Algebra Examples

$\frac{2}{9} \cdot \frac{- x - 1}{x - 10} \div \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2}{9} \cdot \frac{- x - 1}{x - 10} \frac{x^{2} - 11 x + 10}{x^{2} - 1}$

Step 2

Multiply $\frac{2}{9}$ by $\frac{- x - 1}{x - 10}$ .

$\frac{2 (- x - 1)}{9 (x - 10)} \cdot \frac{x^{2} - 11 x + 10}{x^{2} - 1}$

Step 3

Factor $x^{2} - 11 x + 10$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 11$ .

$- 10, - 1$

Step 3.2

Write the factored form using these integers.

$\frac{2 (- x - 1)}{9 (x - 10)} \cdot \frac{(x - 10) (x - 1)}{x^{2} - 1}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{2 (- x - 1)}{9 (x - 10)} \cdot \frac{(x - 10) (x - 1)}{x^{2} - 1^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{2 (- x - 1)}{9 (x - 10)} \cdot \frac{(x - 10) (x - 1)}{(x + 1) (x - 1)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 10$ .

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Step 5.1.1

Factor $x - 10$ out of $9 (x - 10)$ .

$\frac{2 (- x - 1)}{(x - 10) \cdot 9} \cdot \frac{(x - 10) (x - 1)}{(x + 1) (x - 1)}$

Step 5.1.2

Cancel the common factor.

$\frac{2 (- x - 1)}{(x - 10) \cdot 9} \cdot \frac{(x - 10) (x - 1)}{(x + 1) (x - 1)}$

Step 5.1.3

Rewrite the expression.

$\frac{2 (- x - 1)}{9} \cdot \frac{x - 1}{(x + 1) (x - 1)}$

Step 5.2

Multiply $\frac{2 (- x - 1)}{9}$ by $\frac{x - 1}{(x + 1) (x - 1)}$ .

$\frac{2 (- x - 1) (x - 1)}{9 ((x + 1) (x - 1))}$

Step 5.3

Cancel the common factor of $x - 1$ .

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Step 5.3.1

Cancel the common factor.

$\frac{2 (- x - 1) (x - 1)}{9 ((x + 1) (x - 1))}$

Step 5.3.2

Rewrite the expression.

$\frac{2 (- x - 1)}{9 (x + 1)}$

Step 5.4

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

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Step 5.4.1

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

$\frac{2 (- (x) - 1)}{9 (x + 1)}$

Step 5.4.2

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

$\frac{2 (- (x) - 1 (1))}{9 (x + 1)}$

Step 5.4.3

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

$\frac{2 (- (x + 1))}{9 (x + 1)}$

Step 5.4.4

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

$\frac{2 (- 1 (x + 1))}{9 (x + 1)}$

Step 5.4.5

Cancel the common factor.

$\frac{2 (- 1 (x + 1))}{9 (x + 1)}$

Step 5.4.6

Rewrite the expression.

$\frac{2 \cdot (- 1)}{9}$

Step 6

Multiply $2$ by $- 1$ .

$\frac{- 2}{9}$

Step 7

Move the negative in front of the fraction.

$- \frac{2}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{9}$

Decimal Form:

$- 0.\overline{2}$