Pre-Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $2 x - 6$ .

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Step 2.1.1

Factor $2$ out of $2 x$ .

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

Step 2.1.2

Factor $2$ out of $- 6$ .

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

Step 2.1.3

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

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

Step 2.2

Factor $x$ out of $x^{2} + x$ .

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Step 2.2.1

Factor $x$ out of $x^{2}$ .

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

Step 2.2.2

Raise $x$ to the power of $1$ .

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

Step 2.2.3

Factor $x$ out of $x^{1}$ .

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

Step 2.2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 3

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

$- 3, 1$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 3$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Factor $x$ out of $3 x$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

Multiply $\frac{2}{3}$ by $\frac{x + 1}{x + 1}$ .

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

Step 4.4

Cancel the common factor of $x + 1$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

$\frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$