Finite Math Examples

$\frac{3}{5 x} + \frac{1}{6 x} > \frac{2}{3}$

Step 1

Simplify $\frac{3}{5 x} + \frac{1}{6 x}$ .

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Step 1.1

To write $\frac{3}{5 x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{3}{5 x} \cdot \frac{6}{6} + \frac{1}{6 x} > \frac{2}{3}$

Step 1.2

To write $\frac{1}{6 x}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3}{5 x} \cdot \frac{6}{6} + \frac{1}{6 x} \cdot \frac{5}{5} > \frac{2}{3}$

Step 1.3

Write each expression with a common denominator of $30 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{3}{5 x}$ by $\frac{6}{6}$ .

$\frac{3 \cdot 6}{5 x \cdot 6} + \frac{1}{6 x} \cdot \frac{5}{5} > \frac{2}{3}$

Step 1.3.2

Multiply $6$ by $5$ .

$\frac{3 \cdot 6}{30 x} + \frac{1}{6 x} \cdot \frac{5}{5} > \frac{2}{3}$

Step 1.3.3

Multiply $\frac{1}{6 x}$ by $\frac{5}{5}$ .

$\frac{3 \cdot 6}{30 x} + \frac{5}{6 x \cdot 5} > \frac{2}{3}$

Step 1.3.4

Multiply $5$ by $6$ .

$\frac{3 \cdot 6}{30 x} + \frac{5}{30 x} > \frac{2}{3}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3 \cdot 6 + 5}{30 x} > \frac{2}{3}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $3$ by $6$ .

$\frac{18 + 5}{30 x} > \frac{2}{3}$

Step 1.5.2

Add $18$ and $5$ .

$\frac{23}{30 x} > \frac{2}{3}$

Step 2

Multiply both sides by $30 x$ .

$\frac{23}{30 x} (30 x) = \frac{2}{3} (30 x)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{23}{30 x} (30 x)$ .

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$30 \frac{23}{30 x} x = \frac{2}{3} (30 x)$

Step 3.1.1.2

Cancel the common factor of $30$ .

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Step 3.1.1.2.1

Factor $30$ out of $30 x$ .

$30 \frac{23}{30 (x)} x = \frac{2}{3} (30 x)$

Step 3.1.1.2.2

Cancel the common factor.

$30 \frac{23}{30 x} x = \frac{2}{3} (30 x)$

Step 3.1.1.2.3

Rewrite the expression.

$\frac{23}{x} x = \frac{2}{3} (30 x)$

Step 3.1.1.3

Cancel the common factor of $x$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{23}{x} x = \frac{2}{3} (30 x)$

Step 3.1.1.3.2

Rewrite the expression.

$23 = \frac{2}{3} (30 x)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{2}{3} (30 x)$ .

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Step 3.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1

Factor $3$ out of $30 x$ .

$23 = \frac{2}{3} (3 (10 x))$

Step 3.2.1.1.2

Cancel the common factor.

$23 = \frac{2}{3} (3 (10 x))$

Step 3.2.1.1.3

Rewrite the expression.

$23 = 2 (10 x)$

Step 3.2.1.2

Multiply $10$ by $2$ .

$23 = 20 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $20 x = 23$ .

$20 x = 23$

Step 4.2

Divide each term in $20 x = 23$ by $20$ and simplify.

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Step 4.2.1

Divide each term in $20 x = 23$ by $20$ .

$\frac{20 x}{20} = \frac{23}{20}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $20$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} = \frac{23}{20}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{23}{20}$

Step 5

Find the domain of $\frac{23}{30 x} - \frac{2}{3}$ .

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Step 5.1

Set the denominator in $\frac{23}{30 x}$ equal to $0$ to find where the expression is undefined.

$30 x = 0$

Step 5.2

Divide each term in $30 x = 0$ by $30$ and simplify.

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Step 5.2.1

Divide each term in $30 x = 0$ by $30$ .

$\frac{30 x}{30} = \frac{0}{30}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $30$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{30 x}{30} = \frac{0}{30}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{30}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $30$ .

$x = 0$

Step 5.3

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

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

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{23}{20}$

$x > \frac{23}{20}$

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 $x < 0$ to see if it makes the inequality true.

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Step 7.1.1

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

$x = - 2$

Step 7.1.2

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

$\frac{3}{5 (- 2)} + \frac{1}{6 (- 2)} > \frac{2}{3}$

Step 7.1.3

The left side $- 0.38\overline{3}$ is not greater than the right side $0.\overline{6}$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $0 < x < \frac{23}{20}$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $0 < x < \frac{23}{20}$ and see if this value makes the original inequality true.

$x = 1$

Step 7.2.2

Replace $x$ with $1$ in the original inequality.

$\frac{3}{5 (1)} + \frac{1}{6 (1)} > \frac{2}{3}$

Step 7.2.3

The left side $0.7\overline{6}$ is greater than the right side $0.\overline{6}$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $x > \frac{23}{20}$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > \frac{23}{20}$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{3}{5 (4)} + \frac{1}{6 (4)} > \frac{2}{3}$

Step 7.3.3

The left side $0.191\overline{6}$ is not greater than the right side $0.\overline{6}$ , which means that the given statement is false.

False

Step 7.4

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

$x < 0$ False

$0 < x < \frac{23}{20}$ True

$x > \frac{23}{20}$ False

$x < 0$ False

$0 < x < \frac{23}{20}$ True

$x > \frac{23}{20}$ False

Step 8

The solution consists of all of the true intervals.

$0 < x < \frac{23}{20}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$0 < x < \frac{23}{20}$

Interval Notation:

$(0, \frac{23}{20})$

Step 10