Algebra Examples

x^{3} - 4 x^{2} < 0

$x^{3} - 4 x^{2} < 0$

Step 1

Convert the inequality to an equation.

x^{3} - 4 x^{2} = 0

$x^{3} - 4 x^{2} = 0$

Step 2

Factor

x^{2}

$x^{2}$ out of

x^{3} - 4 x^{2}

$x^{3} - 4 x^{2}$ .

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Step 2.1

Factor

x^{2}

$x^{2}$ out of

x^{3}

$x^{3}$ .

x^{2} x - 4 x^{2} = 0

$x^{2} x - 4 x^{2} = 0$

Step 2.2

Factor

x^{2}

$x^{2}$ out of

- 4 x^{2}

$- 4 x^{2}$ .

x^{2} x + x^{2} \cdot - 4 = 0

$x^{2} x + x^{2} \cdot - 4 = 0$

Step 2.3

Factor

x^{2}

$x^{2}$ out of

x^{2} x + x^{2} \cdot - 4

$x^{2} x + x^{2} \cdot - 4$ .

x^{2} (x - 4) = 0

$x^{2} (x - 4) = 0$

x^{2} (x - 4) = 0

$x^{2} (x - 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x^{2} = 0

$x^{2} = 0$

x - 4 = 0

$x - 4 = 0$

Step 4

Set

x^{2}

$x^{2}$ equal to

0

$0$ and solve for

x

$x$ .

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Step 4.1

Set

x^{2}

$x^{2}$ equal to

0

$0$ .

x^{2} = 0

$x^{2} = 0$

Step 4.2

Solve

x^{2} = 0

$x^{2} = 0$ for

x

$x$ .

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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}

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify

\pm \sqrt{0}

$\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

x = \pm \sqrt{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

$x = \pm 0$

Step 4.2.2.3

Plus or minus

0

$0$ is

0

$0$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 5

Set

x - 4

$x - 4$ equal to

0

$0$ and solve for

x

$x$ .

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Step 5.1

Set

x - 4

$x - 4$ equal to

0

$0$ .

x - 4 = 0

$x - 4 = 0$

Step 5.2

Add

4

$4$ to both sides of the equation.

x = 4

$x = 4$

x = 4

$x = 4$

Step 6

The final solution is all the values that make

x^{2} (x - 4) = 0

$x^{2} (x - 4) = 0$ true.

x = 0, 4

$x = 0, 4$

Step 7

Use each root to create test intervals.

x < 0

$x < 0$

0 < x < 4

$0 < x < 4$

x > 4

$x > 4$

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

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

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Step 8.1.1

Choose a value on the interval

x < 0

$x < 0$ and see if this value makes the original inequality true.

x = - 2

$x = - 2$

Step 8.1.2

Replace

x

$x$ with

- 2

$- 2$ in the original inequality.

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

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

Step 8.1.3

The left side

- 24

$- 24$ is less than the right side

0

$0$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval

0 < x < 4

$0 < x < 4$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval

0 < x < 4

$0 < x < 4$ and see if this value makes the original inequality true.

x = 2

$x = 2$

Step 8.2.2

Replace

x

$x$ with

2

$2$ in the original inequality.

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

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

Step 8.2.3

The left side

- 8

$- 8$ is less than the right side

0

$0$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval

x > 4

$x > 4$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval

x > 4

$x > 4$ and see if this value makes the original inequality true.

x = 6

$x = 6$

Step 8.3.2

Replace

x

$x$ with

6

$6$ in the original inequality.

{(6)}^{3} - 4 {(6)}^{2} < 0

${(6)}^{3} - 4 {(6)}^{2} < 0$

Step 8.3.3

The left side

72

$72$ is not less than the right side

0

$0$ , 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

$x < 0$ True

0 < x < 4

$0 < x < 4$ True

x > 4

$x > 4$ False

x < 0

$x < 0$ True

0 < x < 4

$0 < x < 4$ True

x > 4

$x > 4$ False

Step 9

The solution consists of all of the true intervals.

x < 0

$x < 0$ or

0 < x < 4

$0 < x < 4$

Step 10

The result can be shown in multiple forms.

Inequality Form:

x < 0 or 0 < x < 4

$x < 0 or 0 < x < 4$

Interval Notation:

(- \infty, 0) \cup (0, 4)

$(- \infty, 0) \cup (0, 4)$

Step 11