Finite Math Examples

$\frac{22}{12 p} > \frac{33}{12}$

Step 1

Cancel the common factor of $22$ and $12$ .

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Step 1.1

Factor $2$ out of $22$ .

$\frac{2 (11)}{12 p} > \frac{33}{12}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $12 p$ .

$\frac{2 (11)}{2 (6 p)} > \frac{33}{12}$

Step 1.2.2

Cancel the common factor.

$\frac{2 \cdot 11}{2 (6 p)} > \frac{33}{12}$

Step 1.2.3

Rewrite the expression.

$\frac{11}{6 p} > \frac{33}{12}$

Step 2

Multiply both sides by $6 p$ .

$\frac{11}{6 p} (6 p) = \frac{33}{12} (6 p)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{11}{6 p} (6 p)$ .

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$6 \frac{11}{6 p} p = \frac{33}{12} (6 p)$

Step 3.1.1.2

Cancel the common factor of $6$ .

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Step 3.1.1.2.1

Factor $6$ out of $6 p$ .

$6 \frac{11}{6 (p)} p = \frac{33}{12} (6 p)$

Step 3.1.1.2.2

Cancel the common factor.

$6 \frac{11}{6 p} p = \frac{33}{12} (6 p)$

Step 3.1.1.2.3

Rewrite the expression.

$\frac{11}{p} p = \frac{33}{12} (6 p)$

Step 3.1.1.3

Cancel the common factor of $p$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{11}{p} p = \frac{33}{12} (6 p)$

Step 3.1.1.3.2

Rewrite the expression.

$11 = \frac{33}{12} (6 p)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{33}{12} (6 p)$ .

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Step 3.2.1.1

Cancel the common factor of $6$ .

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Step 3.2.1.1.1

Factor $6$ out of $12$ .

$11 = \frac{33}{6 (2)} (6 p)$

Step 3.2.1.1.2

Factor $6$ out of $6 p$ .

$11 = \frac{33}{6 (2)} (6 (p))$

Step 3.2.1.1.3

Cancel the common factor.

$11 = \frac{33}{6 \cdot 2} (6 p)$

Step 3.2.1.1.4

Rewrite the expression.

$11 = \frac{33}{2} p$

Step 3.2.1.2

Combine $\frac{33}{2}$ and $p$ .

$11 = \frac{33 p}{2}$

Step 4

Solve for $p$ .

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Step 4.1

Rewrite the equation as $\frac{33 p}{2} = 11$ .

$\frac{33 p}{2} = 11$

Step 4.2

Multiply both sides of the equation by $\frac{2}{33}$ .

$\frac{2}{33} \cdot \frac{33 p}{2} = \frac{2}{33} \cdot 11$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $\frac{2}{33} \cdot \frac{33 p}{2}$ .

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Step 4.3.1.1.1

Cancel the common factor of $2$ .

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Step 4.3.1.1.1.1

Cancel the common factor.

$\frac{2}{33} \cdot \frac{33 p}{2} = \frac{2}{33} \cdot 11$

Step 4.3.1.1.1.2

Rewrite the expression.

$\frac{1}{33} (33 p) = \frac{2}{33} \cdot 11$

Step 4.3.1.1.2

Cancel the common factor of $33$ .

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Step 4.3.1.1.2.1

Factor $33$ out of $33 p$ .

$\frac{1}{33} (33 (p)) = \frac{2}{33} \cdot 11$

Step 4.3.1.1.2.2

Cancel the common factor.

$\frac{1}{33} (33 p) = \frac{2}{33} \cdot 11$

Step 4.3.1.1.2.3

Rewrite the expression.

$p = \frac{2}{33} \cdot 11$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Cancel the common factor of $11$ .

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Step 4.3.2.1.1

Factor $11$ out of $33$ .

$p = \frac{2}{11 (3)} \cdot 11$

Step 4.3.2.1.2

Cancel the common factor.

$p = \frac{2}{11 \cdot 3} \cdot 11$

Step 4.3.2.1.3

Rewrite the expression.

$p = \frac{2}{3}$

Step 5

Find the domain of $\frac{11}{6 p} - \frac{11}{4}$ .

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Step 5.1

Set the denominator in $\frac{11}{6 p}$ equal to $0$ to find where the expression is undefined.

$6 p = 0$

Step 5.2

Divide each term in $6 p = 0$ by $6$ and simplify.

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Step 5.2.1

Divide each term in $6 p = 0$ by $6$ .

$\frac{6 p}{6} = \frac{0}{6}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{6 p}{6} = \frac{0}{6}$

Step 5.2.2.1.2

Divide $p$ by $1$ .

$p = \frac{0}{6}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $6$ .

$p = 0$

Step 5.3

The domain is all values of $p$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 6

Use each root to create test intervals.

$p < 0$

$0 < p < \frac{2}{3}$

$p > \frac{2}{3}$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $p < 0$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $p < 0$ and see if this value makes the original inequality true.

$p = - 2$

Step 7.1.2

Replace $p$ with $- 2$ in the original inequality.

$\frac{22}{12 (- 2)} > \frac{33}{12}$

Step 7.1.3

The left side $- 0.91\overline{6}$ is not greater than the right side $2.75$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $0 < p < \frac{2}{3}$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $0 < p < \frac{2}{3}$ and see if this value makes the original inequality true.

$p = 0.33$

Step 7.2.2

Replace $p$ with $0.33$ in the original inequality.

$\frac{22}{12 (0.33)} > \frac{33}{12}$

Step 7.2.3

The left side $5.\overline{5}$ is greater than the right side $2.75$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $p > \frac{2}{3}$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $p > \frac{2}{3}$ and see if this value makes the original inequality true.

$p = 3$

Step 7.3.2

Replace $p$ with $3$ in the original inequality.

$\frac{22}{12 (3)} > \frac{33}{12}$

Step 7.3.3

The left side $0.6\overline{1}$ is not greater than the right side $2.75$ , which means that the given statement is false.

False

Step 7.4

Compare the intervals to determine which ones satisfy the original inequality.

$p < 0$ False

$0 < p < \frac{2}{3}$ True

$p > \frac{2}{3}$ False

$p < 0$ False

$0 < p < \frac{2}{3}$ True

$p > \frac{2}{3}$ False

Step 8

The solution consists of all of the true intervals.

$0 < p < \frac{2}{3}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$0 < p < \frac{2}{3}$

Interval Notation:

$(0, \frac{2}{3})$

Step 10