Finite Math Examples

$\frac{x + 5}{x^{2} + x - 12} - \frac{x - 7}{x^{2} + 7 x + 12}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} + x - 12$ using the AC method.

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Step 1.1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{x + 5}{(x - 3) (x + 4)} - \frac{x - 7}{x^{2} + 7 x + 12}$

Step 1.2

Factor $x^{2} + 7 x + 12$ using the AC method.

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Step 1.2.1

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 $12$ and whose sum is $7$ .

$3, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{x + 5}{(x - 3) (x + 4)} - \frac{x - 7}{(x + 3) (x + 4)}$

Step 2

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

$\frac{x + 5}{(x - 3) (x + 4)} \cdot \frac{x + 3}{x + 3} - \frac{x - 7}{(x + 3) (x + 4)}$

Step 3

To write $- \frac{x - 7}{(x + 3) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 5}{(x - 3) (x + 4)} \cdot \frac{x + 3}{x + 3} - \frac{x - 7}{(x + 3) (x + 4)} \cdot \frac{x - 3}{x - 3}$

Step 4

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

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Step 4.1

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

$\frac{(x + 5) (x + 3)}{(x - 3) (x + 4) (x + 3)} - \frac{x - 7}{(x + 3) (x + 4)} \cdot \frac{x - 3}{x - 3}$

Step 4.2

Multiply $\frac{x - 7}{(x + 3) (x + 4)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x + 5) (x + 3)}{(x - 3) (x + 4) (x + 3)} - \frac{(x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 4.3

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

$\frac{(x + 5) (x + 3)}{(x + 3) (x + 4) (x - 3)} - \frac{(x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 5) (x + 3) - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x + 5) (x + 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{x (x + 3) + 5 (x + 3) - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 5 (x + 3) - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 5 x + 5 \cdot 3 - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 3 + 5 x + 5 \cdot 3 - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.2.1.2

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

$\frac{x^{2} + 3 \cdot x + 5 x + 5 \cdot 3 - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.2.1.3

Multiply $5$ by $3$ .

$\frac{x^{2} + 3 x + 5 x + 15 - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.2.2

Add $3 x$ and $5 x$ .

$\frac{x^{2} + 8 x + 15 - (x - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 + (- x - - 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.4

Multiply $- 1$ by $- 7$ .

$\frac{x^{2} + 8 x + 15 + (- x + 7) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.5

Expand $(- x + 7) (x - 3)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 - x (x - 3) + 7 (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.5.2

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 - x \cdot x - x \cdot - 3 + 7 (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Step 6.5.3

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 - x \cdot x - x \cdot - 3 + 7 x + 7 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

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

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Step 6.6.1.1.1

Move $x$ .

$\frac{x^{2} + 8 x + 15 - (x \cdot x) - x \cdot - 3 + 7 x + 7 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Step 6.6.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 8 x + 15 - x^{2} - x \cdot - 3 + 7 x + 7 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Step 6.6.1.2

Multiply $- 3$ by $- 1$ .

$\frac{x^{2} + 8 x + 15 - x^{2} + 3 x + 7 x + 7 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Step 6.6.1.3

Multiply $7$ by $- 3$ .

$\frac{x^{2} + 8 x + 15 - x^{2} + 3 x + 7 x - 21}{(x + 3) (x + 4) (x - 3)}$

Step 6.6.2

Add $3 x$ and $7 x$ .

$\frac{x^{2} + 8 x + 15 - x^{2} + 10 x - 21}{(x + 3) (x + 4) (x - 3)}$

Step 6.7

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

$\frac{8 x + 15 + 0 + 10 x - 21}{(x + 3) (x + 4) (x - 3)}$

Step 6.8

Add $8 x$ and $0$ .

$\frac{8 x + 15 + 10 x - 21}{(x + 3) (x + 4) (x - 3)}$

Step 6.9

Add $8 x$ and $10 x$ .

$\frac{18 x + 15 - 21}{(x + 3) (x + 4) (x - 3)}$

Step 6.10

Subtract $21$ from $15$ .

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

Step 6.11

Factor $6$ out of $18 x - 6$ .

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Step 6.11.1

Factor $6$ out of $18 x$ .

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

Step 6.11.2

Factor $6$ out of $- 6$ .

$\frac{6 (3 x) + 6 (- 1)}{(x + 3) (x + 4) (x - 3)}$

Step 6.11.3

Factor $6$ out of $6 (3 x) + 6 (- 1)$ .

$\frac{6 (3 x - 1)}{(x + 3) (x + 4) (x - 3)}$