Precalculus Examples

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

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.

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

Step 1.1.1.2

Combine the numerators over the common denominator.

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

Step 1.1.1.3

Reorder terms.

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

Step 1.1.1.4

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

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Step 1.1.1.4.1

Subtract $1$ from $1$ .

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

Step 1.1.1.4.2

Add $- x$ and $0$ .

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

Step 1.1.1.5

Move the negative in front of the fraction.

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

Move the negative in front of the fraction.

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

Step 1.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.4

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

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

Step 1.1.5

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

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Step 1.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.5.2

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

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

Step 1.2

Combine into one fraction.

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Step 1.2.1

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

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Subtract $x$ from $x$ .

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

Step 2.2

Add $0$ and $1$ .

$\frac{1}{x}$