Pre-Algebra Examples

$\frac{x + 3}{x^{2} + x - 2} \div \frac{4 x + 12}{x^{2} - 3 x + 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 3}{x^{2} + x - 2} \cdot \frac{x^{2} - 3 x + 2}{4 x + 12}$

Step 2

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 2.2

Write the factored form using these integers.

$\frac{x + 3}{(x - 1) (x + 2)} \cdot \frac{x^{2} - 3 x + 2}{4 x + 12}$

Step 3

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 3.1

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 3.2

Write the factored form using these integers.

$\frac{x + 3}{(x - 1) (x + 2)} \cdot \frac{(x - 2) (x - 1)}{4 x + 12}$

Step 4

Factor $4$ out of $4 x + 12$ .

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Step 4.1

Factor $4$ out of $4 x$ .

$\frac{x + 3}{(x - 1) (x + 2)} \cdot \frac{(x - 2) (x - 1)}{4 (x) + 12}$

Step 4.2

Factor $4$ out of $12$ .

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

Step 4.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

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

Step 5

Cancel the common factor of $x + 3$ .

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Step 5.1

Factor $x + 3$ out of $4 (x + 3)$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $x - 1$ .

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Step 6.1

Factor $x - 1$ out of $(x - 2) (x - 1)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\frac{1}{x + 2}$ by $\frac{x - 2}{4}$ .

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

Step 8

Move $4$ to the left of $x + 2$ .

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