\right)}{x\left(x-6\right)}+\frac{4x}{x\left(x-6\right)}}$$

Since \frac{2\left(x-6\right)}{x\left(x-6\right)} and \frac{4x}{x\left(x-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1}{x-6}}{\frac{2\left(x-6\right)+4x}{x\left(x-6\right)}}$$

Do the multiplications in 2\left(x-6\right)+4x.

$$\frac{\frac{1}{x-6}}{\frac{2x-12+4x}{x\left(x-6\right)}}$$

Combine like terms in 2x-12+4x.

$$\frac{\frac{1}{x-6}}{\frac{6x-12}{x\left(x-6\right)}}$$

Divide \frac{1}{x-6} by \frac{6x-12}{x\left(x-6\right)} by multiplying \frac{1}{x-6} by the reciprocal of \frac{6x-12}{x\left(x-6\right)}.

$$\frac{x\left(x-6\right)}{\left(x-6\right)\left(6x-12\right)}$$

Cancel out x-6 in both numerator and denominator.

$$\frac{x}{6x-12}$$