Solve =(x/x+2-3/2-x-6x/x^2-4):(x-2)^2-1/x^2+x-2

$$= ( \frac { x } { x + 2 } - \frac { 3 } { 2 - x } - \frac { 6 x } { x ^ { 2 } - 4 } ) : \frac { ( x - 2 ) ^ { 2 } - 1 } { x ^ { 2 } + x - 2 }$$

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 2-x is \left(x+2\right)\left(-x+2\right). Multiply \frac{x}{x+2} times \frac{-x+2}{-x+2}. Multiply \frac{3}{2-x} times \frac{x+2}{x+2}.

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

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

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

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

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

Combine like terms in -x^{2}+2x-3x-6.

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

Factor x^{2}-4.

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

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

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

Since \frac{-\left(-x^{2}-x-6\right)}{\left(x-2\right)\left(x+2\right)} and \frac{6x}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(-x^{2}-x-6\right)-6x.

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

Combine like terms in x^{2}+x+6-6x.

$$\frac{\frac{x^{2}-5x+6}{\left(x-2\right)\left(x+2\right)}}{\frac{\left(x-2\right)^{2}-1}{x^{2}+x-2}}$$

Factor the expressions that are not already factored in \frac{x^{2}-5x+6}{\left(x-2\right)\left(x+2\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{\left(x-2\right)^{2}-1}{x^{2}+x-2}.

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

Cancel out x-1 in both numerator and denominator.

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

Divide \frac{x-3}{x+2} by \frac{x-3}{x+2} by multiplying \frac{x-3}{x+2} by the reciprocal of \frac{x-3}{x+2}.

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

Cancel out \left(x-3\right)\left(x+2\right) in both numerator and denominator.

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Factor

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