Algebra Examples

$\frac{x + 6}{5 x + 10} - \frac{x - 2}{4 x + 8}$

Step 1

Simplify each term.

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Step 1.1

Factor $5$ out of $5 x + 10$ .

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Step 1.1.1

Factor $5$ out of $5 x$ .

$\frac{x + 6}{5 (x) + 10} - \frac{x - 2}{4 x + 8}$

Step 1.1.2

Factor $5$ out of $10$ .

$\frac{x + 6}{5 x + 5 \cdot 2} - \frac{x - 2}{4 x + 8}$

Step 1.1.3

Factor $5$ out of $5 x + 5 \cdot 2$ .

$\frac{x + 6}{5 (x + 2)} - \frac{x - 2}{4 x + 8}$

Step 1.2

Factor $4$ out of $4 x + 8$ .

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Step 1.2.1

Factor $4$ out of $4 x$ .

$\frac{x + 6}{5 (x + 2)} - \frac{x - 2}{4 (x) + 8}$

Step 1.2.2

Factor $4$ out of $8$ .

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

Step 1.2.3

Factor $4$ out of $4 x + 4 \cdot 2$ .

$\frac{x + 6}{5 (x + 2)} - \frac{x - 2}{4 (x + 2)}$

Step 2

To write $\frac{x + 6}{5 (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $20 (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x + 6}{5 (x + 2)}$ by $\frac{4}{4}$ .

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

Step 4.2

Multiply $4$ by $5$ .

$\frac{(x + 6) \cdot 4}{20 (x + 2)} - \frac{x - 2}{4 (x + 2)} \cdot \frac{5}{5}$

Step 4.3

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

$\frac{(x + 6) \cdot 4}{20 (x + 2)} - \frac{(x - 2) \cdot 5}{4 (x + 2) \cdot 5}$

Step 4.4

Multiply $5$ by $4$ .

$\frac{(x + 6) \cdot 4}{20 (x + 2)} - \frac{(x - 2) \cdot 5}{20 (x + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 6) \cdot 4 - (x - 2) \cdot 5}{20 (x + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot 4 + 6 \cdot 4 - (x - 2) \cdot 5}{20 (x + 2)}$

Step 6.2

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

$\frac{4 \cdot x + 6 \cdot 4 - (x - 2) \cdot 5}{20 (x + 2)}$

Step 6.3

Multiply $6$ by $4$ .

$\frac{4 \cdot x + 24 - (x - 2) \cdot 5}{20 (x + 2)}$

Step 6.4

Apply the distributive property.

$\frac{4 x + 24 + (- x - - 2) \cdot 5}{20 (x + 2)}$

Step 6.5

Multiply $- 1$ by $- 2$ .

$\frac{4 x + 24 + (- x + 2) \cdot 5}{20 (x + 2)}$

Step 6.6

Apply the distributive property.

$\frac{4 x + 24 - x \cdot 5 + 2 \cdot 5}{20 (x + 2)}$

Step 6.7

Multiply $5$ by $- 1$ .

$\frac{4 x + 24 - 5 x + 2 \cdot 5}{20 (x + 2)}$

Step 6.8

Multiply $2$ by $5$ .

$\frac{4 x + 24 - 5 x + 10}{20 (x + 2)}$

Step 6.9

Subtract $5 x$ from $4 x$ .

$\frac{- x + 24 + 10}{20 (x + 2)}$

Step 6.10

Add $24$ and $10$ .

$\frac{- x + 34}{20 (x + 2)}$

Step 7

Simplify with factoring out.

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Step 7.1

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

$\frac{- (x) + 34}{20 (x + 2)}$

Step 7.2

Rewrite $34$ as $- 1 (- 34)$ .

$\frac{- (x) - 1 (- 34)}{20 (x + 2)}$

Step 7.3

Factor $- 1$ out of $- (x) - 1 (- 34)$ .

$\frac{- (x - 34)}{20 (x + 2)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (x - 34)$ as $- 1 (x - 34)$ .

$\frac{- 1 (x - 34)}{20 (x + 2)}$

Step 7.4.2

Move the negative in front of the fraction.

$- \frac{x - 34}{20 (x + 2)}$