Finite Math Examples

$\frac{\frac{8}{x - 1} + 8}{\frac{8}{x + 1} - 8}$

Step 1

Cancel the common factor of $\frac{8}{x - 1} + 8$ and $\frac{8}{x + 1} - 8$ .

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Step 1.1

Factor $8$ out of $\frac{8}{x - 1}$ .

$\frac{8 \frac{1}{x - 1} + 8}{\frac{8}{x + 1} - 8}$

Step 1.2

Factor $8$ out of $8$ .

$\frac{8 \frac{1}{x - 1} + 8 \cdot 1}{\frac{8}{x + 1} - 8}$

Step 1.3

Factor $8$ out of $8 \frac{1}{x - 1} + 8 \cdot 1$ .

$\frac{8 (\frac{1}{x - 1} + 1)}{\frac{8}{x + 1} - 8}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $8$ out of $\frac{8}{x + 1}$ .

$\frac{8 (\frac{1}{x - 1} + 1)}{8 \frac{1}{x + 1} - 8}$

Step 1.4.2

Factor $8$ out of $- 8$ .

$\frac{8 (\frac{1}{x - 1} + 1)}{8 \frac{1}{x + 1} + 8 \cdot - 1}$

Step 1.4.3

Factor $8$ out of $8 \frac{1}{x + 1} + 8 \cdot - 1$ .

$\frac{8 (\frac{1}{x - 1} + 1)}{8 (\frac{1}{x + 1} - 1)}$

Step 1.4.4

Cancel the common factor.

$\frac{8 (\frac{1}{x - 1} + 1)}{8 (\frac{1}{x + 1} - 1)}$

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

Combine the opposite terms in $x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$ .

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Step 5.2.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 5.2.2

Subtract $1 x$ from $1 x$ .

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

Step 5.2.3

Add $x \cdot x$ and $0$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $x$ by $x$ .

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

Step 5.3.2

Multiply $- 1$ by $1$ .

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

Step 5.4

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

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

Step 5.5

Subtract $1$ from $1$ .

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

Step 5.6

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

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

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

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

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

Step 5.7.3

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

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

Step 5.7.4

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

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

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

Combine the opposite terms in $x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$ .

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Step 6.2.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 6.2.2

Subtract $1 x$ from $1 x$ .

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

Step 6.2.3

Add $x \cdot x$ and $0$ .

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

Step 6.3

Simplify each term.

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Step 6.3.1

Multiply $x$ by $x$ .

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

Step 6.3.2

Multiply $- 1$ by $1$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

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

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

Step 6.6

Multiply $- 1$ by $- 1$ .

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

Step 6.7

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

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

Step 6.8

Add $- 1$ and $1$ .

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

Step 6.9

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

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

Step 6.10

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

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Step 6.10.1

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

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

Step 6.10.2

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

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

Step 6.10.3

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

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

Step 6.10.4

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

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

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.

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