Pre-Algebra Examples

$\frac{x + 3}{x^{2} - 2 x - 8} - \frac{x - 5}{x^{2} - 12 x + 32}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 2 x - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{x + 3}{(x - 4) (x + 2)} - \frac{x - 5}{x^{2} - 12 x + 32}$

Step 1.2

Factor $x^{2} - 12 x + 32$ 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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{x + 3}{(x - 4) (x + 2)} - \frac{x - 5}{(x - 8) (x - 4)}$

Step 2

To write $\frac{x + 3}{(x - 4) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{x + 3}{(x - 4) (x + 2)} \cdot \frac{x - 8}{x - 8} - \frac{x - 5}{(x - 8) (x - 4)}$

Step 3

To write $- \frac{x - 5}{(x - 8) (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{x + 3}{(x - 4) (x + 2)} \cdot \frac{x - 8}{x - 8} - \frac{x - 5}{(x - 8) (x - 4)} \cdot \frac{x + 2}{x + 2}$

Step 4

Write each expression with a common denominator of $(x - 4) (x + 2) (x - 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x + 3}{(x - 4) (x + 2)}$ by $\frac{x - 8}{x - 8}$ .

$\frac{(x + 3) (x - 8)}{(x - 4) (x + 2) (x - 8)} - \frac{x - 5}{(x - 8) (x - 4)} \cdot \frac{x + 2}{x + 2}$

Step 4.2

Multiply $\frac{x - 5}{(x - 8) (x - 4)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{(x + 3) (x - 8)}{(x - 4) (x + 2) (x - 8)} - \frac{(x - 5) (x + 2)}{(x - 8) (x - 4) (x + 2)}$

Step 4.3

Reorder the factors of $(x - 4) (x + 2) (x - 8)$ .

$\frac{(x + 3) (x - 8)}{(x + 2) (x - 4) (x - 8)} - \frac{(x - 5) (x + 2)}{(x - 8) (x - 4) (x + 2)}$

Step 4.4

Reorder the factors of $(x - 8) (x - 4) (x + 2)$ .

$\frac{(x + 3) (x - 8)}{(x + 2) (x - 4) (x - 8)} - \frac{(x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 3) (x - 8) - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x + 3) (x - 8)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{x (x - 8) + 3 (x - 8) - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 8 + 3 (x - 8) - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 8 + 3 x + 3 \cdot - 8 - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

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 $x$ .

$\frac{x^{2} + x \cdot - 8 + 3 x + 3 \cdot - 8 - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.2.1.2

Move $- 8$ to the left of $x$ .

$\frac{x^{2} - 8 \cdot x + 3 x + 3 \cdot - 8 - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.2.1.3

Multiply $3$ by $- 8$ .

$\frac{x^{2} - 8 x + 3 x - 24 - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.2.2

Add $- 8 x$ and $3 x$ .

$\frac{x^{2} - 5 x - 24 - (x - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.3

Apply the distributive property.

$\frac{x^{2} - 5 x - 24 + (- x - - 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.4

Multiply $- 1$ by $- 5$ .

$\frac{x^{2} - 5 x - 24 + (- x + 5) (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.5

Expand $(- x + 5) (x + 2)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{x^{2} - 5 x - 24 - x (x + 2) + 5 (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.5.2

Apply the distributive property.

$\frac{x^{2} - 5 x - 24 - x \cdot x - x \cdot 2 + 5 (x + 2)}{(x + 2) (x - 4) (x - 8)}$

Step 6.5.3

Apply the distributive property.

$\frac{x^{2} - 5 x - 24 - x \cdot x - x \cdot 2 + 5 x + 5 \cdot 2}{(x + 2) (x - 4) (x - 8)}$

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 $x$ .

$\frac{x^{2} - 5 x - 24 - (x \cdot x) - x \cdot 2 + 5 x + 5 \cdot 2}{(x + 2) (x - 4) (x - 8)}$

Step 6.6.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 5 x - 24 - x^{2} - x \cdot 2 + 5 x + 5 \cdot 2}{(x + 2) (x - 4) (x - 8)}$

Step 6.6.1.2

Multiply $2$ by $- 1$ .

$\frac{x^{2} - 5 x - 24 - x^{2} - 2 x + 5 x + 5 \cdot 2}{(x + 2) (x - 4) (x - 8)}$

Step 6.6.1.3

Multiply $5$ by $2$ .

$\frac{x^{2} - 5 x - 24 - x^{2} - 2 x + 5 x + 10}{(x + 2) (x - 4) (x - 8)}$

Step 6.6.2

Add $- 2 x$ and $5 x$ .

$\frac{x^{2} - 5 x - 24 - x^{2} + 3 x + 10}{(x + 2) (x - 4) (x - 8)}$

Step 6.7

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

$\frac{- 5 x - 24 + 0 + 3 x + 10}{(x + 2) (x - 4) (x - 8)}$

Step 6.8

Add $- 5 x$ and $0$ .

$\frac{- 5 x - 24 + 3 x + 10}{(x + 2) (x - 4) (x - 8)}$

Step 6.9

Add $- 5 x$ and $3 x$ .

$\frac{- 2 x - 24 + 10}{(x + 2) (x - 4) (x - 8)}$

Step 6.10

Add $- 24$ and $10$ .

$\frac{- 2 x - 14}{(x + 2) (x - 4) (x - 8)}$

Step 6.11

Factor $2$ out of $- 2 x - 14$ .

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Step 6.11.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) - 14}{(x + 2) (x - 4) (x - 8)}$

Step 6.11.2

Factor $2$ out of $- 14$ .

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

Step 6.11.3

Factor $2$ out of $2 (- x) + 2 (- 7)$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- x$ .

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

Step 7.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{2 (- (x) - 1 (7))}{(x + 2) (x - 4) (x - 8)}$

Step 7.3

Factor $- 1$ out of $- (x) - 1 (7)$ .

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

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (x + 7)$ as $- 1 (x + 7)$ .

$\frac{2 (- 1 (x + 7))}{(x + 2) (x - 4) (x - 8)}$

Step 7.4.2

Move the negative in front of the fraction.

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