Precalculus Examples

$x {(x + 5)}^{2} (x - 8) < 0$

Step 1

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

$x = 0$

${(x + 5)}^{2} = 0$

$x - 8 = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 3.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 3.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

Set $x - 8$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.2

Add $8$ to both sides of the equation.

$x = 8$

Step 5

The final solution is all the values that make $x {(x + 5)}^{2} (x - 8) < 0$ true.

$x = 0, - 5, 8$

Step 6

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 0$

$0 < x < 8$

$x > 8$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

$x = - 8$

Step 7.1.2

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

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

Step 7.1.3

The left side $1152$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

$x = - 2$

Step 7.2.2

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

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

Step 7.2.3

The left side $180$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $0 < x < 8$ to see if it makes the inequality true.

Tap for more steps...

Step 7.3.1

Choose a value on the interval $0 < x < 8$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

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

$(4) {((4) + 5)}^{2} ((4) - 8) < 0$

Step 7.3.3

The left side $- 1296$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

Test a value on the interval $x > 8$ to see if it makes the inequality true.

Tap for more steps...

Step 7.4.1

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Step 7.4.2

Replace $x$ with $10$ in the original inequality.

$(10) {((10) + 5)}^{2} ((10) - 8) < 0$

Step 7.4.3

The left side $4500$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.5

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

$x < - 5$ False

$- 5 < x < 0$ False

$0 < x < 8$ True

$x > 8$ False

$x < - 5$ False

$- 5 < x < 0$ False

$0 < x < 8$ True

$x > 8$ False

Step 8

The solution consists of all of the true intervals.

$0 < x < 8$

Step 9

Convert the inequality to interval notation.

$(0, 8)$

Step 10