Pre-Algebra Examples

$\frac{\frac{11}{x - 11}}{\frac{1}{x - 11} - \frac{1}{x}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{11}{x - 11} \cdot \frac{1}{\frac{1}{x - 11} - \frac{1}{x}}$

Step 2

Simplify the denominator.

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Step 2.1

To write $\frac{1}{x - 11}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{1}{x - 11} \cdot \frac{x}{x} - \frac{1}{x}}$

Step 2.2

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 11}{x - 11}$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{1}{x - 11} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{x - 11}{x - 11}}$

Step 2.3

Write each expression with a common denominator of $(x - 11) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{x - 11}$ by $\frac{x}{x}$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{x}{(x - 11) x} - \frac{1}{x} \cdot \frac{x - 11}{x - 11}}$

Step 2.3.2

Multiply $\frac{1}{x}$ by $\frac{x - 11}{x - 11}$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{x}{(x - 11) x} - \frac{x - 11}{x (x - 11)}}$

Step 2.3.3

Reorder the factors of $(x - 11) x$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{x}{x (x - 11)} - \frac{x - 11}{x (x - 11)}}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{11}{x - 11} \cdot \frac{1}{\frac{x - (x - 11)}{x (x - 11)}}$

Step 2.5

Rewrite $\frac{x - (x - 11)}{x (x - 11)}$ in a factored form.

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Step 2.5.1

Apply the distributive property.

$\frac{11}{x - 11} \cdot \frac{1}{\frac{x - x - - 11}{x (x - 11)}}$

Step 2.5.2

Multiply $- 1$ by $- 11$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{x - x + 11}{x (x - 11)}}$

Step 2.5.3

Subtract $x$ from $x$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{0 + 11}{x (x - 11)}}$

Step 2.5.4

Add $0$ and $11$ .

$\frac{11}{x - 11} \cdot \frac{1}{\frac{11}{x (x - 11)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{11}{x - 11} (1 \frac{x (x - 11)}{11})$

Step 4

Multiply $\frac{x (x - 11)}{11}$ by $1$ .

$\frac{11}{x - 11} \cdot \frac{x (x - 11)}{11}$

Step 5

Cancel the common factor of $11$ .

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Step 5.1

Cancel the common factor.

$\frac{11}{x - 11} \cdot \frac{x (x - 11)}{11}$

Step 5.2

Rewrite the expression.

$\frac{1}{x - 11} (x (x - 11))$

Step 6

Cancel the common factor of $x - 11$ .

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Step 6.1

Factor $x - 11$ out of $x (x - 11)$ .

$\frac{1}{x - 11} ((x - 11) x)$

Step 6.2

Cancel the common factor.

$\frac{1}{x - 11} ((x - 11) x)$

Step 6.3

Rewrite the expression.

$x$