Calculus Examples

$x^{3} < 9 x$

Step 1

Subtract $9 x$ from both sides of the inequality.

$x^{3} - 9 x < 0$

Step 2

Convert the inequality to an equation.

$x^{3} - 9 x = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $x$ out of $x^{3} - 9 x$ .

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Step 3.1.1

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

$x \cdot x^{2} - 9 x = 0$

Step 3.1.2

Factor $x$ out of $- 9 x$ .

$x \cdot x^{2} + x \cdot - 9 = 0$

Step 3.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$x (x^{2} - 9) = 0$

Step 3.2

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

$x (x^{2} - 3^{2}) = 0$

Step 3.3

Factor.

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Step 3.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$x ((x + 3) (x - 3)) = 0$

Step 3.3.2

Remove unnecessary parentheses.

$x (x + 3) (x - 3) = 0$

Step 4

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

$x = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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Step 6.1

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

$x + 3 = 0$

Step 6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

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

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Step 7.1

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

$x - 3 = 0$

Step 7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 8

The final solution is all the values that make $x (x + 3) (x - 3) = 0$ true.

$x = 0, - 3, 3$

Step 9

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 0$

$0 < x < 3$

$x > 3$

Step 10

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 10.1

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

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Step 10.1.1

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

$x = - 6$

Step 10.1.2

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

${(- 6)}^{3} < 9 (- 6)$

Step 10.1.3

The left side $- 216$ is less than the right side $- 54$ , which means that the given statement is always true.

True

Step 10.2

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

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Step 10.2.1

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

$x = - 2$

Step 10.2.2

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

${(- 2)}^{3} < 9 (- 2)$

Step 10.2.3

The left side $- 8$ is not less than the right side $- 18$ , which means that the given statement is false.

False

Step 10.3

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

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Step 10.3.1

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

$x = 2$

Step 10.3.2

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

${(2)}^{3} < 9 (2)$

Step 10.3.3

The left side $8$ is less than the right side $18$ , which means that the given statement is always true.

True

Step 10.4

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

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Step 10.4.1

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

$x = 6$

Step 10.4.2

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

${(6)}^{3} < 9 (6)$

Step 10.4.3

The left side $216$ is not less than the right side $54$ , which means that the given statement is false.

False

Step 10.5

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

$x < - 3$ True

$- 3 < x < 0$ False

$0 < x < 3$ True

$x > 3$ False

$x < - 3$ True

$- 3 < x < 0$ False

$0 < x < 3$ True

$x > 3$ False

Step 11

The solution consists of all of the true intervals.

$x < - 3$ or $0 < x < 3$

Step 12

Convert the inequality to interval notation.

$(- \infty, - 3) \cup (0, 3)$

Step 13