Algebra Examples

$\frac{\frac{5}{x + 1} + \frac{2 x}{x + 3}}{\frac{x}{x + 1} + \frac{7}{x + 3}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 1) (x + 3)$ .

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Step 1.1

Multiply $\frac{\frac{5}{x + 1} + \frac{2 x}{x + 3}}{\frac{x}{x + 1} + \frac{7}{x + 3}}$ by $\frac{(x + 1) (x + 3)}{(x + 1) (x + 3)}$ .

$\frac{(x + 1) (x + 3)}{(x + 1) (x + 3)} \cdot \frac{\frac{5}{x + 1} + \frac{2 x}{x + 3}}{\frac{x}{x + 1} + \frac{7}{x + 3}}$

Step 1.2

Combine.

$\frac{(x + 1) (x + 3) (\frac{5}{x + 1} + \frac{2 x}{x + 3})}{(x + 1) (x + 3) (\frac{x}{x + 1} + \frac{7}{x + 3})}$

Step 2

Apply the distributive property.

$\frac{(x + 1) (x + 3) \frac{5}{x + 1} + (x + 1) (x + 3) \frac{2 x}{x + 3}}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 1$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(x + 1) (x + 3) \frac{5}{x + 1} + (x + 1) (x + 3) \frac{2 x}{x + 3}}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.1.2

Rewrite the expression.

$\frac{(x + 3) \cdot 5 + (x + 1) (x + 3) \frac{2 x}{x + 3}}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.2

Cancel the common factor of $x + 3$ .

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Step 3.2.1

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

$\frac{(x + 3) \cdot 5 + (x + 3) (x + 1) \frac{2 x}{x + 3}}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.2.2

Cancel the common factor.

$\frac{(x + 3) \cdot 5 + (x + 3) (x + 1) \frac{2 x}{x + 3}}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.2.3

Rewrite the expression.

$\frac{(x + 3) \cdot 5 + (x + 1) (2 x)}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x + 3) \cdot 5 + (x + 1) (2 x)}{(x + 1) (x + 3) \frac{x}{x + 1} + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.3.2

Rewrite the expression.

$\frac{(x + 3) \cdot 5 + (x + 1) (2 x)}{(x + 3) x + (x + 1) (x + 3) \frac{7}{x + 3}}$

Step 3.4

Cancel the common factor of $x + 3$ .

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Step 3.4.1

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

$\frac{(x + 3) \cdot 5 + (x + 1) (2 x)}{(x + 3) x + (x + 3) (x + 1) \frac{7}{x + 3}}$

Step 3.4.2

Cancel the common factor.

$\frac{(x + 3) \cdot 5 + (x + 1) (2 x)}{(x + 3) x + (x + 3) (x + 1) \frac{7}{x + 3}}$

Step 3.4.3

Rewrite the expression.

$\frac{(x + 3) \cdot 5 + (x + 1) (2 x)}{(x + 3) x + (x + 1) \cdot 7}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x \cdot 5 + 3 \cdot 5 + (x + 1) \cdot 2 x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.2

Move $5$ to the left of $x$ .

$\frac{5 \cdot x + 3 \cdot 5 + (x + 1) \cdot 2 x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.3

Multiply $3$ by $5$ .

$\frac{5 \cdot x + 15 + (x + 1) \cdot 2 x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.4

Apply the distributive property.

$\frac{5 x + 15 + (x \cdot 2 + 1 \cdot 2) x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.5

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

$\frac{5 x + 15 + (2 \cdot x + 1 \cdot 2) x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.6

Multiply $2$ by $1$ .

$\frac{5 x + 15 + (2 \cdot x + 2) x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.7

Apply the distributive property.

$\frac{5 x + 15 + 2 x \cdot x + 2 x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.8

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

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Step 4.8.1

Move $x$ .

$\frac{5 x + 15 + 2 (x \cdot x) + 2 x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.8.2

Multiply $x$ by $x$ .

$\frac{5 x + 15 + 2 x^{2} + 2 x}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.9

Add $5 x$ and $2 x$ .

$\frac{7 x + 15 + 2 x^{2}}{(x + 3) x + (x + 1) \cdot 7}$

Step 4.10

Reorder terms.

$\frac{2 x^{2} + 7 x + 15}{(x + 3) x + (x + 1) \cdot 7}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{2 x^{2} + 7 x + 15}{x \cdot x + 3 x + (x + 1) \cdot 7}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 7 x + 15}{x^{2} + 3 x + (x + 1) \cdot 7}$

Step 5.3

Apply the distributive property.

$\frac{2 x^{2} + 7 x + 15}{x^{2} + 3 x + x \cdot 7 + 1 \cdot 7}$

Step 5.4

Move $7$ to the left of $x$ .

$\frac{2 x^{2} + 7 x + 15}{x^{2} + 3 x + 7 \cdot x + 1 \cdot 7}$

Step 5.5

Multiply $7$ by $1$ .

$\frac{2 x^{2} + 7 x + 15}{x^{2} + 3 x + 7 \cdot x + 7}$

Step 5.6

Add $3 x$ and $7 x$ .

$\frac{2 x^{2} + 7 x + 15}{x^{2} + 10 x + 7}$