Algebra Examples

$\frac{x + 7}{2 x + 6} - \frac{- 2 x + 20}{x^{2} - 7 x - 30}$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 x + 6$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{x + 7}{2 (x) + 6} - \frac{- 2 x + 20}{x^{2} - 7 x - 30}$

Step 1.1.2

Factor $2$ out of $6$ .

$\frac{x + 7}{2 x + 2 \cdot 3} - \frac{- 2 x + 20}{x^{2} - 7 x - 30}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{x + 7}{2 (x + 3)} - \frac{- 2 x + 20}{x^{2} - 7 x - 30}$

Step 1.2

Factor $2$ out of $- 2 x + 20$ .

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Step 1.2.1

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

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- x) + 20}{x^{2} - 7 x - 30}$

Step 1.2.2

Factor $2$ out of $20$ .

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- x) + 2 (10)}{x^{2} - 7 x - 30}$

Step 1.2.3

Factor $2$ out of $2 (- x) + 2 (10)$ .

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- x + 10)}{x^{2} - 7 x - 30}$

Step 1.3

Factor $x^{2} - 7 x - 30$ using the AC method.

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Step 1.3.1

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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Step 1.3.2

Write the factored form using these integers.

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- x + 10)}{(x - 10) (x + 3)}$

Step 1.4

Cancel the common factor of $- x + 10$ and $x - 10$ .

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Step 1.4.1

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

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- (x) + 10)}{(x - 10) (x + 3)}$

Step 1.4.2

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- (x) - 1 (- 10))}{(x - 10) (x + 3)}$

Step 1.4.3

Factor $- 1$ out of $- (x) - 1 (- 10)$ .

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- (x - 10))}{(x - 10) (x + 3)}$

Step 1.4.4

Rewrite $- (x - 10)$ as $- 1 (x - 10)$ .

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- 1 (x - 10))}{(x - 10) (x + 3)}$

Step 1.4.5

Cancel the common factor.

$\frac{x + 7}{2 (x + 3)} - \frac{2 (- 1 (x - 10))}{(x - 10) (x + 3)}$

Step 1.4.6

Rewrite the expression.

$\frac{x + 7}{2 (x + 3)} - \frac{2 \cdot (- 1)}{x + 3}$

Step 1.5

Multiply $2$ by $- 1$ .

$\frac{x + 7}{2 (x + 3)} - \frac{- 2}{x + 3}$

Step 1.6

Move the negative in front of the fraction.

$\frac{x + 7}{2 (x + 3)} - - \frac{2}{x + 3}$

Step 1.7

Multiply $- - \frac{2}{x + 3}$ .

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Step 1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{x + 7}{2 (x + 3)} + 1 \frac{2}{x + 3}$

Step 1.7.2

Multiply $\frac{2}{x + 3}$ by $1$ .

$\frac{x + 7}{2 (x + 3)} + \frac{2}{x + 3}$

Step 2

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

$\frac{x + 7}{2 (x + 3)} + \frac{2}{x + 3} \cdot \frac{2}{2}$

Step 3

Write each expression with a common denominator of $2 (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{x + 3}$ by $\frac{2}{2}$ .

$\frac{x + 7}{2 (x + 3)} + \frac{2 \cdot 2}{(x + 3) \cdot 2}$

Step 3.2

Reorder the factors of $(x + 3) \cdot 2$ .

$\frac{x + 7}{2 (x + 3)} + \frac{2 \cdot 2}{2 (x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x + 7 + 2 \cdot 2}{2 (x + 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $2$ .

$\frac{x + 7 + 4}{2 (x + 3)}$

Step 5.2

Add $7$ and $4$ .

$\frac{x + 11}{2 (x + 3)}$