Precalculus Examples

\frac{x - 2}{x + 4} \geq 0

$\frac{x - 2}{x + 4} \geq 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to

0

$0$ and solving.

x - 2 = 0

$x - 2 = 0$

x + 4 = 0

$x + 4 = 0$

Step 2

Add

2

$2$ to both sides of the equation.

x = 2

$x = 2$

Step 3

Subtract

4

$4$ from both sides of the equation.

x = - 4

$x = - 4$

Step 4

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

x = 2

$x = 2$

x = - 4

$x = - 4$

Step 5

Consolidate the solutions.

x = 2, - 4

$x = 2, - 4$

Step 6

Find the domain of

\frac{x - 2}{x + 4}

$\frac{x - 2}{x + 4}$ .

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Step 6.1

Set the denominator in

\frac{x - 2}{x + 4}

$\frac{x - 2}{x + 4}$ equal to

0

$0$ to find where the expression is undefined.

x + 4 = 0

$x + 4 = 0$

Step 6.2

Subtract

4

$4$ from both sides of the equation.

x = - 4

$x = - 4$

Step 6.3

The domain is all values of

x

$x$ that make the expression defined.

(- \infty, - 4) \cup (- 4, \infty)

$(- \infty, - 4) \cup (- 4, \infty)$

(- \infty, - 4) \cup (- 4, \infty)

$(- \infty, - 4) \cup (- 4, \infty)$

Step 7

Use each root to create test intervals.

x < - 4

$x < - 4$

- 4 < x < 2

$- 4 < x < 2$

x > 2

$x > 2$

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 < - 4

$x < - 4$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval

x < - 4

$x < - 4$ and see if this value makes the original inequality true.

x = - 6

$x = - 6$

Step 8.1.2

Replace

x

$x$ with

- 6

$- 6$ in the original inequality.

\frac{(- 6) - 2}{(- 6) + 4} \geq 0

$\frac{(- 6) - 2}{(- 6) + 4} \geq 0$

Step 8.1.3

The left side

4

$4$ is greater than the right side

0

$0$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval

- 4 < x < 2

$- 4 < x < 2$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval

- 4 < x < 2

$- 4 < x < 2$ and see if this value makes the original inequality true.

x = 0

$x = 0$

Step 8.2.2

Replace

x

$x$ with

0

$0$ in the original inequality.

\frac{(0) - 2}{(0) + 4} \geq 0

$\frac{(0) - 2}{(0) + 4} \geq 0$

Step 8.2.3

The left side

- 0.5

$- 0.5$ is less than the right side

0

$0$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval

x > 2

$x > 2$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval

x > 2

$x > 2$ and see if this value makes the original inequality true.

x = 4

$x = 4$

Step 8.3.2

Replace

x

$x$ with

4

$4$ in the original inequality.

\frac{(4) - 2}{(4) + 4} \geq 0

$\frac{(4) - 2}{(4) + 4} \geq 0$

Step 8.3.3

The left side

0.25

$0.25$ is greater than the right side

0

$0$ , which means that the given statement is always true.

True

Step 8.4

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

x < - 4

$x < - 4$ True

- 4 < x < 2

$- 4 < x < 2$ False

x > 2

$x > 2$ True

x < - 4

$x < - 4$ True

- 4 < x < 2

$- 4 < x < 2$ False

x > 2

$x > 2$ True

Step 9

The solution consists of all of the true intervals.

x < - 4

$x < - 4$ or

x \geq 2

$x \geq 2$

Step 10

The result can be shown in multiple forms.

Inequality Form:

x < - 4 or x \geq 2

$x < - 4 or x \geq 2$

Interval Notation:

(- \infty, - 4) \cup [2, \infty)

$(- \infty, - 4) \cup [2, \infty)$

Step 11

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