Calculus Examples

$\int_{- 2}^{4} | 2 x - 3 | d x$

Step 1

Split up the integral depending on where $2 x - 3$ is positive and negative.

$\int_{- 2}^{\frac{3}{2}} - 2 x + 3 d x + \int_{\frac{3}{2}}^{4} 2 x - 3 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{\frac{3}{2}} - 2 x d x + \int_{- 2}^{\frac{3}{2}} 3 d x + \int_{\frac{3}{2}}^{4} 2 x - 3 d x$

Step 3

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

$- 2 \int_{- 2}^{\frac{3}{2}} x d x + \int_{- 2}^{\frac{3}{2}} 3 d x + \int_{\frac{3}{2}}^{4} 2 x - 3 d x$

Step 4

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

$- 2 (\frac{1}{2} x^{2}]_{- 2}^{\frac{3}{2}}) + \int_{- 2}^{\frac{3}{2}} 3 d x + \int_{\frac{3}{2}}^{4} 2 x - 3 d x$

Step 5

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

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + \int_{- 2}^{\frac{3}{2}} 3 d x + \int_{\frac{3}{2}}^{4} 2 x - 3 d x$

Step 6

Apply the constant rule.

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + 3 x]_{- 2}^{\frac{3}{2}} + \int_{\frac{3}{2}}^{4} 2 x - 3 d x$

Step 7

Split the single integral into multiple integrals.

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + 3 x]_{- 2}^{\frac{3}{2}} + \int_{\frac{3}{2}}^{4} 2 x d x + \int_{\frac{3}{2}}^{4} - 3 d x$

Step 8

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

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + 3 x]_{- 2}^{\frac{3}{2}} + 2 \int_{\frac{3}{2}}^{4} x d x + \int_{\frac{3}{2}}^{4} - 3 d x$

Step 9

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

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + 3 x]_{- 2}^{\frac{3}{2}} + 2 (\frac{1}{2} x^{2}]_{\frac{3}{2}}^{4}) + \int_{\frac{3}{2}}^{4} - 3 d x$

Step 10

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

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + 3 x]_{- 2}^{\frac{3}{2}} + 2 (\frac{x^{2}}{2}]_{\frac{3}{2}}^{4}) + \int_{\frac{3}{2}}^{4} - 3 d x$

Step 11

Apply the constant rule.

$- 2 (\frac{x^{2}}{2}]_{- 2}^{\frac{3}{2}}) + 3 x]_{- 2}^{\frac{3}{2}} + 2 (\frac{x^{2}}{2}]_{\frac{3}{2}}^{4}) + - 3 x]_{\frac{3}{2}}^{4}$

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $\frac{x^{2}}{2}$ at $\frac{3}{2}$ and at $- 2$ .

$- 2 ((\frac{{(\frac{3}{2})}^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + 3 x]_{- 2}^{\frac{3}{2}} + 2 (\frac{x^{2}}{2}]_{\frac{3}{2}}^{4}) + - 3 x]_{\frac{3}{2}}^{4}$

Step 12.2

Evaluate $3 x$ at $\frac{3}{2}$ and at $- 2$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{x^{2}}{2}]_{\frac{3}{2}}^{4}) + - 3 x]_{\frac{3}{2}}^{4}$

Step 12.3

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $\frac{3}{2}$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 ((\frac{4^{2}}{2}) - \frac{{(\frac{3}{2})}^{2}}{2}) + - 3 x]_{\frac{3}{2}}^{4}$

Step 12.4

Evaluate $- 3 x$ at $4$ and at $\frac{3}{2}$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) + - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5

Simplify.

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Step 12.5.1

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

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{4}{2}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.2

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

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Step 12.5.2.1

Factor $2$ out of $4$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2 \cdot 2}{2}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.2.2

Cancel the common factors.

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Step 12.5.2.2.1

Factor $2$ out of $2$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2 \cdot 2}{2 (1)}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.2.2.2

Cancel the common factor.

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.2.2.3

Rewrite the expression.

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2}{1}) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.2.2.4

Divide $2$ by $1$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 1 \cdot 2) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.3

Multiply $- 1$ by $2$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 3 (\frac{3}{2}) - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.4

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

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{3 \cdot 3}{2} - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.5

Multiply $3$ by $3$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{9}{2} - 3 \cdot - 2 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.6

Multiply $- 3$ by $- 2$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{9}{2} + 6 + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.7

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{9}{2} + 6 \cdot \frac{2}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.8

Combine $6$ and $\frac{2}{2}$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{9}{2} + \frac{6 \cdot 2}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.9

Combine the numerators over the common denominator.

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{9 + 6 \cdot 2}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.10

Simplify the numerator.

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Step 12.5.10.1

Multiply $6$ by $2$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{9 + 12}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.10.2

Add $9$ and $12$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + \frac{21}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.11

To write $- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) \cdot \frac{2}{2} + \frac{21}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.12

Combine $- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2)$ and $\frac{2}{2}$ .

$\frac{- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) \cdot 2}{2} + \frac{21}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.13

Combine the numerators over the common denominator.

$\frac{- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) \cdot 2 + 21}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.14

Multiply $2$ by $- 2$ .

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (\frac{4^{2}}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.15

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

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (\frac{16}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.16

Cancel the common factor of $16$ and $2$ .

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Step 12.5.16.1

Factor $2$ out of $16$ .

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (\frac{2 \cdot 8}{2} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.16.2

Cancel the common factors.

