Finite Math Examples

$\frac{6 - \frac{6}{x}}{6 - \frac{6}{x + 1}}$

Step 1

Cancel the common factor of $6 - \frac{6}{x}$ and $6 - \frac{6}{x + 1}$ .

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Step 1.1

Factor $6$ out of $6$ .

$\frac{6 \cdot 1 - \frac{6}{x}}{6 - \frac{6}{x + 1}}$

Step 1.2

Factor $6$ out of $- \frac{6}{x}$ .

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

Step 1.3

Factor $6$ out of $6 (1) + 6 (- \frac{1}{x})$ .

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

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $6$ out of $6$ .

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

Step 1.4.2

Factor $6$ out of $- \frac{6}{x + 1}$ .

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

Step 1.4.3

Factor $6$ out of $6 (1) + 6 (- \frac{1}{x + 1})$ .

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x + 1$ .

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Step 4.2.1

Move the leading negative in $- \frac{1}{x + 1}$ into the numerator.

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Multiply $x$ by $1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.2

Multiply $x + 1$ by $1$ .

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

Step 6.3

Subtract $1$ from $1$ .

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

Step 6.4

Add $x$ and $0$ .

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

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.4

Add $1$ and $1$ .

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