Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

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

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.4.3

Subtract $x$ from $x$ .

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

Step 2.4.4

Add $0$ and $1$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $x - 1$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$x$