Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Multiply $x$ by $1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

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 $b = 1$ .

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

Step 6

Cancel the common factor of $x + 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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