Algebra Examples

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

Step 1

Write $f (x) = \frac{\frac{1}{x} - 1}{1 - \frac{1}{1 + \frac{1}{x - 1}}}$ as an equation.

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

Step 2

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

Tap for more steps...

Step 2.1

Multiply the numerator and denominator of the fraction by $x (1 + \frac{1}{x - 1})$ .

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

Combine.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Simplify by cancelling.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.1.1

Cancel the common factor.

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

Step 2.3.1.2

Rewrite the expression.

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

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

Cancel the common factor.

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

Step 2.3.2.4

Rewrite the expression.

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

Step 2.4

Multiply the numerator and denominator of the fraction by $x - 1$ .

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Combine.

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

Step 2.5

Apply the distributive property.

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

Step 2.6

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

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

Step 2.7

Simplify the numerator.

Tap for more steps...

Step 2.7.1

Multiply $x - 1$ by $1$ .

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

Step 2.7.2

Apply the distributive property.

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

Step 2.7.3

Multiply $x$ by $x$ .

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

Step 2.7.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.7.5

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

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

Step 2.7.6

Combine the numerators over the common denominator.

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

Step 2.7.7

Simplify the numerator.

Tap for more steps...

Step 2.7.7.1

Add $- 1$ and $1$ .

$y = \frac{x - 1 + 1 + (x^{2} - x) \frac{x + 0}{x - 1} \cdot - 1}{(x - 1) x (1 + \frac{1}{x - 1}) \cdot 1 + (x - 1) x \cdot - 1}$

Step 2.7.7.2

Add $x$ and $0$ .

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

Step 2.7.8

Multiply $x^{2} - x$ by $\frac{x}{x - 1}$ .

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

Step 2.7.9

Simplify the numerator.

Tap for more steps...

Step 2.7.9.1

Factor $x$ out of $x^{2} - x$ .

Tap for more steps...

Step 2.7.9.1.1

Factor $x$ out of $x^{2}$ .

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

Step 2.7.9.1.2

Factor $x$ out of $- x$ .

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

Step 2.7.9.1.3

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

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

Step 2.7.9.2

Combine exponents.

Tap for more steps...

Step 2.7.9.2.1

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

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

Step 2.7.9.2.2

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

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

Step 2.7.9.2.3

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

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

Step 2.7.9.2.4

Add $1$ and $1$ .

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

Step 2.7.10

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 2.7.10.1

Cancel the common factor.

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

Step 2.7.10.2

Divide $x^{2}$ by $1$ .

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

Step 2.7.11

Move $- 1$ to the left of $x^{2}$ .

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

Step 2.7.12

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

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

Step 2.7.13

Add $- 1$ and $1$ .

$y = \frac{x + 0 - x^{2}}{(x - 1) x (1 + \frac{1}{x - 1}) \cdot 1 + (x - 1) x \cdot - 1}$

Step 2.7.14

Add $x$ and $0$ .

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

Step 2.7.15

Factor $x$ out of $x - x^{2}$ .

Tap for more steps...

Step 2.7.15.1

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

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

Step 2.7.15.2

Factor $x$ out of $x^{1}$ .

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

Step 2.7.15.3

Factor $x$ out of $- x^{2}$ .

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

Step 2.7.15.4

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

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

Step 2.8

Simplify the denominator.

Tap for more steps...

Step 2.8.1

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

Tap for more steps...

Step 2.8.1.1

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

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

Step 2.8.1.2

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

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

Step 2.8.1.3

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

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

Step 2.8.2

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

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

Step 2.8.3

Combine the opposite terms in $1 + \frac{1}{x - 1} - 1$ .

Tap for more steps...

Step 2.8.3.1

Subtract $1$ from $1$ .

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

Step 2.8.3.2

Add $0$ and $\frac{1}{x - 1}$ .

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

Step 2.8.4

Combine $\frac{1}{x - 1}$ and $x$ .

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

Step 2.9

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

Tap for more steps...

Step 2.9.1

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

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

Step 2.9.2

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

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

Step 2.9.3

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

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

Step 2.9.4

Reorder terms.

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

Step 2.9.5

Cancel the common factor.

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

Step 2.9.6

Rewrite the expression.

$y = \frac{x \cdot (- 1)}{\frac{x}{x - 1}}$

Step 2.10

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.11

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.11.1

Cancel the common factor.

$y = x \cdot - 1 \frac{x - 1}{x}$

Step 2.11.2

Rewrite the expression.

$y = - (x - 1)$

Step 2.12

Apply the distributive property.

$y = - x - - 1$

Step 2.13

Multiply $- 1$ by $- 1$ .

$y = - x + 1$