Finite Math Examples

$\frac{2}{x - 9} - \frac{1}{x - 3} - \frac{1}{x - 4}$

Step 1

To write $\frac{2}{x - 9}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{2}{x - 9} \cdot \frac{x - 3}{x - 3} - \frac{1}{x - 3} - \frac{1}{x - 4}$

Step 2

To write $- \frac{1}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 9}{x - 9}$ .

$\frac{2}{x - 9} \cdot \frac{x - 3}{x - 3} - \frac{1}{x - 3} \cdot \frac{x - 9}{x - 9} - \frac{1}{x - 4}$

Step 3

Write each expression with a common denominator of $(x - 9) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{x - 9}$ by $\frac{x - 3}{x - 3}$ .

$\frac{2 (x - 3)}{(x - 9) (x - 3)} - \frac{1}{x - 3} \cdot \frac{x - 9}{x - 9} - \frac{1}{x - 4}$

Step 3.2

Multiply $\frac{1}{x - 3}$ by $\frac{x - 9}{x - 9}$ .

$\frac{2 (x - 3)}{(x - 9) (x - 3)} - \frac{x - 9}{(x - 3) (x - 9)} - \frac{1}{x - 4}$

Step 3.3

Reorder the factors of $(x - 9) (x - 3)$ .

$\frac{2 (x - 3)}{(x - 3) (x - 9)} - \frac{x - 9}{(x - 3) (x - 9)} - \frac{1}{x - 4}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 (x - 3) - (x - 9)}{(x - 3) (x - 9)} - \frac{1}{x - 4}$

Step 5

To write $\frac{2 (x - 3) - (x - 9)}{(x - 3) (x - 9)}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{2 (x - 3) - (x - 9)}{(x - 3) (x - 9)} \cdot \frac{x - 4}{x - 4} - \frac{1}{x - 4}$

Step 6

To write $- \frac{1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{(x - 3) (x - 9)}{(x - 3) (x - 9)}$ .

$\frac{2 (x - 3) - (x - 9)}{(x - 3) (x - 9)} \cdot \frac{x - 4}{x - 4} - \frac{1}{x - 4} \cdot \frac{(x - 3) (x - 9)}{(x - 3) (x - 9)}$

Step 7

Write each expression with a common denominator of $(x - 3) (x - 9) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{2 (x - 3) - (x - 9)}{(x - 3) (x - 9)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(2 (x - 3) - (x - 9)) (x - 4)}{(x - 3) (x - 9) (x - 4)} - \frac{1}{x - 4} \cdot \frac{(x - 3) (x - 9)}{(x - 3) (x - 9)}$

Step 7.2

Multiply $\frac{1}{x - 4}$ by $\frac{(x - 3) (x - 9)}{(x - 3) (x - 9)}$ .

$\frac{(2 (x - 3) - (x - 9)) (x - 4)}{(x - 3) (x - 9) (x - 4)} - \frac{(x - 3) (x - 9)}{(x - 4) ((x - 3) (x - 9))}$

Step 7.3

Reorder the factors of $(x - 3) (x - 9) (x - 4)$ .

$\frac{(2 (x - 3) - (x - 9)) (x - 4)}{(x - 3) (x - 4) (x - 9)} - \frac{(x - 3) (x - 9)}{(x - 4) ((x - 3) (x - 9))}$

Step 7.4

Reorder the factors of $(x - 4) ((x - 3) (x - 9))$ .

$\frac{(2 (x - 3) - (x - 9)) (x - 4)}{(x - 3) (x - 4) (x - 9)} - \frac{(x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 8

Combine the numerators over the common denominator.

$\frac{(2 (x - 3) - (x - 9)) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9

Rewrite $\frac{(2 (x - 3) - (x - 9)) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$ in a factored form.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot - 3 - (x - 9)) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.1.2

Multiply $2$ by $- 3$ .

$\frac{(2 x - 6 - (x - 9)) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.1.3

Apply the distributive property.

$\frac{(2 x - 6 - x - - 9) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.1.4

Multiply $- 1$ by $- 9$ .

$\frac{(2 x - 6 - x + 9) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.2

Subtract $x$ from $2 x$ .

$\frac{(x - 6 + 9) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.3

Add $- 6$ and $9$ .

$\frac{(x + 3) (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.4

Expand $(x + 3) (x - 4)$ using the FOIL Method.

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Step 9.4.1

Apply the distributive property.

$\frac{x (x - 4) + 3 (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.4.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 + 3 (x - 4) - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.4.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 + 3 x + 3 \cdot - 4 - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.5

Simplify and combine like terms.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 4 + 3 x + 3 \cdot - 4 - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.5.1.2

Move $- 4$ to the left of $x$ .

$\frac{x^{2} - 4 x + 3 x + 3 \cdot - 4 - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.5.1.3

Multiply $3$ by $- 4$ .

$\frac{x^{2} - 4 x + 3 x - 12 - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.5.2

Add $- 4 x$ and $3 x$ .

$\frac{x^{2} - x - 12 - (x - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.6

Apply the distributive property.

$\frac{x^{2} - x - 12 + (- x - - 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.7

Multiply $- 1$ by $- 3$ .

$\frac{x^{2} - x - 12 + (- x + 3) (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.8

Expand $(- x + 3) (x - 9)$ using the FOIL Method.

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Step 9.8.1

Apply the distributive property.

$\frac{x^{2} - x - 12 - x (x - 9) + 3 (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.8.2

Apply the distributive property.

$\frac{x^{2} - x - 12 - x \cdot x - x \cdot - 9 + 3 (x - 9)}{(x - 3) (x - 4) (x - 9)}$

Step 9.8.3

Apply the distributive property.

$\frac{x^{2} - x - 12 - x \cdot x - x \cdot - 9 + 3 x + 3 \cdot - 9}{(x - 3) (x - 4) (x - 9)}$

Step 9.9

Simplify and combine like terms.

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Step 9.9.1

Simplify each term.

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Step 9.9.1.1

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

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Step 9.9.1.1.1

Move $x$ .

$\frac{x^{2} - x - 12 - (x \cdot x) - x \cdot - 9 + 3 x + 3 \cdot - 9}{(x - 3) (x - 4) (x - 9)}$

Step 9.9.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - x - 12 - x^{2} - x \cdot - 9 + 3 x + 3 \cdot - 9}{(x - 3) (x - 4) (x - 9)}$

Step 9.9.1.2

Multiply $- 9$ by $- 1$ .

$\frac{x^{2} - x - 12 - x^{2} + 9 x + 3 x + 3 \cdot - 9}{(x - 3) (x - 4) (x - 9)}$

Step 9.9.1.3

Multiply $3$ by $- 9$ .

$\frac{x^{2} - x - 12 - x^{2} + 9 x + 3 x - 27}{(x - 3) (x - 4) (x - 9)}$

Step 9.9.2

Add $9 x$ and $3 x$ .

$\frac{x^{2} - x - 12 - x^{2} + 12 x - 27}{(x - 3) (x - 4) (x - 9)}$

Step 9.10

Subtract $x^{2}$ from $x^{2}$ .

$\frac{- x - 12 + 0 + 12 x - 27}{(x - 3) (x - 4) (x - 9)}$

Step 9.11

Add $- x$ and $0$ .

$\frac{- x - 12 + 12 x - 27}{(x - 3) (x - 4) (x - 9)}$

Step 9.12

Add $- x$ and $12 x$ .

$\frac{11 x - 12 - 27}{(x - 3) (x - 4) (x - 9)}$

Step 9.13

Subtract $27$ from $- 12$ .

$\frac{11 x - 39}{(x - 3) (x - 4) (x - 9)}$