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

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

Evaluate

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

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{x}{1-x} times \frac{x}{x}. Multiply \frac{1-x}{x} times \frac{-x+1}{-x+1}.

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

Since \frac{xx}{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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Expand

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

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{x}{1-x} times \frac{x}{x}. Multiply \frac{1-x}{x} times \frac{-x+1}{-x+1}.

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

Since \frac{xx}{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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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