Precalculus Examples

$\frac{\frac{5}{x + 1} - 5}{\frac{5}{x - 1} + 5}$

Step 1

Cancel the common factor of $\frac{5}{x + 1} - 5$ and $\frac{5}{x - 1} + 5$ .

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Step 1.1

Factor $5$ out of $\frac{5}{x + 1}$ .

$\frac{5 \frac{1}{x + 1} - 5}{\frac{5}{x - 1} + 5}$

Step 1.2

Factor $5$ out of $- 5$ .

$\frac{5 \frac{1}{x + 1} + 5 \cdot - 1}{\frac{5}{x - 1} + 5}$

Step 1.3

Factor $5$ out of $5 \frac{1}{x + 1} + 5 \cdot - 1$ .

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

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $5$ out of $\frac{5}{x - 1}$ .

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

Step 1.4.2

Factor $5$ out of $5$ .

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

Step 1.4.3

Factor $5$ out of $5 \frac{1}{x - 1} + 5 (1)$ .

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

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

Step 2

Multiply the numerator and denominator of the fraction by $(x + 1) (x - 1)$ .

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $x + 1$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 1$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 5.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.2.1.4

Multiply $x$ by $1$ .

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

Step 5.2.1.5

Multiply $- 1$ by $1$ .

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

Step 5.2.2

Add $- x$ and $x$ .

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

Step 5.2.3

Add $x^{2}$ and $0$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Move $- 1$ to the left of $x^{2}$ .

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

Step 5.5

Multiply $- 1$ by $- 1$ .

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

Step 5.6

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

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

Step 5.7

Add $- 1$ and $1$ .

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

Step 5.8

Add $x - x^{2}$ and $0$ .

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

Step 5.9

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

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Step 5.9.1

Raise $x$ to the power of $1$ .

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

Step 5.9.2

Factor $x$ out of $x^{1}$ .

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

Step 5.9.3

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

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

Step 5.9.4

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

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

Step 6

Simplify the denominator.

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Step 6.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 6.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.2.1.4

Multiply $x$ by $1$ .

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

Step 6.2.1.5

Multiply $- 1$ by $1$ .

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

Step 6.2.2

Add $- x$ and $x$ .

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

Step 6.2.3

Add $x^{2}$ and $0$ .

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

Step 6.3

Multiply $x^{2} - 1$ by $1$ .

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

Step 6.4

Subtract $1$ from $1$ .

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

Step 6.5

Add $x + x^{2}$ and $0$ .

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

Step 6.6

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

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Step 6.6.1

Raise $x$ to the power of $1$ .

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

Step 6.6.2

Factor $x$ out of $x^{1}$ .

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

Step 6.6.3

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

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

Step 6.6.4

Factor $x$ out of $x \cdot 1 + x \cdot x$ .

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

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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