Finite Math Examples

$\frac{x - 2}{3} < \frac{5}{x}$

Step 1

Subtract $\frac{5}{x}$ from both sides of the inequality.

$\frac{x - 2}{3} - \frac{5}{x} < 0$

Step 2

Simplify $\frac{x - 2}{3} - \frac{5}{x}$ .

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Step 2.1

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

$\frac{x - 2}{3} \cdot \frac{x}{x} - \frac{5}{x} < 0$

Step 2.2

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

$\frac{x - 2}{3} \cdot \frac{x}{x} - \frac{5}{x} \cdot \frac{3}{3} < 0$

Step 2.3

Write each expression with a common denominator of $3 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{x - 2}{3}$ by $\frac{x}{x}$ .

$\frac{(x - 2) x}{3 x} - \frac{5}{x} \cdot \frac{3}{3} < 0$

Step 2.3.2

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

$\frac{(x - 2) x}{3 x} - \frac{5 \cdot 3}{x \cdot 3} < 0$

Step 2.3.3

Reorder the factors of $x \cdot 3$ .

$\frac{(x - 2) x}{3 x} - \frac{5 \cdot 3}{3 x} < 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(x - 2) x - 5 \cdot 3}{3 x} < 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{x \cdot x - 2 x - 5 \cdot 3}{3 x} < 0$

Step 2.5.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 2 x - 5 \cdot 3}{3 x} < 0$

Step 2.5.3

Multiply $- 5$ by $3$ .

$\frac{x^{2} - 2 x - 15}{3 x} < 0$

Step 2.5.4

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 2.5.4.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 2.5.4.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{3 x} < 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$3 x = 0$

$x - 5 = 0$

$x + 3 = 0$

Step 4

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $3$ .

$x = 0$

Step 5

Add $5$ to both sides of the equation.

$x = 5$

Step 6

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

$x = 5$

$x = - 3$

Step 8

Consolidate the solutions.

$x = 0, 5, - 3$

Step 9

Find the domain of $\frac{(x - 5) (x + 3)}{3 x}$ .

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Step 9.1

Set the denominator in $\frac{(x - 5) (x + 3)}{3 x}$ equal to $0$ to find where the expression is undefined.

$3 x = 0$

Step 9.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 9.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $3$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 9.3

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

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

Step 10

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 0$

$0 < x < 5$

$x > 5$

Step 11

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 11.1

Test a value on the interval $x < - 3$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

Step 11.1.2

Replace $x$ with $- 6$ in the original inequality.

$\frac{(- 6) - 2}{3} < \frac{5}{- 6}$

Step 11.1.3

The left side $- 2.\overline{6}$ is less than the right side $- 0.8\overline{3}$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $- 3 < x < 0$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- 3 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 11.2.2

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

$\frac{(- 2) - 2}{3} < \frac{5}{- 2}$

Step 11.2.3

The left side $- 1.\overline{3}$ is not less than the right side $- 2.5$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $0 < x < 5$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $0 < x < 5$ and see if this value makes the original inequality true.

$x = 2$

Step 11.3.2

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

$\frac{(2) - 2}{3} < \frac{5}{2}$

Step 11.3.3

The left side $0$ is less than the right side $2.5$ , which means that the given statement is always true.

True

Step 11.4

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 11.4.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 11.4.2

Replace $x$ with $8$ in the original inequality.

$\frac{(8) - 2}{3} < \frac{5}{8}$

Step 11.4.3

The left side $2$ is not less than the right side $0.625$ , which means that the given statement is false.

False

Step 11.5

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

$x < - 3$ True

$- 3 < x < 0$ False

$0 < x < 5$ True

$x > 5$ False

$x < - 3$ True

$- 3 < x < 0$ False

$0 < x < 5$ True

$x > 5$ False

Step 12

The solution consists of all of the true intervals.

$x < - 3$ or $0 < x < 5$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < - 3 or 0 < x < 5$

Interval Notation:

$(- \infty, - 3) \cup (0, 5)$

Step 14