Precalculus Examples

$(x - 2) (x + 4) < 0$

Step 1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 4 = 0$

Step 2

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4

The final solution is all the values that make $(x - 2) (x + 4) < 0$ true.

$x = 2, - 4$

Step 5

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 2$

$x > 2$

Step 6

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 6.1

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

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Step 6.1.1

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

$x = - 6$

Step 6.1.2

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

$((- 6) - 2) ((- 6) + 4) < 0$

Step 6.1.3

The left side $16$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.2

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

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Step 6.2.1

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

$x = 0$

Step 6.2.2

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

$((0) - 2) ((0) + 4) < 0$

Step 6.2.3

The left side $- 8$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.3

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

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Step 6.3.1

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

$x = 4$

Step 6.3.2

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

$((4) - 2) ((4) + 4) < 0$

Step 6.3.3

The left side $16$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.4

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

$x < - 4$ False

$- 4 < x < 2$ True

$x > 2$ False

$x < - 4$ False

$- 4 < x < 2$ True

$x > 2$ False

Step 7

The solution consists of all of the true intervals.

$- 4 < x < 2$

Step 8

Convert the inequality to interval notation.

$(- 4, 2)$

Step 9