Algebra Examples

$\frac{11}{x^{2} - 11 x} - \frac{x}{11 x - 121}$

Step 1

Simplify each term.

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Step 1.1

Factor $x$ out of $x^{2} - 11 x$ .

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Step 1.1.1

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

$\frac{11}{x \cdot x - 11 x} - \frac{x}{11 x - 121}$

Step 1.1.2

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

$\frac{11}{x \cdot x + x \cdot - 11} - \frac{x}{11 x - 121}$

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 11$ .

$\frac{11}{x (x - 11)} - \frac{x}{11 x - 121}$

Step 1.2

Factor $11$ out of $11 x - 121$ .

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Step 1.2.1

Factor $11$ out of $11 x$ .

$\frac{11}{x (x - 11)} - \frac{x}{11 (x) - 121}$

Step 1.2.2

Factor $11$ out of $- 121$ .

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

Step 1.2.3

Factor $11$ out of $11 x + 11 \cdot - 11$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $11$ by $11$ .

$\frac{121 - x \cdot x}{11 x (x - 11)}$

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$\frac{121 - (x \cdot x)}{11 x (x - 11)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{121 - x^{2}}{11 x (x - 11)}$

Step 6.3

Rewrite $121 - x^{2}$ in a factored form.

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Step 6.3.1

Rewrite $121$ as $11^{2}$ .

$\frac{11^{2} - x^{2}}{11 x (x - 11)}$

Step 6.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 11$ and $b = x$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $11 - x$ and $x - 11$ .

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Step 7.1.1

Rewrite $11$ as $- 1 (- 11)$ .

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

Step 7.1.2

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

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

Step 7.1.3

Factor $- 1$ out of $- 1 (- 11) - (x)$ .

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

Step 7.1.4

Reorder terms.

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

Step 7.1.5

Cancel the common factor.

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

Step 7.1.6

Rewrite the expression.

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

Step 7.2

Simplify the expression.

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Step 7.2.1

Move $- 1$ to the left of $11 + x$ .

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

Step 7.2.2

Move the negative in front of the fraction.

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