Pre-Algebra Examples

$\frac{3}{3 x - 9} + \frac{x - 2}{3 - x}$

Step 1

Simplify terms.

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Step 1.1

Cancel the common factor of $3$ and $3 x - 9$ .

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Step 1.1.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 1}{3 x - 9} + \frac{x - 2}{3 - x}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{3 \cdot 1}{3 (x) - 9} + \frac{x - 2}{3 - x}$

Step 1.1.2.2

Factor $3$ out of $- 9$ .

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

Step 1.1.2.3

Factor $3$ out of $3 (x) + 3 (- 3)$ .

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

Step 1.1.2.4

Cancel the common factor.

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

Step 1.1.2.5

Rewrite the expression.

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

Step 1.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.3

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

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

Step 1.4

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

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

Step 1.5

Simplify the expression.

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Step 1.5.1

Move a negative from the denominator of $\frac{x - 2}{- 1 (- 3 + x)}$ to the numerator.

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

Step 1.5.2

Reorder terms.

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $- 1$ by $- 2$ .

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

Step 2.3

Add $1$ and $2$ .

$\frac{- x + 3}{x - 3}$

Step 3

Cancel the common factor of $- x + 3$ and $x - 3$ .

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Step 3.1

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

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

Step 3.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 3.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 3.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

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

Step 3.5

Cancel the common factor.

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

Step 3.6

Divide $- 1$ by $1$ .

$- 1$