Pre-Algebra Examples

$\frac{\frac{x - 1}{x^{2} + 4 x + 3}}{\frac{4 x - 4}{x^{2} + x - 6}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 1}{x^{2} + 4 x + 3} \cdot \frac{x^{2} + x - 6}{4 x - 4}$

Step 2

Factor $x^{2} + 4 x + 3$ 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 $4$ .

$1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{x - 1}{(x + 1) (x + 3)} \cdot \frac{x^{2} + x - 6}{4 x - 4}$

Step 3

Factor $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 3.2

Write the factored form using these integers.

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

Step 4

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

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Step 4.1

Factor $4$ out of $4 x$ .

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

Step 4.2

Factor $4$ out of $- 4$ .

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

Step 4.3

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

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

Step 5

Cancel the common factor of $x - 1$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $x + 3$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

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

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

Step 8

Move $4$ to the left of $x + 1$ .

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