Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Reorder terms.

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

Step 2.6

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

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Step 2.6.1

Subtract $x$ from $x$ .

$\frac{1}{\frac{0 - 1}{x - 1}}$

Step 2.6.2

Subtract $1$ from $0$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.2

Move the negative in front of the fraction.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (x - 1))$

Step 5

Multiply $x - 1$ by $1$ .

$- (x - 1)$

Step 6

Apply the distributive property.

$- x - - 1$

Step 7

Multiply $- 1$ by $- 1$ .

$- x + 1$