Solve 1/x-1/x+1/frac{1{x}+1/x+1}

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

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in x+1-x.

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

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

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

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

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

Combine like terms in x+1+x.

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

Combine like terms in x+1-x.

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

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

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

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

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

Combine like terms in x+1+x.

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

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

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

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

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