Calculus Examples

$f (x) = 8 x^{4} + 2 x^{3} + 4 x - 9$ , $[- 2, - 1]$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

$f (x)$ is continuous on $[- 2, - 1]$ .

$f (x)$ is continuous

Step 3

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{- 1 + 2} (\int_{- 2}^{- 1} 8 x^{4} + 2 x^{3} + 4 x - 9 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{- 1 + 2} (\int_{- 2}^{- 1} 8 x^{4} d x + \int_{- 2}^{- 1} 2 x^{3} d x + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 6

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$A (x) = \frac{1}{- 1 + 2} (8 \int_{- 2}^{- 1} x^{4} d x + \int_{- 2}^{- 1} 2 x^{3} d x + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 7

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{1}{5} x^{5}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} 2 x^{3} d x + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 8

Combine $\frac{1}{5}$ and $x^{5}$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} 2 x^{3} d x + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 9

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 \int_{- 2}^{- 1} x^{3} d x + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 10

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 (\frac{1}{4} x^{4}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 11

Combine $\frac{1}{4}$ and $x^{4}$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 (\frac{x^{4}}{4}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} 4 x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 12

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 (\frac{x^{4}}{4}]_{- 2}^{- 1}) + 4 \int_{- 2}^{- 1} x d x + \int_{- 2}^{- 1} - 9 d x)$

Step 13

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 (\frac{x^{4}}{4}]_{- 2}^{- 1}) + 4 (\frac{1}{2} x^{2}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} - 9 d x)$

Step 14

Combine $\frac{1}{2}$ and $x^{2}$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 (\frac{x^{4}}{4}]_{- 2}^{- 1}) + 4 (\frac{x^{2}}{2}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} - 9 d x)$

Step 15

Apply the constant rule.

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{x^{5}}{5}]_{- 2}^{- 1}) + 2 (\frac{x^{4}}{4}]_{- 2}^{- 1}) + 4 (\frac{x^{2}}{2}]_{- 2}^{- 1}) + - 9 x]_{- 2}^{- 1})$

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $\frac{x^{5}}{5}$ at $- 1$ and at $- 2$ .

$A (x) = \frac{1}{- 1 + 2} (8 ((\frac{{(- 1)}^{5}}{5}) - \frac{{(- 2)}^{5}}{5}) + 2 (\frac{x^{4}}{4}]_{- 2}^{- 1}) + 4 (\frac{x^{2}}{2}]_{- 2}^{- 1}) + - 9 x]_{- 2}^{- 1})$

Step 16.2

Evaluate $\frac{x^{4}}{4}$ at $- 1$ and at $- 2$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{x^{2}}{2}]_{- 2}^{- 1}) + - 9 x]_{- 2}^{- 1})$

Step 16.3

Evaluate $\frac{x^{2}}{2}$ at $- 1$ and at $- 2$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 ((\frac{{(- 1)}^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + - 9 x]_{- 2}^{- 1})$

Step 16.4

Evaluate $- 9 x$ at $- 1$ and at $- 2$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5

Simplify.

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Step 16.5.1

Raise $- 1$ to the power of $5$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{- 1}{5} - \frac{{(- 2)}^{5}}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.2

Move the negative in front of the fraction.

$A (x) = \frac{1}{- 1 + 2} (8 (- \frac{1}{5} - \frac{{(- 2)}^{5}}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.3

Raise $- 2$ to the power of $5$ .

$A (x) = \frac{1}{- 1 + 2} (8 (- \frac{1}{5} - \frac{- 32}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.4

Move the negative in front of the fraction.

$A (x) = \frac{1}{- 1 + 2} (8 (- \frac{1}{5} + \frac{32}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.5

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{- 1 + 2} (8 (- \frac{1}{5} + 1 (\frac{32}{5})) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.6

Multiply $\frac{32}{5}$ by $1$ .

$A (x) = \frac{1}{- 1 + 2} (8 (- \frac{1}{5} + \frac{32}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.7

Combine the numerators over the common denominator.

