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

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

Evaluate

$$-\frac{3x^{2}+3x+1}{x\left(2x+1\right)}$$

Short Solution Steps

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

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

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

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

Combine like terms in x+1+x.

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

Divide \frac{x+1}{x} by \frac{2x+1}{x} by multiplying \frac{x+1}{x} by the reciprocal of \frac{2x+1}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

Combine like terms in x+x+1.

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

Divide \frac{2x+1}{x+1} by \frac{x}{x+1} by multiplying \frac{2x+1}{x+1} by the reciprocal of \frac{x}{x+1}.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

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

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

Do the multiplications in \left(x+1\right)x-\left(2x+1\right)\left(2x+1\right).

$$\frac{x^{2}+x-4x^{2}-2x-2x-1}{x\left(2x+1\right)}$$

Combine like terms in x^{2}+x-4x^{2}-2x-2x-1.

$$\frac{-3x^{2}-3x-1}{x\left(2x+1\right)}$$

Expand x\left(2x+1\right).

$$\frac{-3x^{2}-3x-1}{2x^{2}+x}$$

Expand

$$-\frac{3x^{2}+3x+1}{x\left(2x+1\right)}$$

Short Solution Steps

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

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

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

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

Combine like terms in x+1+x.

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

Divide \frac{x+1}{x} by \frac{2x+1}{x} by multiplying \frac{x+1}{x} by the reciprocal of \frac{2x+1}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

Combine like terms in x+x+1.

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

Divide \frac{2x+1}{x+1} by \frac{x}{x+1} by multiplying \frac{2x+1}{x+1} by the reciprocal of \frac{x}{x+1}.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

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

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

Do the multiplications in \left(x+1\right)x-\left(2x+1\right)\left(2x+1\right).

$$\frac{x^{2}+x-4x^{2}-2x-2x-1}{x\left(2x+1\right)}$$

Combine like terms in x^{2}+x-4x^{2}-2x-2x-1.

$$\frac{-3x^{2}-3x-1}{x\left(2x+1\right)}$$

Expand x\left(2x+1\right).

$$\frac{-3x^{2}-3x-1}{2x^{2}+x}$$