Solve (x-x/x+1)div(1+1/x^2-1)

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

Evaluate

$$x-1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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

Cancel out \left(x+1\right)x^{2} in both numerator and denominator.

$$x-1$$

Expand

$$x-1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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

Cancel out \left(x+1\right)x^{2} in both numerator and denominator.

$$x-1$$