Precalculus Examples

$x^{3} - 8 x^{2} < 0$

Step 1

Convert the inequality to an equation.

$x^{3} - 8 x^{2} = 0$

Step 2

Factor $x^{2}$ out of $x^{3} - 8 x^{2}$ .

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Step 2.1

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

$x^{2} x - 8 x^{2} = 0$

Step 2.2

Factor $x^{2}$ out of $- 8 x^{2}$ .

$x^{2} x + x^{2} \cdot - 8 = 0$

Step 2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 8$ .

$x^{2} (x - 8) = 0$

Step 3

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

$x^{2} = 0$

$x - 8 = 0$

Step 4

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

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Step 4.1

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

$x^{2} = 0$

Step 4.2

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

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Step 4.2.1

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

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

$x = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

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

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Step 5.1

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

$x - 8 = 0$

Step 5.2

Add $8$ to both sides of the equation.

$x = 8$

Step 6

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

$x = 0, 8$

Step 7

Use each root to create test intervals.

$x < 0$

$0 < x < 8$

$x > 8$

Step 8

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 8.1

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

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Step 8.1.1

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

$x = - 2$

Step 8.1.2

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

${(- 2)}^{3} - 8 {(- 2)}^{2} < 0$

Step 8.1.3

The left side $- 40$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.2

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

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Step 8.2.1

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

$x = 4$

Step 8.2.2

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

${(4)}^{3} - 8 {(4)}^{2} < 0$

Step 8.2.3

The left side $- 64$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

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

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Step 8.3.1

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

$x = 10$

Step 8.3.2

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

${(10)}^{3} - 8 {(10)}^{2} < 0$

Step 8.3.3

The left side $200$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < 0$ True

$0 < x < 8$ True

$x > 8$ False

$x < 0$ True

$0 < x < 8$ True

$x > 8$ False

Step 9

The solution consists of all of the true intervals.

$x < 0$ or $0 < x < 8$

Step 10

Convert the inequality to interval notation.

$(- \infty, 0) \cup (0, 8)$

Step 11