Solve (3-x)+(1/x-1)/1-(3-x)*frac{1{(x-1)}}

$$\frac { ( 3 - x ) + ( \frac { 1 } { x - 1 } ) } { 1 - ( 3 - x ) \cdot \frac { 1 } { ( x - 1 ) } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3-x times \frac{x-1}{x-1}.

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

Since \frac{\left(3-x\right)\left(x-1\right)}{x-1} and \frac{1}{x-1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3-x\right)\left(x-1\right)+1.

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

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

$$\frac{\frac{4x-2-x^{2}}{x-1}}{1-\left(3-x\right)\times \frac{1}{x-1}}$$

Express \left(3-x\right)\times \frac{1}{x-1} as a single fraction.

$$\frac{\frac{4x-2-x^{2}}{x-1}}{1-\frac{3-x}{x-1}}$$

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

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{x-1}{x-1}-\frac{3-x}{x-1}}$$

Since \frac{x-1}{x-1} and \frac{3-x}{x-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{x-1-\left(3-x\right)}{x-1}}$$

Do the multiplications in x-1-\left(3-x\right).

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{x-1-3+x}{x-1}}$$

Combine like terms in x-1-3+x.

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{2x-4}{x-1}}$$

Divide \frac{4x-2-x^{2}}{x-1} by \frac{2x-4}{x-1} by multiplying \frac{4x-2-x^{2}}{x-1} by the reciprocal of \frac{2x-4}{x-1}.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{-x^{2}+4x-2}{2x-4}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3-x times \frac{x-1}{x-1}.

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

Since \frac{\left(3-x\right)\left(x-1\right)}{x-1} and \frac{1}{x-1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3-x\right)\left(x-1\right)+1.

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

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

$$\frac{\frac{4x-2-x^{2}}{x-1}}{1-\left(3-x\right)\times \frac{1}{x-1}}$$

Express \left(3-x\right)\times \frac{1}{x-1} as a single fraction.

$$\frac{\frac{4x-2-x^{2}}{x-1}}{1-\frac{3-x}{x-1}}$$

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

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{x-1}{x-1}-\frac{3-x}{x-1}}$$

Since \frac{x-1}{x-1} and \frac{3-x}{x-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{x-1-\left(3-x\right)}{x-1}}$$

Do the multiplications in x-1-\left(3-x\right).

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{x-1-3+x}{x-1}}$$

Combine like terms in x-1-3+x.

$$\frac{\frac{4x-2-x^{2}}{x-1}}{\frac{2x-4}{x-1}}$$

Divide \frac{4x-2-x^{2}}{x-1} by \frac{2x-4}{x-1} by multiplying \frac{4x-2-x^{2}}{x-1} by the reciprocal of \frac{2x-4}{x-1}.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{-x^{2}+4x-2}{2x-4}$$