Solve [x-1/x^2+x-x+1/x^2-x]:4/x^2-1

$$[ \frac { x - 1 } { x ^ { 2 } + x } - \frac { x + 1 } { x ^ { 2 } - x } ] : \frac { 4 } { x ^ { 2 } - 1 }$$

Evaluate

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Short Solution Steps

Factor x^{2}+x. Factor x^{2}-x.

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

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

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

Since \frac{\left(x-1\right)\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{\left(x+1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in x^{2}-x-x+1-x^{2}-x-1-x.

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

Cancel out x in both numerator and denominator.

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

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

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

Cancel out 4 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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Factor

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