Pre-Algebra Examples

$\frac{5}{6 (x + 3)} + \frac{x + 4}{2 x}$

Step 1

To write $\frac{5}{6 (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{5}{6 (x + 3)} \cdot \frac{x}{x} + \frac{x + 4}{2 x}$

Step 2

To write $\frac{x + 4}{2 x}$ as a fraction with a common denominator, multiply by $\frac{3 (x + 3)}{3 (x + 3)}$ .

$\frac{5}{6 (x + 3)} \cdot \frac{x}{x} + \frac{x + 4}{2 x} \cdot \frac{3 (x + 3)}{3 (x + 3)}$

Step 3

Write each expression with a common denominator of $6 (x + 3) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{6 (x + 3)}$ by $\frac{x}{x}$ .

$\frac{5 x}{6 (x + 3) x} + \frac{x + 4}{2 x} \cdot \frac{3 (x + 3)}{3 (x + 3)}$

Step 3.2

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

$\frac{5 x}{6 (x + 3) x} + \frac{(x + 4) (3 (x + 3))}{2 x (3 (x + 3))}$

Step 3.3

Multiply $3$ by $2$ .

$\frac{5 x}{6 (x + 3) x} + \frac{(x + 4) (3 (x + 3))}{6 x (x + 3)}$

Step 3.4

Reorder the factors of $6 (x + 3) x$ .

$\frac{5 x}{6 x (x + 3)} + \frac{(x + 4) (3 (x + 3))}{6 x (x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 x + (x + 4) (3 (x + 3))}{6 x (x + 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{5 x + (x \cdot 3 + 4 \cdot 3) (x + 3)}{6 x (x + 3)}$

Step 5.2

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

$\frac{5 x + (3 \cdot x + 4 \cdot 3) (x + 3)}{6 x (x + 3)}$

Step 5.3

Multiply $4$ by $3$ .

$\frac{5 x + (3 \cdot x + 12) (x + 3)}{6 x (x + 3)}$

Step 5.4

Expand $(3 x + 12) (x + 3)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

$\frac{5 x + 3 x (x + 3) + 12 (x + 3)}{6 x (x + 3)}$

Step 5.4.2

Apply the distributive property.

$\frac{5 x + 3 x \cdot x + 3 x \cdot 3 + 12 (x + 3)}{6 x (x + 3)}$

Step 5.4.3

Apply the distributive property.

$\frac{5 x + 3 x \cdot x + 3 x \cdot 3 + 12 x + 12 \cdot 3}{6 x (x + 3)}$

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.5.1.1.1

Move $x$ .

$\frac{5 x + 3 (x \cdot x) + 3 x \cdot 3 + 12 x + 12 \cdot 3}{6 x (x + 3)}$

Step 5.5.1.1.2

Multiply $x$ by $x$ .

$\frac{5 x + 3 x^{2} + 3 x \cdot 3 + 12 x + 12 \cdot 3}{6 x (x + 3)}$

Step 5.5.1.2

Multiply $3$ by $3$ .

$\frac{5 x + 3 x^{2} + 9 x + 12 x + 12 \cdot 3}{6 x (x + 3)}$

Step 5.5.1.3

Multiply $12$ by $3$ .

$\frac{5 x + 3 x^{2} + 9 x + 12 x + 36}{6 x (x + 3)}$

Step 5.5.2

Add $9 x$ and $12 x$ .

$\frac{5 x + 3 x^{2} + 21 x + 36}{6 x (x + 3)}$

Step 5.6

Add $5 x$ and $21 x$ .

$\frac{3 x^{2} + 26 x + 36}{6 x (x + 3)}$