Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the denominator.

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Step 1.1.1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.1.1.2

Combine the numerators over the common denominator.

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

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.1.2.2

Combine the numerators over the common denominator.

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

Step 1.1.2.3

Rewrite $\frac{\frac{x + 1}{x} + 1}{\frac{x + 1}{x}}$ in a factored form.

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Step 1.1.2.3.1

Write $1$ as a fraction with a common denominator.

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

Step 1.1.2.3.2

Combine the numerators over the common denominator.

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

Step 1.1.2.3.3

Reorder terms.

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

Step 1.1.2.3.4

Add $x$ and $x$ .

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

Step 1.1.2.4

Multiply the numerator by the reciprocal of the denominator.

$1 + \frac{1}{\frac{2 x + 1}{x} \cdot \frac{x}{x + 1}}$

Step 1.1.2.5

Cancel the common factor of $x$ .

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Step 1.1.2.5.1

Cancel the common factor.

$1 + \frac{1}{\frac{2 x + 1}{x} \cdot \frac{x}{x + 1}}$

Step 1.1.2.5.2

Rewrite the expression.

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

Step 1.1.2.6

Multiply $2 x + 1$ by $\frac{1}{x + 1}$ .

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

Step 1.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.4

Multiply $\frac{x + 1}{2 x + 1}$ by $1$ .

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

Step 1.2

Combine into one fraction.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Add $2 x$ and $x$ .

$\frac{3 x + 1 + 1}{2 x + 1}$

Step 2.2

Add $1$ and $1$ .

$\frac{3 x + 2}{2 x + 1}$