Solve 1/1+x+1-x/x

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand x\left(x+1\right).

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand x\left(x+1\right).

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