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{- 1 + 32}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.8

Add $- 1$ and $32$ .

$A (x) = \frac{1}{- 1 + 2} (8 (\frac{31}{5}) + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.9

Combine $8$ and $\frac{31}{5}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{8 \cdot 31}{5} + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.10

Multiply $8$ by $31$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.11

Raise $- 1$ to the power of $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - \frac{{(- 2)}^{4}}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.12

Raise $- 2$ to the power of $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - \frac{16}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.13

Cancel the common factor of $16$ and $4$ .

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Step 16.5.13.1

Factor $4$ out of $16$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - \frac{4 \cdot 4}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.13.2

Cancel the common factors.

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Step 16.5.13.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - \frac{4 \cdot 4}{4 (1)}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.13.2.2

Cancel the common factor.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - \frac{4 \cdot 4}{4 \cdot 1}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.13.2.3

Rewrite the expression.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - \frac{4}{1}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.13.2.4

Divide $4$ by $1$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - 1 \cdot 4) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.14

Multiply $- 1$ by $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - 4) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.15

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} - 4 \cdot \frac{4}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.16

Combine $- 4$ and $\frac{4}{4}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1}{4} + \frac{- 4 \cdot 4}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.17

Combine the numerators over the common denominator.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1 - 4 \cdot 4}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.18

Simplify the numerator.

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Step 16.5.18.1

Multiply $- 4$ by $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{1 - 16}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.18.2

Subtract $16$ from $1$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (\frac{- 15}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.19

Move the negative in front of the fraction.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + 2 (- \frac{15}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.20

Multiply $- 1$ by $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} - 2 (\frac{15}{4}) + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.21

Combine $- 2$ and $\frac{15}{4}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + \frac{- 2 \cdot 15}{4} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.22

Multiply $- 2$ by $15$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + \frac{- 30}{4} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.23

Cancel the common factor of $- 30$ and $4$ .

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Step 16.5.23.1

Factor $2$ out of $- 30$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + \frac{2 (- 15)}{4} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.23.2

Cancel the common factors.

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Step 16.5.23.2.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + \frac{2 \cdot - 15}{2 \cdot 2} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.23.2.2

Cancel the common factor.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + \frac{2 \cdot - 15}{2 \cdot 2} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.23.2.3

Rewrite the expression.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} + \frac{- 15}{2} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.24

Move the negative in front of the fraction.

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} - \frac{15}{2} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.25

To write $\frac{248}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} \cdot \frac{2}{2} - \frac{15}{2} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.26

To write $- \frac{15}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248}{5} \cdot \frac{2}{2} - \frac{15}{2} \cdot \frac{5}{5} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.27

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.5.27.1

Multiply $\frac{248}{5}$ by $\frac{2}{2}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248 \cdot 2}{5 \cdot 2} - \frac{15}{2} \cdot \frac{5}{5} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.27.2

Multiply $5$ by $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248 \cdot 2}{10} - \frac{15}{2} \cdot \frac{5}{5} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.27.3

Multiply $\frac{15}{2}$ by $\frac{5}{5}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248 \cdot 2}{10} - \frac{15 \cdot 5}{2 \cdot 5} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.27.4

Multiply $2$ by $5$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{248 \cdot 2}{10} - \frac{15 \cdot 5}{10} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.28

Combine the numerators over the common denominator.

$A (x) = \frac{1}{- 1 + 2} (\frac{248 \cdot 2 - 15 \cdot 5}{10} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.29

Simplify the numerator.

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Step 16.5.29.1

Multiply $248$ by $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{496 - 15 \cdot 5}{10} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.29.2

Multiply $- 15$ by $5$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{496 - 75}{10} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.29.3

Subtract $75$ from $496$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{{(- 1)}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.30

Raise $- 1$ to the power of $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - \frac{{(- 2)}^{2}}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.31

Raise $- 2$ to the power of $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - \frac{4}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.32

Cancel the common factor of $4$ and $2$ .

