Precalculus Examples

$x {(x + 2)}^{2} (x - 3) \leq 0$

Step 1

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

$x = 0$

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

$x - 3 = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

The final solution is all the values that make $x {(x + 2)}^{2} (x - 3) \leq 0$ true.

$x = 0, - 2, 3$

Step 6

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 3$

$x > 3$

Step 7

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 7.1

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

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Step 7.1.1

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

$x = - 4$

Step 7.1.2

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

$(- 4) {((- 4) + 2)}^{2} ((- 4) - 3) \leq 0$

Step 7.1.3

The left side $112$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 7.2.2

Replace $x$ with $- 1$ in the original inequality.

$(- 1) {((- 1) + 2)}^{2} ((- 1) - 3) \leq 0$

Step 7.2.3

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

False

Step 7.3

Test a value on the interval $0 < x < 3$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $0 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 7.3.2

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

$(2) {((2) + 2)}^{2} ((2) - 3) \leq 0$

Step 7.3.3

The left side $- 32$ 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 $x > 3$ to see if it makes the inequality true.

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Step 7.4.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 7.4.2

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

$(6) {((6) + 2)}^{2} ((6) - 3) \leq 0$

Step 7.4.3

The left side $1152$ is greater 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.

$x < - 2$ False

$- 2 < x < 0$ False

$0 < x < 3$ True

$x > 3$ False

$x < - 2$ False

$- 2 < x < 0$ False

$0 < x < 3$ True

$x > 3$ False

Step 8

The solution consists of all of the true intervals.

$0 \leq x \leq 3$ or $x = - 2$

Step 9

Combine the intervals.

$x = - 2 or 0 \leq x \leq 3$

Step 10

Convert the inequality to interval notation.

$[- 2, - 2] \cup [0, 3]$

Step 11