Precalculus Examples

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

Step 1

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

$x - 2 = 0$

$x + 4 = 0$

Step 2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4

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

$x = 2$

$x = - 4$

Step 5

Consolidate the solutions.

$x = 2, - 4$

Step 6

Find the domain of $\frac{x - 2}{x + 4}$ .

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Step 6.1

Set the denominator in $\frac{x - 2}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6.3

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

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

Step 7

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 2$

$x > 2$

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

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Step 8.1.1

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

$x = - 6$

Step 8.1.2

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

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

Step 8.1.3

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval $- 4 < x < 2$ to see if it makes the inequality true.

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Step 8.2.1

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

$x = 0$

Step 8.2.2

Replace $x$ with $0$ in the original inequality.

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

Step 8.2.3

The left side $- 0.5$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 8.3.2

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

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

Step 8.3.3

The left side $0.25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.4

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

$x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

$x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

Step 9

The solution consists of all of the true intervals.

$x < - 4$ or $x \geq 2$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < - 4 or x \geq 2$

Interval Notation:

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

Step 11

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