Solve x+1+1/x-1/x^2+x+1+frac{1{x-1}}

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+x+1 times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in x^{3}-x^{2}+x^{2}-x+x-1+1.

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

Divide \frac{x^{2}}{x-1} by \frac{x^{3}}{x-1} by multiplying \frac{x^{2}}{x-1} by the reciprocal of \frac{x^{3}}{x-1}.

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+x+1 times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in x^{3}-x^{2}+x^{2}-x+x-1+1.

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

Divide \frac{x^{2}}{x-1} by \frac{x^{3}}{x-1} by multiplying \frac{x^{2}}{x-1} by the reciprocal of \frac{x^{3}}{x-1}.

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

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

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