Solve 1-x/2left(x/x+1-x/x-1right)

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in 1-x.

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

Cancel out 2\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

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in 1-x.

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

Cancel out 2\left(x-1\right) in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

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