Pre-Algebra Examples

$\frac{x - 1}{x^{2} - x - 6} \div \frac{x + 3}{3 x + 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x - 1}{x^{2} - x - 6} \cdot \frac{3 x + 6}{x + 3}$

Step 2

Factor $x^{2} - x - 6$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

$\frac{x - 1}{(x - 3) (x + 2)} \cdot \frac{3 x + 6}{x + 3}$

Step 3

Factor $3$ out of $3 x + 6$ .

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Step 3.1

Factor $3$ out of $3 x$ .

$\frac{x - 1}{(x - 3) (x + 2)} \cdot \frac{3 (x) + 6}{x + 3}$

Step 3.2

Factor $3$ out of $6$ .

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

Step 3.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

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

Step 4

Cancel the common factor of $x + 2$ .

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Step 4.1

Factor $x + 2$ out of $(x - 3) (x + 2)$ .

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

Step 4.2

Factor $x + 2$ out of $3 (x + 2)$ .

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

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

Step 5

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

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

Step 6

Move $3$ to the left of $x - 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $3$ by $- 1$ .

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

Step 9

Split the fraction $\frac{3 x - 3}{(x - 3) (x + 3)}$ into two fractions.

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

Step 10

Move the negative in front of the fraction.

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