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Step 16.5.32.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - \frac{2 \cdot 2}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.32.2

Cancel the common factors.

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Step 16.5.32.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - \frac{2 \cdot 2}{2 (1)}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.32.2.2

Cancel the common factor.

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - \frac{2 \cdot 2}{2 \cdot 1}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.32.2.3

Rewrite the expression.

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - \frac{2}{1}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.32.2.4

Divide $2$ by $1$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - 1 \cdot 2) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.33

Multiply $- 1$ by $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - 2) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.34

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} - 2 \cdot \frac{2}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.35

Combine $- 2$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1}{2} + \frac{- 2 \cdot 2}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.36

Combine the numerators over the common denominator.

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1 - 2 \cdot 2}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.37

Simplify the numerator.

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Step 16.5.37.1

Multiply $- 2$ by $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{1 - 4}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.37.2

Subtract $4$ from $1$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (\frac{- 3}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.38

Move the negative in front of the fraction.

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + 4 (- \frac{3}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.39

Multiply $- 1$ by $4$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} - 4 (\frac{3}{2}) - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.40

Combine $- 4$ and $\frac{3}{2}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{- 4 \cdot 3}{2} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.41

Multiply $- 4$ by $3$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{- 12}{2} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.42

Cancel the common factor of $- 12$ and $2$ .

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Step 16.5.42.1

Factor $2$ out of $- 12$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{2 \cdot - 6}{2} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.42.2

Cancel the common factors.

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Step 16.5.42.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{2 \cdot - 6}{2 (1)} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.42.2.2

Cancel the common factor.

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{2 \cdot - 6}{2 \cdot 1} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.42.2.3

Rewrite the expression.

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{- 6}{1} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.42.2.4

Divide $- 6$ by $1$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} - 6 - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.43

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} - 6 \cdot \frac{10}{10} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.44

Combine $- 6$ and $\frac{10}{10}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421}{10} + \frac{- 6 \cdot 10}{10} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.45

Combine the numerators over the common denominator.

$A (x) = \frac{1}{- 1 + 2} (\frac{421 - 6 \cdot 10}{10} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.46

Simplify the numerator.

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Step 16.5.46.1

Multiply $- 6$ by $10$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{421 - 60}{10} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.46.2

Subtract $60$ from $421$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361}{10} - 9 \cdot - 1 + 9 \cdot - 2)$

Step 16.5.47

Multiply $- 9$ by $- 1$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361}{10} + 9 + 9 \cdot - 2)$

Step 16.5.48

Multiply $9$ by $- 2$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361}{10} + 9 - 18)$

Step 16.5.49

Subtract $18$ from $9$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361}{10} - 9)$

Step 16.5.50

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361}{10} - 9 \cdot \frac{10}{10})$

Step 16.5.51

Combine $- 9$ and $\frac{10}{10}$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361}{10} + \frac{- 9 \cdot 10}{10})$

Step 16.5.52

Combine the numerators over the common denominator.

$A (x) = \frac{1}{- 1 + 2} (\frac{361 - 9 \cdot 10}{10})$

Step 16.5.53

Simplify the numerator.

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Step 16.5.53.1

Multiply $- 9$ by $10$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{361 - 90}{10})$

Step 16.5.53.2

Subtract $90$ from $361$ .

$A (x) = \frac{1}{- 1 + 2} (\frac{271}{10})$

Step 17

Add $- 1$ and $2$ .

$A (x) = \frac{1}{1} \cdot \frac{271}{10}$

Step 18

Cancel the common factor of $1$ .

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Step 18.1

Cancel the common factor.

$A (x) = \frac{1}{1} \cdot \frac{271}{10}$

Step 18.2

Rewrite the expression.

$A (x) = 1 \cdot \frac{271}{10}$

Step 19

Multiply $\frac{271}{10}$ by $1$ .

$A (x) = \frac{271}{10}$

Step 20