Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $3$ and $3 - \frac{3}{x + 3}$ .

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Step 1.1.1

Factor $3$ out of $3$ .

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

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $3$ out of $3$ .

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

Step 1.1.2.2

Factor $3$ out of $- \frac{3}{x + 3}$ .

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

Step 1.1.2.3

Factor $3$ out of $3 (1) + 3 (- \frac{1}{x + 3})$ .

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

Step 1.1.2.4

Cancel the common factor.

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

Step 1.1.2.5

Rewrite the expression.

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Subtract $1$ from $3$ .

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

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4

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

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

Step 1.5

Cancel the common factor of $3$ and $3 + \frac{3}{x + 3}$ .

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Step 1.5.1

Factor $3$ out of $3$ .

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $3$ out of $3$ .

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

Step 1.5.2.2

Factor $3$ out of $\frac{3}{x + 3}$ .

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

Step 1.5.2.3

Factor $3$ out of $3 \cdot 1 + 3 \frac{1}{x + 3}$ .

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

Step 1.5.2.4

Cancel the common factor.

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

Step 1.5.2.5

Rewrite the expression.

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

Step 1.6

Simplify the denominator.

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Step 1.6.1

Write $1$ as a fraction with a common denominator.

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

Step 1.6.2

Combine the numerators over the common denominator.

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

Step 1.6.3

Add $3$ and $1$ .

$\frac{x + 3}{x + 2} - \frac{1}{\frac{x + 4}{x + 3}}$

Step 1.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{x + 3}{x + 2} - (1 \frac{x + 3}{x + 4})$

Step 1.8

Multiply $\frac{x + 3}{x + 4}$ by $1$ .

$\frac{x + 3}{x + 2} - \frac{x + 3}{x + 4}$

Step 2

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

$\frac{x + 3}{x + 2} \cdot \frac{x + 4}{x + 4} - \frac{x + 3}{x + 4}$

Step 3

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

$\frac{x + 3}{x + 2} \cdot \frac{x + 4}{x + 4} - \frac{x + 3}{x + 4} \cdot \frac{x + 2}{x + 2}$

Step 4

Write each expression with a common denominator of $(x + 2) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x + 4) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $x + 3$ out of $(x + 3) (x + 4) - (x + 3) (x + 2)$ .

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Step 6.1.1

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

$\frac{(x + 3) (x + 4) + (x + 3) (- 1 (x + 2))}{(x + 2) (x + 4)}$

Step 6.1.2

Factor $x + 3$ out of $(x + 3) (x + 4) + (x + 3) (- 1 (x + 2))$ .

$\frac{(x + 3) (x + 4 - 1 (x + 2))}{(x + 2) (x + 4)}$

Step 6.2

Apply the distributive property.

$\frac{(x + 3) (x + 4 - 1 x - 1 \cdot 2)}{(x + 2) (x + 4)}$

Step 6.3

Rewrite $- 1 x$ as $- x$ .

$\frac{(x + 3) (x + 4 - x - 1 \cdot 2)}{(x + 2) (x + 4)}$

Step 6.4

Multiply $- 1$ by $2$ .

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

Step 6.5

Subtract $x$ from $x$ .

$\frac{(x + 3) (0 + 4 - 2)}{(x + 2) (x + 4)}$

Step 6.6

Add $0$ and $4$ .

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

Step 6.7

Subtract $2$ from $4$ .

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

Step 7

Move $2$ to the left of $x + 3$ .

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