Solve =x/x-4-(x-4/x-2+x-2/x-4-2):1/x-2

$$= \frac { x } { x - 4 } - ( \frac { x - 4 } { x - 2 } + \frac { x - 2 } { x - 4 } - 2 ) : \frac { 1 } { x - 2 }$$

Evaluate

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Short Solution Steps

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

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

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

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

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

$$\frac{x}{x-4}-\frac{\frac{x^{2}-4x-4x+16+x^{2}-2x-2x+4}{\left(x-4\right)\left(x-2\right)}-2}{\frac{1}{x-2}}$$

Combine like terms in x^{2}-4x-4x+16+x^{2}-2x-2x+4.

$$\frac{x}{x-4}-\frac{\frac{2x^{2}-12x+20}{\left(x-4\right)\left(x-2\right)}-2}{\frac{1}{x-2}}$$

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

$$\frac{x}{x-4}-\frac{\frac{2x^{2}-12x+20}{\left(x-4\right)\left(x-2\right)}-\frac{2\left(x-4\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)}}{\frac{1}{x-2}}$$

Since \frac{2x^{2}-12x+20}{\left(x-4\right)\left(x-2\right)} and \frac{2\left(x-4\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x}{x-4}-\frac{\frac{2x^{2}-12x+20-2\left(x-4\right)\left(x-2\right)}{\left(x-4\right)\left(x-2\right)}}{\frac{1}{x-2}}$$

Do the multiplications in 2x^{2}-12x+20-2\left(x-4\right)\left(x-2\right).

$$\frac{x}{x-4}-\frac{\frac{2x^{2}-12x+20-2x^{2}+4x+8x-16}{\left(x-4\right)\left(x-2\right)}}{\frac{1}{x-2}}$$

Combine like terms in 2x^{2}-12x+20-2x^{2}+4x+8x-16.

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

Divide \frac{4}{\left(x-4\right)\left(x-2\right)} by \frac{1}{x-2} by multiplying \frac{4}{\left(x-4\right)\left(x-2\right)} by the reciprocal of \frac{1}{x-2}.

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

Cancel out x-2 in both numerator and denominator.

$$\frac{x}{x-4}-\frac{4}{x-4}$$

Since \frac{x}{x-4} and \frac{4}{x-4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-4}{x-4}$$

Cancel out x-4 in both numerator and denominator.

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Factor

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