Algebra Examples

$\frac{\frac{1}{3 x^{2} - 3}}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3 x^{2} - 3} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $3$ out of $3 x^{2} - 3$ .

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Step 2.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{1}{3 (x^{2}) - 3} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 2.1.2

Factor $3$ out of $- 3$ .

$\frac{1}{3 (x^{2}) + 3 (- 1)} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 2.1.3

Factor $3$ out of $3 (x^{2}) + 3 (- 1)$ .

$\frac{1}{3 (x^{2} - 1)} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$\frac{1}{3 (x^{2} - 1^{2})} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{x^{2} - 3 x - 4}}$

Step 3

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 3.2

Write the factored form using these integers.

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5}{x + 1} - \frac{x + 4}{(x - 4) (x + 1)}}$

Step 4

Simplify the denominator.

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Step 4.1

To write $\frac{5}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5}{x + 1} \cdot \frac{x - 4}{x - 4} - \frac{x + 4}{(x - 4) (x + 1)}}$

Step 4.2

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

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Step 4.2.1

Multiply $\frac{5}{x + 1}$ by $\frac{x - 4}{x - 4}$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 (x - 4)}{(x + 1) (x - 4)} - \frac{x + 4}{(x - 4) (x + 1)}}$

Step 4.2.2

Reorder the factors of $(x - 4) (x + 1)$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 (x - 4)}{(x + 1) (x - 4)} - \frac{x + 4}{(x + 1) (x - 4)}}$

Step 4.3

Combine the numerators over the common denominator.

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 (x - 4) - (x + 4)}{(x + 1) (x - 4)}}$

Step 4.4

Rewrite $\frac{5 (x - 4) - (x + 4)}{(x + 1) (x - 4)}$ in a factored form.

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Step 4.4.1

Apply the distributive property.

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 x + 5 \cdot - 4 - (x + 4)}{(x + 1) (x - 4)}}$

Step 4.4.2

Multiply $5$ by $- 4$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 x - 20 - (x + 4)}{(x + 1) (x - 4)}}$

Step 4.4.3

Apply the distributive property.

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 x - 20 - x - 1 \cdot 4}{(x + 1) (x - 4)}}$

Step 4.4.4

Multiply $- 1$ by $4$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{5 x - 20 - x - 4}{(x + 1) (x - 4)}}$

Step 4.4.5

Subtract $x$ from $5 x$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{4 x - 20 - 4}{(x + 1) (x - 4)}}$

Step 4.4.6

Subtract $4$ from $- 20$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{4 x - 24}{(x + 1) (x - 4)}}$

Step 4.4.7

Factor $4$ out of $4 x - 24$ .

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Step 4.4.7.1

Factor $4$ out of $4 x$ .

$\frac{1}{3 (x + 1) (x - 1)} \cdot \frac{1}{\frac{4 (x) - 24}{(x + 1) (x - 4)}}$

Step 4.4.7.2

Factor $4$ out of $- 24$ .

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

Step 4.4.7.3

Factor $4$ out of $4 x + 4 \cdot - 6$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Multiply $\frac{(x + 1) (x - 4)}{4 (x - 6)}$ by $1$ .

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

Step 7

Cancel the common factor of $x + 1$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Multiply $\frac{1}{3 (x - 1)}$ by $\frac{x - 4}{4 (x - 6)}$ .

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

Step 9

Multiply $4$ by $3$ .

$\frac{x - 4}{12 (x - 1) (x - 6)}$