Algebra Examples

$\frac{9}{x^{2} - 9 x} - \frac{x}{9 x - 81}$

Step 1

Simplify each term.

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Step 1.1

Factor $x$ out of $x^{2} - 9 x$ .

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Step 1.1.1

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

$\frac{9}{x \cdot x - 9 x} - \frac{x}{9 x - 81}$

Step 1.1.2

Factor $x$ out of $- 9 x$ .

$\frac{9}{x \cdot x + x \cdot - 9} - \frac{x}{9 x - 81}$

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 9$ .

$\frac{9}{x (x - 9)} - \frac{x}{9 x - 81}$

Step 1.2

Factor $9$ out of $9 x - 81$ .

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Step 1.2.1

Factor $9$ out of $9 x$ .

$\frac{9}{x (x - 9)} - \frac{x}{9 (x) - 81}$

Step 1.2.2

Factor $9$ out of $- 81$ .

$\frac{9}{x (x - 9)} - \frac{x}{9 x + 9 \cdot - 9}$

Step 1.2.3

Factor $9$ out of $9 x + 9 \cdot - 9$ .

$\frac{9}{x (x - 9)} - \frac{x}{9 (x - 9)}$

Step 2

To write $\frac{9}{x (x - 9)}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{9}{x (x - 9)} \cdot \frac{9}{9} - \frac{x}{9 (x - 9)}$

Step 3

To write $- \frac{x}{9 (x - 9)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{9}{x (x - 9)} \cdot \frac{9}{9} - \frac{x}{9 (x - 9)} \cdot \frac{x}{x}$

Step 4

Write each expression with a common denominator of $x (x - 9) \cdot 9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9}{x (x - 9)}$ by $\frac{9}{9}$ .

$\frac{9 \cdot 9}{x (x - 9) \cdot 9} - \frac{x}{9 (x - 9)} \cdot \frac{x}{x}$

Step 4.2

Multiply $\frac{x}{9 (x - 9)}$ by $\frac{x}{x}$ .

$\frac{9 \cdot 9}{x (x - 9) \cdot 9} - \frac{x \cdot x}{9 (x - 9) x}$

Step 4.3

Reorder the factors of $x (x - 9) \cdot 9$ .

$\frac{9 \cdot 9}{9 x (x - 9)} - \frac{x \cdot x}{9 (x - 9) x}$

Step 4.4

Reorder the factors of $9 (x - 9) x$ .

$\frac{9 \cdot 9}{9 x (x - 9)} - \frac{x \cdot x}{9 x (x - 9)}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 \cdot 9 - x \cdot x}{9 x (x - 9)}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $9 \cdot 9$ as ${(3 \cdot 3)}^{2}$ .

$\frac{{(3 \cdot 3)}^{2} - x \cdot x}{9 x (x - 9)}$

Step 6.2

Rewrite $x \cdot x$ as $x^{2}$ .

$\frac{{(3 \cdot 3)}^{2} - x^{2}}{9 x (x - 9)}$

Step 6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 \cdot 3$ and $b = x$ .

$\frac{(3 \cdot 3 + x) (3 \cdot 3 - x)}{9 x (x - 9)}$

Step 6.4

Simplify.

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Step 6.4.1

Multiply $3$ by $3$ .

$\frac{(9 + x) (3 \cdot 3 - x)}{9 x (x - 9)}$

Step 6.4.2

Multiply $3$ by $3$ .

$\frac{(9 + x) (9 - x)}{9 x (x - 9)}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $9 - x$ and $x - 9$ .

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Step 7.1.1

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{(9 + x) (- 1 (- 9) - x)}{9 x (x - 9)}$

Step 7.1.2

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

$\frac{(9 + x) (- 1 (- 9) - (x))}{9 x (x - 9)}$

Step 7.1.3

Factor $- 1$ out of $- 1 (- 9) - (x)$ .

$\frac{(9 + x) (- 1 (- 9 + x))}{9 x (x - 9)}$

Step 7.1.4

Reorder terms.

$\frac{(9 + x) (- 1 (x - 9))}{9 x (x - 9)}$

Step 7.1.5

Cancel the common factor.

$\frac{(9 + x) (- 1 (x - 9))}{9 x (x - 9)}$

Step 7.1.6

Rewrite the expression.

$\frac{(9 + x) \cdot (- 1)}{9 x}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Move $- 1$ to the left of $9 + x$ .

$\frac{- 1 \cdot (9 + x)}{9 x}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{9 + x}{9 x}$