Precalculus Examples

$\frac{33 x}{x - 1} + \frac{66}{x^{2} - 1} - \frac{33 x}{x + 1}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{33 x}{x - 1} + \frac{66}{x^{2} - 1^{2}} - \frac{33 x}{x + 1}$

Step 1.2

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 $b = 1$ .

$\frac{33 x}{x - 1} + \frac{66}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 2

To write $\frac{33 x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{33 x}{x - 1} \cdot \frac{x + 1}{x + 1} + \frac{66}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 3

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

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Step 3.1

Multiply $\frac{33 x}{x - 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{33 x (x + 1)}{(x - 1) (x + 1)} + \frac{66}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 3.2

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

$\frac{33 x (x + 1)}{(x + 1) (x - 1)} + \frac{66}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 4

Combine the numerators over the common denominator.

$\frac{33 x (x + 1) + 66}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $33$ out of $33 x (x + 1) + 66$ .

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Step 5.1.1

Factor $33$ out of $33 x (x + 1)$ .

$\frac{33 (x (x + 1)) + 66}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 5.1.2

Factor $33$ out of $66$ .

$\frac{33 (x (x + 1)) + 33 \cdot 2}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 5.1.3

Factor $33$ out of $33 (x (x + 1)) + 33 \cdot 2$ .

$\frac{33 (x (x + 1) + 2)}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 5.2

Apply the distributive property.

$\frac{33 (x \cdot x + x \cdot 1 + 2)}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 5.3

Multiply $x$ by $x$ .

$\frac{33 (x^{2} + x \cdot 1 + 2)}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 5.4

Multiply $x$ by $1$ .

$\frac{33 (x^{2} + x + 2)}{(x + 1) (x - 1)} - \frac{33 x}{x + 1}$

Step 6

To write $- \frac{33 x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{33 (x^{2} + x + 2)}{(x + 1) (x - 1)} - \frac{33 x}{x + 1} \cdot \frac{x - 1}{x - 1}$

Step 7

Simplify terms.

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Step 7.1

Multiply $\frac{33 x}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{33 (x^{2} + x + 2)}{(x + 1) (x - 1)} - \frac{33 x (x - 1)}{(x + 1) (x - 1)}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{33 (x^{2} + x + 2) - 33 x (x - 1)}{(x + 1) (x - 1)}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $33$ out of $33 (x^{2} + x + 2) - 33 x (x - 1)$ .

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Step 8.1.1

Factor $33$ out of $- 33 x (x - 1)$ .

$\frac{33 (x^{2} + x + 2) + 33 (- x (x - 1))}{(x + 1) (x - 1)}$

Step 8.1.2

Factor $33$ out of $33 (x^{2} + x + 2) + 33 (- x (x - 1))$ .

$\frac{33 (x^{2} + x + 2 - x (x - 1))}{(x + 1) (x - 1)}$

Step 8.2

Apply the distributive property.

$\frac{33 (x^{2} + x + 2 - x \cdot x - x \cdot - 1)}{(x + 1) (x - 1)}$

Step 8.3

Multiply $x$ by $x$ by adding the exponents.

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Step 8.3.1

Move $x$ .

$\frac{33 (x^{2} + x + 2 - (x \cdot x) - x \cdot - 1)}{(x + 1) (x - 1)}$

Step 8.3.2

Multiply $x$ by $x$ .

$\frac{33 (x^{2} + x + 2 - x^{2} - x \cdot - 1)}{(x + 1) (x - 1)}$

Step 8.4

Multiply $- x \cdot - 1$ .

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Step 8.4.1

Multiply $- 1$ by $- 1$ .

$\frac{33 (x^{2} + x + 2 - x^{2} + 1 x)}{(x + 1) (x - 1)}$

Step 8.4.2

Multiply $x$ by $1$ .

$\frac{33 (x^{2} + x + 2 - x^{2} + x)}{(x + 1) (x - 1)}$

Step 8.5

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

$\frac{33 (x + 2 + 0 + x)}{(x + 1) (x - 1)}$

Step 8.6

Add $x$ and $0$ .

$\frac{33 (x + 2 + x)}{(x + 1) (x - 1)}$

Step 8.7

Add $x$ and $x$ .

$\frac{33 (2 x + 2)}{(x + 1) (x - 1)}$

Step 8.8

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

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Step 8.8.1

Factor $2$ out of $2 x$ .

$\frac{33 (2 (x) + 2)}{(x + 1) (x - 1)}$

Step 8.8.2

Factor $2$ out of $2$ .

$\frac{33 (2 (x) + 2 (1))}{(x + 1) (x - 1)}$

Step 8.8.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$\frac{33 (2 (x + 1))}{(x + 1) (x - 1)}$

$\frac{33 \cdot 2 (x + 1)}{(x + 1) (x - 1)}$

Step 8.9

Multiply $33$ by $2$ .

$\frac{66 (x + 1)}{(x + 1) (x - 1)}$

Step 9

Cancel the common factor of $x + 1$ .

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Step 9.1

Cancel the common factor.

$\frac{66 (x + 1)}{(x + 1) (x - 1)}$

Step 9.2

Rewrite the expression.

$\frac{66}{x - 1}$