Finite Math Examples

$\frac{x + 5}{2 x} < 0$

Step 1

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

$2 x = 0$

$x + 5 = 0$

Step 2

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

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

$x = 0$

$x = - 5$

Step 5

Consolidate the solutions.

$x = 0, - 5$

Step 6

Find the domain of $\frac{x + 5}{2 x}$ .

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Step 6.1

Set the denominator in $\frac{x + 5}{2 x}$ equal to $0$ to find where the expression is undefined.

$2 x = 0$

Step 6.2

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $0$ by $2$ .

$x = 0$

Step 6.3

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

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

Step 7

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 0$

$x > 0$

Step 8

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 8.1

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

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Step 8.1.1

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

$x = - 8$

Step 8.1.2

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

$\frac{(- 8) + 5}{2 (- 8)} < 0$

Step 8.1.3

The left side $0.1875$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.2

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

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Step 8.2.1

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

$x = - 2$

Step 8.2.2

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

$\frac{(- 2) + 5}{2 (- 2)} < 0$

Step 8.2.3

The left side $- 0.75$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

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

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Step 8.3.1

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

$x = 2$

Step 8.3.2

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

$\frac{(2) + 5}{2 (2)} < 0$

Step 8.3.3

The left side $1.75$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < - 5$ False

$- 5 < x < 0$ True

$x > 0$ False

$x < - 5$ False

$- 5 < x < 0$ True

$x > 0$ False

Step 9

The solution consists of all of the true intervals.

$- 5 < x < 0$

Step 10

Convert the inequality to interval notation.

$(- 5, 0)$

Step 11