Finite Math Examples

$\frac{6}{x - 4} + \frac{x - 5}{x - 3}$

Step 1

To write $\frac{6}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{6}{x - 4} \cdot \frac{x - 3}{x - 3} + \frac{x - 5}{x - 3}$

Step 2

To write $\frac{x - 5}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{6}{x - 4} \cdot \frac{x - 3}{x - 3} + \frac{x - 5}{x - 3} \cdot \frac{x - 4}{x - 4}$

Step 3

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

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Step 3.1

Multiply $\frac{6}{x - 4}$ by $\frac{x - 3}{x - 3}$ .

$\frac{6 (x - 3)}{(x - 4) (x - 3)} + \frac{x - 5}{x - 3} \cdot \frac{x - 4}{x - 4}$

Step 3.2

Multiply $\frac{x - 5}{x - 3}$ by $\frac{x - 4}{x - 4}$ .

$\frac{6 (x - 3)}{(x - 4) (x - 3)} + \frac{(x - 5) (x - 4)}{(x - 3) (x - 4)}$

Step 3.3

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

$\frac{6 (x - 3)}{(x - 3) (x - 4)} + \frac{(x - 5) (x - 4)}{(x - 3) (x - 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{6 (x - 3) + (x - 5) (x - 4)}{(x - 3) (x - 4)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{6 x + 6 \cdot - 3 + (x - 5) (x - 4)}{(x - 3) (x - 4)}$

Step 5.2

Multiply $6$ by $- 3$ .

$\frac{6 x - 18 + (x - 5) (x - 4)}{(x - 3) (x - 4)}$

Step 5.3

Expand $(x - 5) (x - 4)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{6 x - 18 + x (x - 4) - 5 (x - 4)}{(x - 3) (x - 4)}$

Step 5.3.2

Apply the distributive property.

$\frac{6 x - 18 + x \cdot x + x \cdot - 4 - 5 (x - 4)}{(x - 3) (x - 4)}$

Step 5.3.3

Apply the distributive property.

$\frac{6 x - 18 + x \cdot x + x \cdot - 4 - 5 x - 5 \cdot - 4}{(x - 3) (x - 4)}$

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $x$ by $x$ .

$\frac{6 x - 18 + x^{2} + x \cdot - 4 - 5 x - 5 \cdot - 4}{(x - 3) (x - 4)}$

Step 5.4.1.2

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

$\frac{6 x - 18 + x^{2} - 4 \cdot x - 5 x - 5 \cdot - 4}{(x - 3) (x - 4)}$

Step 5.4.1.3

Multiply $- 5$ by $- 4$ .

$\frac{6 x - 18 + x^{2} - 4 x - 5 x + 20}{(x - 3) (x - 4)}$

Step 5.4.2

Subtract $5 x$ from $- 4 x$ .

$\frac{6 x - 18 + x^{2} - 9 x + 20}{(x - 3) (x - 4)}$

Step 5.5

Subtract $9 x$ from $6 x$ .

$\frac{- 3 x - 18 + x^{2} + 20}{(x - 3) (x - 4)}$

Step 5.6

Add $- 18$ and $20$ .

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

Step 5.7

Reorder terms.

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

Step 5.8

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 5.8.1

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 5.8.2

Write the factored form using these integers.

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