Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x + 2) \cdot 3 x$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $3$ by $3$ .

$\frac{9 x + (x + 2) \cdot 3 x \frac{2}{3}}{(x + 2) \cdot 3 x \frac{2 x}{x + 2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

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

$\frac{9 x + 3 ((x + 2) x) \frac{2}{3}}{(x + 2) \cdot 3 x \frac{2 x}{x + 2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.3.2

Cancel the common factor.

$\frac{9 x + 3 ((x + 2) x) \frac{2}{3}}{(x + 2) \cdot 3 x \frac{2 x}{x + 2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.3.3

Rewrite the expression.

$\frac{9 x + (x + 2) x \cdot 2}{(x + 2) \cdot 3 x \frac{2 x}{x + 2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.4

Cancel the common factor of $x + 2$ .

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Step 3.4.1

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

$\frac{9 x + (x + 2) x \cdot 2}{(x + 2) (3 x) \frac{2 x}{x + 2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.4.2

Cancel the common factor.

$\frac{9 x + (x + 2) x \cdot 2}{(x + 2) (3 x) \frac{2 x}{x + 2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.4.3

Rewrite the expression.

$\frac{9 x + (x + 2) x \cdot 2}{3 x (2 x) + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.5

Multiply $2$ by $3$ .

$\frac{9 x + (x + 2) x \cdot 2}{6 x \cdot x + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.6

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

$\frac{9 x + (x + 2) x \cdot 2}{6 (x^{1} x) + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.7

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

$\frac{9 x + (x + 2) x \cdot 2}{6 (x^{1} x^{1}) + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{1 + 1} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.9

Add $1$ and $1$ .

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{2} + (x + 2) \cdot 3 x (- \frac{1}{x})}$

Step 3.10

Cancel the common factor of $x$ .

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Step 3.10.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{2} + (x + 2) \cdot 3 x \frac{- 1}{x}}$

Step 3.10.2

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

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{2} + x ((x + 2) \cdot 3) \frac{- 1}{x}}$

Step 3.10.3

Cancel the common factor.

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{2} + x ((x + 2) \cdot 3) \frac{- 1}{x}}$

Step 3.10.4

Rewrite the expression.

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{2} + (x + 2) \cdot 3 \cdot - 1}$

Step 3.11

Multiply $- 1$ by $3$ .

$\frac{9 x + (x + 2) x \cdot 2}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $9 x + (x + 2) x \cdot 2$ .

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Step 4.1.1

Factor $x$ out of $9 x$ .

$\frac{x \cdot 9 + (x + 2) x \cdot 2}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4.1.2

Factor $x$ out of $(x + 2) x \cdot 2$ .

$\frac{x \cdot 9 + x ((x + 2) \cdot 2)}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4.1.3

Factor $x$ out of $x \cdot 9 + x ((x + 2) \cdot 2)$ .

$\frac{x (9 + (x + 2) \cdot 2)}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4.2

Apply the distributive property.

$\frac{x (9 + x \cdot 2 + 2 \cdot 2)}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4.3

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

$\frac{x (9 + 2 \cdot x + 2 \cdot 2)}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4.4

Multiply $2$ by $2$ .

$\frac{x (9 + 2 \cdot x + 4)}{6 x^{2} + (x + 2) \cdot - 3}$

Step 4.5

Add $9$ and $4$ .

$\frac{x (2 x + 13)}{6 x^{2} + (x + 2) \cdot - 3}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $3$ out of $6 x^{2} + (x + 2) \cdot - 3$ .

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Step 5.1.1

Factor $3$ out of $6 x^{2}$ .

$\frac{x (2 x + 13)}{3 (2 x^{2}) + (x + 2) \cdot - 3}$

Step 5.1.2

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

$\frac{x (2 x + 13)}{3 (2 x^{2}) + 3 ((x + 2) \cdot - 1)}$

Step 5.1.3

Factor $3$ out of $3 (2 x^{2}) + 3 ((x + 2) \cdot - 1)$ .

$\frac{x (2 x + 13)}{3 (2 x^{2} + (x + 2) \cdot - 1)}$

Step 5.2

Apply the distributive property.

$\frac{x (2 x + 13)}{3 (2 x^{2} + x \cdot - 1 + 2 \cdot - 1)}$

Step 5.3

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

$\frac{x (2 x + 13)}{3 (2 x^{2} - 1 \cdot x + 2 \cdot - 1)}$

Step 5.4

Multiply $2$ by $- 1$ .

$\frac{x (2 x + 13)}{3 (2 x^{2} - 1 \cdot x - 2)}$

Step 5.5

Simplify each term.

$\frac{x (2 x + 13)}{3 (2 x^{2} - x - 2)}$