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

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

Evaluate

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

Short Solution Steps

Factor 1-x^{2}.

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

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

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

Since \frac{x-1}{\left(x-1\right)\left(x+1\right)} and \frac{-2x}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-1-2x.

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

Extract the negative sign in -x-1.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

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

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

Multiply \frac{-1}{x-1} times \frac{1-x}{x} by multiplying numerator times numerator and denominator times denominator.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

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

Expand

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

Short Solution Steps

Factor 1-x^{2}.

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

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

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

Since \frac{x-1}{\left(x-1\right)\left(x+1\right)} and \frac{-2x}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-1-2x.

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

Extract the negative sign in -x-1.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

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

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

Multiply \frac{-1}{x-1} times \frac{1-x}{x} by multiplying numerator times numerator and denominator times denominator.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

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