Algebra Examples

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

Step 1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 2

Factor $- 1$ out of $- x$ .

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

Step 3

Factor $- 1$ out of $- 1 (- 1) - (x)$ .

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

Step 4

Reorder terms.

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

Step 5

To write $\frac{x - 3}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{x - 3}{x - 1} \cdot \frac{- 1}{- 1} - \frac{2}{- 1 (x - 1)}$

Step 6

Write each expression with a common denominator of $(x - 1) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{x - 3}{x - 1}$ by $\frac{- 1}{- 1}$ .

$\frac{(x - 3) \cdot - 1}{(x - 1) \cdot - 1} - \frac{2}{- 1 (x - 1)}$

Step 6.2

Reorder the factors of $(x - 1) \cdot - 1$ .

$\frac{(x - 3) \cdot - 1}{- (x - 1)} - \frac{2}{- 1 (x - 1)}$

Step 7

Combine the numerators over the common denominator.

$\frac{(x - 3) \cdot - 1 - 2}{- (x - 1)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{x \cdot - 1 - 3 \cdot - 1 - 2}{- (x - 1)}$

Step 8.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x - 3 \cdot - 1 - 2}{- (x - 1)}$

Step 8.3

Multiply $- 3$ by $- 1$ .

$\frac{- 1 \cdot x + 3 - 2}{- (x - 1)}$

Step 8.4

Rewrite $- 1 x$ as $- x$ .

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

Step 8.5

Subtract $2$ from $3$ .

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Cancel the common factor of $- x + 1$ and $x - 1$ .

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Step 9.1.1

Factor $- 1$ out of $- x$ .

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

Step 9.1.2

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (x) - 1 (- 1)}{- (x - 1)}$

Step 9.1.3

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$\frac{- (x - 1)}{- (x - 1)}$

Step 9.1.4

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{- 1 (x - 1)}{- (x - 1)}$

Step 9.1.5

Cancel the common factor.

$\frac{- 1 (x - 1)}{- (x - 1)}$

Step 9.1.6

Rewrite the expression.

$\frac{- 1}{- 1}$

Step 9.1.7

Move the negative one from the denominator of $\frac{- 1}{- 1}$ .

$- 1 \cdot - 1$

Step 9.2

Simplify the expression.

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Step 9.2.1

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$- - 1$

Step 9.2.2

Multiply $- 1$ by $- 1$ .

$1$