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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{x}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Factor 1-x^{2}.

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

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)}.

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

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, subtract them by subtracting their numerators.

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

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

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

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

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

Multiply \frac{x+1}{x} times \frac{-x^{2}}{\left(x-1\right)\left(-x-1\right)} by multiplying numerator times numerator and denominator times denominator.

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

Extract the negative sign in x+1.

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

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

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

Multiply -1 and -1 to get 1.

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

Expand

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

Short Solution Steps

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

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

Since \frac{x}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Factor 1-x^{2}.

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

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)}.

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

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, subtract them by subtracting their numerators.

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

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

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

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

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

Multiply \frac{x+1}{x} times \frac{-x^{2}}{\left(x-1\right)\left(-x-1\right)} by multiplying numerator times numerator and denominator times denominator.

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

Extract the negative sign in x+1.

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

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

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

Multiply -1 and -1 to get 1.

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