Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 3$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 3$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{x + 3}$ into the numerator.

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

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

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

Step 4.5

Multiply $3$ by $3$ .

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

Step 4.6

Add $3 x$ and $3 x$ .

$\frac{x + 2}{x^{2} + 6 x + 9 - 1}$

Step 4.7

Subtract $1$ from $9$ .

$\frac{x + 2}{x^{2} + 6 x + 8}$

Step 4.8

Factor $x^{2} + 6 x + 8$ using the AC method.

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Step 4.8.1

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 $8$ and whose sum is $6$ .

$2, 4$

Step 4.8.2

Write the factored form using these integers.

$\frac{x + 2}{(x + 2) (x + 4)}$

Step 5

Cancel the common factor of $x + 2$ .

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Step 5.1

Cancel the common factor.

$\frac{x + 2}{(x + 2) (x + 4)}$

Step 5.2

Rewrite the expression.

$\frac{1}{x + 4}$