Algebra Examples

$\frac{x - 10}{x - 4} - \frac{x + 2}{4 - x}$

Step 1

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x - 10}{x - 4} - \frac{x + 2}{- 1 (- 4) - x}$

Step 2

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

$\frac{x - 10}{x - 4} - \frac{x + 2}{- 1 (- 4) - (x)}$

Step 3

Factor $- 1$ out of $- 1 (- 4) - (x)$ .

$\frac{x - 10}{x - 4} - \frac{x + 2}{- 1 (- 4 + x)}$

Step 4

Reorder terms.

$\frac{x - 10}{x - 4} - \frac{x + 2}{- 1 (x - 4)}$

Step 5

To write $\frac{x - 10}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{x - 10}{x - 4} \cdot \frac{- 1}{- 1} - \frac{x + 2}{- 1 (x - 4)}$

Step 6

Write each expression with a common denominator of $(x - 4) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{x - 10}{x - 4}$ by $\frac{- 1}{- 1}$ .

$\frac{(x - 10) \cdot - 1}{(x - 4) \cdot - 1} - \frac{x + 2}{- 1 (x - 4)}$

Step 6.2

Reorder the factors of $(x - 4) \cdot - 1$ .

$\frac{(x - 10) \cdot - 1}{- (x - 4)} - \frac{x + 2}{- 1 (x - 4)}$

Step 7

Combine the numerators over the common denominator.

$\frac{(x - 10) \cdot - 1 - (x + 2)}{- (x - 4)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{x \cdot - 1 - 10 \cdot - 1 - (x + 2)}{- (x - 4)}$

Step 8.2

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

$\frac{- 1 \cdot x - 10 \cdot - 1 - (x + 2)}{- (x - 4)}$

Step 8.3

Multiply $- 10$ by $- 1$ .

$\frac{- 1 \cdot x + 10 - (x + 2)}{- (x - 4)}$

Step 8.4

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

$\frac{- x + 10 - (x + 2)}{- (x - 4)}$

Step 8.5

Apply the distributive property.

$\frac{- x + 10 - x - 1 \cdot 2}{- (x - 4)}$

Step 8.6

Multiply $- 1$ by $2$ .

$\frac{- x + 10 - x - 2}{- (x - 4)}$

Step 8.7

Subtract $x$ from $- x$ .

$\frac{- 2 x + 10 - 2}{- (x - 4)}$

Step 8.8

Subtract $2$ from $10$ .

$\frac{- 2 x + 8}{- (x - 4)}$

Step 8.9

Factor $2$ out of $- 2 x + 8$ .

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Step 8.9.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) + 8}{- (x - 4)}$

Step 8.9.2

Factor $2$ out of $8$ .

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

Step 8.9.3

Factor $2$ out of $2 (- x) + 2 (4)$ .

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Cancel the common factor of $- x + 4$ and $x - 4$ .

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Step 9.1.1

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

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

Step 9.1.2

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 9.1.3

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{2 (- (x - 4))}{- (x - 4)}$

Step 9.1.4

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 9.1.5

Cancel the common factor.

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

Step 9.1.6

Rewrite the expression.

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

Step 9.1.7

Move the negative one from the denominator of $\frac{2 \cdot (- 1)}{- 1}$ .

$- 1 \cdot (2 \cdot (- 1))$

Step 9.2

Rewrite $- 1 \cdot (2 \cdot (- 1))$ as $- (2 \cdot (- 1))$ .

$- (2 \cdot (- 1))$

Step 10

Multiply $- (2 \cdot (- 1))$ .

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Step 10.1

Multiply $2$ by $- 1$ .

$- - 2$

Step 10.2

Multiply $- 1$ by $- 2$ .

$2$