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Step 12.5.16.2.1

Factor $2$ out of $2$ .

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (\frac{2 \cdot 8}{2 (1)} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.16.2.2

Cancel the common factor.

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.16.2.3

Rewrite the expression.

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (\frac{8}{1} - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.16.2.4

Divide $8$ by $1$ .

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.17

To write $2 (8 - \frac{{(\frac{3}{2})}^{2}}{2})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot \frac{2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.18

Combine $2 (8 - \frac{{(\frac{3}{2})}^{2}}{2})$ and $\frac{2}{2}$ .

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21}{2} + \frac{2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.19

Combine the numerators over the common denominator.

$\frac{- 4 (\frac{{(\frac{3}{2})}^{2}}{2} - 2) + 21 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20

Simplify the numerator.

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Step 12.5.20.1

Divide $3$ by $2$ .

$\frac{- 4 (\frac{{1.5}^{2}}{2} - 2) + 21 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.2

Raise $1.5$ to the power of $2$ .

$\frac{- 4 (\frac{2.25}{2} - 2) + 21 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.3

Divide $2.25$ by $2$ .

$\frac{- 4 (1.125 - 2) + 21 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.4

Subtract $2$ from $1.125$ .

$\frac{- 4 \cdot - 0.875 + 21 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.5

Multiply $- 4$ by $- 0.875$ .

$\frac{3.5 + 21 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.6

Add $3.5$ and $21$ .

$\frac{24.5 + 2 (8 - \frac{{(\frac{3}{2})}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.7

Divide $3$ by $2$ .

$\frac{24.5 + 2 (8 - \frac{{1.5}^{2}}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.8

Raise $1.5$ to the power of $2$ .

$\frac{24.5 + 2 (8 - \frac{2.25}{2}) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.9

Divide $2.25$ by $2$ .

$\frac{24.5 + 2 (8 - 1 \cdot 1.125) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.10

Multiply $- 1$ by $1.125$ .

$\frac{24.5 + 2 (8 - 1.125) \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.11

Subtract $1.125$ from $8$ .

$\frac{24.5 + 2 \cdot 6.875 \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.12

Multiply $2$ by $6.875$ .

$\frac{24.5 + 13.75 \cdot 2}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.13

Multiply $13.75$ by $2$ .

$\frac{24.5 + 27.5}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.20.14

Add $24.5$ and $27.5$ .

$\frac{52}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.21

Cancel the common factor of $52$ and $2$ .

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Step 12.5.21.1

Factor $2$ out of $52$ .

$\frac{2 \cdot 26}{2} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.21.2

Cancel the common factors.

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Step 12.5.21.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 26}{2 (1)} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.21.2.2

Cancel the common factor.

$\frac{2 \cdot 26}{2 \cdot 1} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.21.2.3

Rewrite the expression.

$\frac{26}{1} - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.21.2.4

Divide $26$ by $1$ .

$26 - 3 \cdot 4 + 3 (\frac{3}{2})$

Step 12.5.22

Multiply $- 3$ by $4$ .

$26 - 12 + 3 (\frac{3}{2})$

Step 12.5.23

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

$26 - 12 + \frac{3 \cdot 3}{2}$

Step 12.5.24

Multiply $3$ by $3$ .

$26 - 12 + \frac{9}{2}$

Step 12.5.25

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$26 - 12 \cdot \frac{2}{2} + \frac{9}{2}$

Step 12.5.26

Combine $- 12$ and $\frac{2}{2}$ .

$26 + \frac{- 12 \cdot 2}{2} + \frac{9}{2}$

Step 12.5.27

Combine the numerators over the common denominator.

$26 + \frac{- 12 \cdot 2 + 9}{2}$

Step 12.5.28

Simplify the numerator.

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Step 12.5.28.1

Multiply $- 12$ by $2$ .

$26 + \frac{- 24 + 9}{2}$

Step 12.5.28.2

Add $- 24$ and $9$ .

$26 + \frac{- 15}{2}$

Step 12.5.29

Move the negative in front of the fraction.

$26 - \frac{15}{2}$

Step 12.5.30

To write $26$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$26 \cdot \frac{2}{2} - \frac{15}{2}$

Step 12.5.31

Combine $26$ and $\frac{2}{2}$ .

$\frac{26 \cdot 2}{2} - \frac{15}{2}$

Step 12.5.32

Combine the numerators over the common denominator.

$\frac{26 \cdot 2 - 15}{2}$

Step 12.5.33

Simplify the numerator.

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Step 12.5.33.1

Multiply $26$ by $2$ .

$\frac{52 - 15}{2}$

Step 12.5.33.2

Subtract $15$ from $52$ .

$\frac{37}{2}$

Step 12.5.34

Cancel the common factor of $37$ and $2$ .

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Step 12.5.34.1

Rewrite $37$ as $1 (37)$ .

$\frac{1 (37)}{2}$

Step 12.5.34.2

Cancel the common factors.

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Step 12.5.34.2.1

Rewrite $2$ as $1 (2)$ .

$\frac{1 \cdot 37}{1 \cdot 2}$

Step 12.5.34.2.2

Cancel the common factor.

$\frac{1 \cdot 37}{1 \cdot 2}$

Step 12.5.34.2.3

Rewrite the expression.

$\frac{37}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{37}{2}$

Decimal Form:

$18.5$

Mixed Number Form:

$18 \frac{1}{2}$

Step 14