Pre-Algebra Examples

$\frac{2 x}{x + 3} + \frac{7}{3 x + 1}$

Step 1

To write $\frac{2 x}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{3 x + 1}{3 x + 1}$ .

$\frac{2 x}{x + 3} \cdot \frac{3 x + 1}{3 x + 1} + \frac{7}{3 x + 1}$

Step 2

To write $\frac{7}{3 x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{2 x}{x + 3} \cdot \frac{3 x + 1}{3 x + 1} + \frac{7}{3 x + 1} \cdot \frac{x + 3}{x + 3}$

Step 3

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

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Step 3.1

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

$\frac{2 x (3 x + 1)}{(x + 3) (3 x + 1)} + \frac{7}{3 x + 1} \cdot \frac{x + 3}{x + 3}$

Step 3.2

Multiply $\frac{7}{3 x + 1}$ by $\frac{x + 3}{x + 3}$ .

$\frac{2 x (3 x + 1)}{(x + 3) (3 x + 1)} + \frac{7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 3.3

Reorder the factors of $(x + 3) (3 x + 1)$ .

$\frac{2 x (3 x + 1)}{(3 x + 1) (x + 3)} + \frac{7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 x (3 x + 1) + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{2 x (3 x) + 2 x \cdot 1 + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5.2

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 3 x \cdot x + 2 x \cdot 1 + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5.3

Multiply $2$ by $1$ .

$\frac{2 \cdot 3 x \cdot x + 2 x + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5.4

Simplify each term.

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Step 5.4.1

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

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Step 5.4.1.1

Move $x$ .

$\frac{2 \cdot 3 (x \cdot x) + 2 x + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5.4.1.2

Multiply $x$ by $x$ .

$\frac{2 \cdot 3 x^{2} + 2 x + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5.4.2

Multiply $2$ by $3$ .

$\frac{6 x^{2} + 2 x + 7 (x + 3)}{(3 x + 1) (x + 3)}$

Step 5.5

Apply the distributive property.

$\frac{6 x^{2} + 2 x + 7 x + 7 \cdot 3}{(3 x + 1) (x + 3)}$

Step 5.6

Multiply $7$ by $3$ .

$\frac{6 x^{2} + 2 x + 7 x + 21}{(3 x + 1) (x + 3)}$

Step 5.7

Add $2 x$ and $7 x$ .

$\frac{6 x^{2} + 9 x + 21}{(3 x + 1) (x + 3)}$

Step 5.8

Factor $3$ out of $6 x^{2} + 9 x + 21$ .

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Step 5.8.1

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

$\frac{3 (2 x^{2}) + 9 x + 21}{(3 x + 1) (x + 3)}$

Step 5.8.2

Factor $3$ out of $9 x$ .

$\frac{3 (2 x^{2}) + 3 (3 x) + 21}{(3 x + 1) (x + 3)}$

Step 5.8.3

Factor $3$ out of $21$ .

$\frac{3 (2 x^{2}) + 3 (3 x) + 3 (7)}{(3 x + 1) (x + 3)}$

Step 5.8.4

Factor $3$ out of $3 (2 x^{2}) + 3 (3 x)$ .

$\frac{3 (2 x^{2} + 3 x) + 3 (7)}{(3 x + 1) (x + 3)}$

Step 5.8.5

Factor $3$ out of $3 (2 x^{2} + 3 x) + 3 (7)$ .

$\frac{3 (2 x^{2} + 3 x + 7)}{(3 x + 1) (x + 3)}$