Solve the Inequality for X 4X^2(X^2-9)<=0

$4 X^{2} (X^{2} - 9) \leq 0$

Step 1

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

$X^{2} = 0$

$X^{2} - 9 = 0$

Step 2

Set $X^{2}$ equal to $0$ and solve for $X$ .

Set $X^{2}$ equal to $0$ .

$X^{2} = 0$

Solve $X^{2} = 0$ for $X$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$X = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$X = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$X = \pm 0$

Plus or minus $0$ is $0$ .

$X = 0$

Step 3

Set $X^{2} - 9$ equal to $0$ and solve for $X$ .

Set $X^{2} - 9$ equal to $0$ .

$X^{2} - 9 = 0$

Solve $X^{2} - 9 = 0$ for $X$ .

Add $9$ to both sides of the equation.

$X^{2} = 9$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$X = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$X = \pm \sqrt{3^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$X = \pm 3$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$X = 3$

Next, use the negative value of the $\pm$ to find the second solution.

$X = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$X = 3, - 3$

Step 4

The final solution is all the values that make $4 X^{2} (X^{2} - 9) \leq 0$ true.

$X = 0, 3, - 3$

Step 5

Use each root to create test intervals.

$X < - 3$

$- 3 < X < 0$

$0 < X < 3$

$X > 3$

Step 6

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

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

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

$X = - 6$

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

$4 {(- 6)}^{2} ({(- 6)}^{2} - 9) \leq 0$

The left side $3888$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$X = - 2$

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

$4 {(- 2)}^{2} ({(- 2)}^{2} - 9) \leq 0$

The left side $- 80$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$X = 2$

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

$4 {(2)}^{2} ({(2)}^{2} - 9) \leq 0$

The left side $- 80$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $X > 3$ to see if it makes the inequality true.

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

$X = 6$

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

$4 {(6)}^{2} ({(6)}^{2} - 9) \leq 0$

The left side $3888$ is greater than the right side $0$ , which means that the given statement is false.

False

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

$X < - 3$ False

$- 3 < X < 0$ True

$0 < X < 3$ True

$X > 3$ False

$X < - 3$ False

$- 3 < X < 0$ True

$0 < X < 3$ True

$X > 3$ False

Step 7

The solution consists of all of the true intervals.

$- 3 \leq X \leq 0$ or $0 \leq X \leq 3$

Step 8

Combine the intervals.

$- 3 \leq X \leq 3$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq X \leq 3$

Interval Notation:

$[- 3, 3]$

Step 10

Solve the Inequality for X 4X^2(X^2-9)<=0

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