Algebra Examples

$\frac{\frac{4}{x + 3} - \frac{7}{x - 3}}{\frac{6}{x + 3}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{4}{x + 3} - \frac{7}{x - 3}) \frac{x + 3}{6}$

Step 2

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

$(\frac{4}{x + 3} \cdot \frac{x - 3}{x - 3} - \frac{7}{x - 3}) \frac{x + 3}{6}$

Step 3

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

$(\frac{4}{x + 3} \cdot \frac{x - 3}{x - 3} - \frac{7}{x - 3} \cdot \frac{x + 3}{x + 3}) \frac{x + 3}{6}$

Step 4

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

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Step 4.1

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

$(\frac{4 (x - 3)}{(x + 3) (x - 3)} - \frac{7}{x - 3} \cdot \frac{x + 3}{x + 3}) \frac{x + 3}{6}$

Step 4.2

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

$(\frac{4 (x - 3)}{(x + 3) (x - 3)} - \frac{7 (x + 3)}{(x - 3) (x + 3)}) \frac{x + 3}{6}$

Step 4.3

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

$(\frac{4 (x - 3)}{(x + 3) (x - 3)} - \frac{7 (x + 3)}{(x + 3) (x - 3)}) \frac{x + 3}{6}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (x - 3) - 7 (x + 3)}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{4 x + 4 \cdot - 3 - 7 (x + 3)}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.2

Multiply $4$ by $- 3$ .

$\frac{4 x - 12 - 7 (x + 3)}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.3

Apply the distributive property.

$\frac{4 x - 12 - 7 x - 7 \cdot 3}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.4

Multiply $- 7$ by $3$ .

$\frac{4 x - 12 - 7 x - 21}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.5

Subtract $7 x$ from $4 x$ .

$\frac{- 3 x - 12 - 21}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.6

Subtract $21$ from $- 12$ .

$\frac{- 3 x - 33}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.7

Factor $3$ out of $- 3 x - 33$ .

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Step 6.7.1

Factor $3$ out of $- 3 x$ .

$\frac{3 (- x) - 33}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.7.2

Factor $3$ out of $- 33$ .

$\frac{3 (- x) + 3 (- 11)}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 6.7.3

Factor $3$ out of $3 (- x) + 3 (- 11)$ .

$\frac{3 (- x - 11)}{(x + 3) (x - 3)} \cdot \frac{x + 3}{6}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $3$ .

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Step 7.1.1

Factor $3$ out of $6$ .

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

Step 7.1.2

Cancel the common factor.

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

Step 7.1.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x + 3$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$\frac{- x - 11}{x - 3} \cdot \frac{1}{2}$

Step 7.3

Multiply $\frac{- x - 11}{x - 3}$ by $\frac{1}{2}$ .

$\frac{- x - 11}{(x - 3) \cdot 2}$

Step 7.4

Move $2$ to the left of $x - 3$ .

$\frac{- x - 11}{2 \cdot (x - 3)}$

Step 7.5

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

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

Step 7.6

Rewrite $- 11$ as $- 1 (11)$ .

$\frac{- (x) - 1 (11)}{2 (x - 3)}$

Step 7.7

Factor $- 1$ out of $- (x) - 1 (11)$ .

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

Step 7.8

Simplify the expression.

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Step 7.8.1

Rewrite $- (x + 11)$ as $- 1 (x + 11)$ .

$\frac{- 1 (x + 11)}{2 (x - 3)}$

Step 7.8.2

Move the negative in front of the fraction.

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