Pre-Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{x - 3}{(x - 1) (x + 3)} \cdot \frac{4 x - 4}{x - 2}$

Step 3

Factor $4$ out of $4 x - 4$ .

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Step 3.1

Factor $4$ out of $4 x$ .

$\frac{x - 3}{(x - 1) (x + 3)} \cdot \frac{4 (x) - 4}{x - 2}$

Step 3.2

Factor $4$ out of $- 4$ .

$\frac{x - 3}{(x - 1) (x + 3)} \cdot \frac{4 (x) + 4 (- 1)}{x - 2}$

Step 3.3

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

$\frac{x - 3}{(x - 1) (x + 3)} \cdot \frac{4 (x - 1)}{x - 2}$

Step 4

Cancel the common factor of $x - 1$ .

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Step 4.1

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

$\frac{x - 3}{(x - 1) (x + 3)} \cdot \frac{(x - 1) \cdot 4}{x - 2}$

Step 4.2

Cancel the common factor.

$\frac{x - 3}{(x - 1) (x + 3)} \cdot \frac{(x - 1) \cdot 4}{x - 2}$

Step 4.3

Rewrite the expression.

$\frac{x - 3}{x + 3} \cdot \frac{4}{x - 2}$

Step 5

Multiply $\frac{x - 3}{x + 3}$ by $\frac{4}{x - 2}$ .

$\frac{(x - 3) \cdot 4}{(x + 3) (x - 2)}$

Step 6

Move $4$ to the left of $x - 3$ .

$\frac{4 (x - 3)}{(x + 3) (x - 2)}$

Step 7

Apply the distributive property.

$\frac{4 x + 4 \cdot - 3}{(x + 3) (x - 2)}$

Step 8

Multiply $4$ by $- 3$ .

$\frac{4 x - 12}{(x + 3) (x - 2)}$

Step 9

Split the fraction $\frac{4 x - 12}{(x + 3) (x - 2)}$ into two fractions.

$\frac{4 x}{(x + 3) (x - 2)} + \frac{- 12}{(x + 3) (x - 2)}$

Step 10

Move the negative in front of the fraction.

$\frac{4 x}{(x + 3) (x - 2)} - \frac{12}{(x + 3) (x - 2)}$