Finite Math Examples

$\frac{\frac{9}{x + 3} - \frac{3}{x + 7}}{\frac{x + 9}{x + 3}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{9}{x + 3} - \frac{3}{x + 7}) \frac{x + 3}{x + 9}$

Step 2

To write $\frac{9}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$(\frac{9}{x + 3} \cdot \frac{x + 7}{x + 7} - \frac{3}{x + 7}) \frac{x + 3}{x + 9}$

Step 3

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

$(\frac{9}{x + 3} \cdot \frac{x + 7}{x + 7} - \frac{3}{x + 7} \cdot \frac{x + 3}{x + 3}) \frac{x + 3}{x + 9}$

Step 4

Write each expression with a common denominator of $(x + 3) (x + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9}{x + 3}$ by $\frac{x + 7}{x + 7}$ .

$(\frac{9 (x + 7)}{(x + 3) (x + 7)} - \frac{3}{x + 7} \cdot \frac{x + 3}{x + 3}) \frac{x + 3}{x + 9}$

Step 4.2

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

$(\frac{9 (x + 7)}{(x + 3) (x + 7)} - \frac{3 (x + 3)}{(x + 7) (x + 3)}) \frac{x + 3}{x + 9}$

Step 4.3

Reorder the factors of $(x + 7) (x + 3)$ .

$(\frac{9 (x + 7)}{(x + 3) (x + 7)} - \frac{3 (x + 3)}{(x + 3) (x + 7)}) \frac{x + 3}{x + 9}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 (x + 7) - 3 (x + 3)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $3$ out of $9 (x + 7) - 3 (x + 3)$ .

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Step 6.1.1

Factor $3$ out of $9 (x + 7)$ .

$\frac{3 (3 (x + 7)) - 3 (x + 3)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.1.2

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

$\frac{3 (3 (x + 7)) + 3 (- (x + 3))}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.1.3

Factor $3$ out of $3 (3 (x + 7)) + 3 (- (x + 3))$ .

$\frac{3 (3 (x + 7) - (x + 3))}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.2

Apply the distributive property.

$\frac{3 (3 x + 3 \cdot 7 - (x + 3))}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.3

Multiply $3$ by $7$ .

$\frac{3 (3 x + 21 - (x + 3))}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.4

Apply the distributive property.

$\frac{3 (3 x + 21 - x - 1 \cdot 3)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.5

Multiply $- 1$ by $3$ .

$\frac{3 (3 x + 21 - x - 3)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.6

Subtract $x$ from $3 x$ .

$\frac{3 (2 x + 21 - 3)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.7

Subtract $3$ from $21$ .

$\frac{3 (2 x + 18)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.8

Factor $2$ out of $2 x + 18$ .

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Step 6.8.1

Factor $2$ out of $2 x$ .

$\frac{3 (2 (x) + 18)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.8.2

Factor $2$ out of $18$ .

$\frac{3 (2 x + 2 \cdot 9)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.8.3

Factor $2$ out of $2 x + 2 \cdot 9$ .

$\frac{3 (2 (x + 9))}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

$\frac{3 \cdot 2 (x + 9)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6.9

Multiply $3$ by $2$ .

$\frac{6 (x + 9)}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $x + 9$ .

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Step 7.1.1

Factor $x + 9$ out of $6 (x + 9)$ .

$\frac{(x + 9) \cdot 6}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 7.1.2

Cancel the common factor.

$\frac{(x + 9) \cdot 6}{(x + 3) (x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 7.1.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x + 3$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$\frac{6}{x + 7}$