Solve (x/x-2-1/x-2)divx^3+x^2-2x/x^2-4

$$( \frac { x } { x - 2 } - \frac { 1 } { x - 2 } ) \div \frac { x ^ { 3 } + x ^ { 2 } - 2 x } { x ^ { 2 } - 4 }$$

Evaluate

$$\frac{1}{x}$$

Short Solution Steps

Since \frac{x}{x-2} and \frac{1}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-1}{x-2}}{\frac{x^{3}+x^{2}-2x}{x^{2}-4}}$$

Factor the expressions that are not already factored in \frac{x^{3}+x^{2}-2x}{x^{2}-4}.

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

Cancel out x+2 in both numerator and denominator.

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

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

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

Cancel out \left(x-2\right)\left(x-1\right) in both numerator and denominator.

$$\frac{1}{x}$$

Expand

$$\frac{1}{x}$$

Short Solution Steps

Since \frac{x}{x-2} and \frac{1}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-1}{x-2}}{\frac{x^{3}+x^{2}-2x}{x^{2}-4}}$$

Factor the expressions that are not already factored in \frac{x^{3}+x^{2}-2x}{x^{2}-4}.

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

Cancel out x+2 in both numerator and denominator.

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

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

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

Cancel out \left(x-2\right)\left(x-1\right) in both numerator and denominator.

$$\frac{1}{x}$$