Solve the Inequality for x x^3>=6x^2

$x^{3} \geq 6 x^{2}$

Step 1

Subtract $6 x^{2}$ from both sides of the inequality.

$x^{3} - 6 x^{2} \geq 0$

Step 2

Convert the inequality to an equation.

$x^{3} - 6 x^{2} = 0$

Step 3

Factor $x^{2}$ out of $x^{3} - 6 x^{2}$ .

Factor $x^{2}$ out of $x^{3}$ .

$x^{2} x - 6 x^{2} = 0$

Factor $x^{2}$ out of $- 6 x^{2}$ .

$x^{2} x + x^{2} \cdot - 6 = 0$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 6$ .

$x^{2} (x - 6) = 0$

Step 4

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

$x^{2} = 0$

$x - 6 = 0$

Step 5

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 6

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 7

The final solution is all the values that make $x^{2} (x - 6) = 0$ true.

$x = 0, 6$

Step 8

Use each root to create test intervals.

$x < 0$

$0 < x < 6$

$x > 6$

Step 9

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 $x < 0$ to see if it makes the inequality true.

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

$x = - 2$

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

${(- 2)}^{3} \geq 6 {(- 2)}^{2}$

The left side $- 8$ is less than the right side $24$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 6$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

${(3)}^{3} \geq 6 {(3)}^{2}$

The left side $27$ is less than the right side $54$ , which means that the given statement is false.

False

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

${(8)}^{3} \geq 6 {(8)}^{2}$

The left side $512$ is greater than the right side $384$ , which means that the given statement is always true.

True

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

$x < 0$ False

$0 < x < 6$ False

$x > 6$ True

$x < 0$ False

$0 < x < 6$ False

$x > 6$ True

Step 10

The solution consists of all of the true intervals.

$x \geq 6$ or $x = 0$

Step 11

Combine the intervals.

$x = 0 or x \geq 6$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x = 0 or x \geq 6$

Interval Notation:

$[0, 0] \cup [6, \infty)$

Step 13

Solve the Inequality for x x^3>=6x^2

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