Calculus Examples

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

Step 1

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

$x + 3 = 0$

$x - 3 = 0$

Step 2

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

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Step 2.1

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

$x + 3 = 0$

Step 2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3

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

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Step 3.1

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

$x - 3 = 0$

Step 3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4

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

$x = - 3, 3$

Step 5

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 3$

$x > 3$

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

$x = - 6$

Step 6.1.2

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

$((- 6) + 3) ((- 6) - 3) > 0$

Step 6.1.3

The left side $27$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.2

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

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Step 6.2.1

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

$x = 0$

Step 6.2.2

Replace $x$ with $0$ in the original inequality.

$((0) + 3) ((0) - 3) > 0$

Step 6.2.3

The left side $- 9$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 6.3

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

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Step 6.3.1

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

$x = 6$

Step 6.3.2

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

$((6) + 3) ((6) - 3) > 0$

Step 6.3.3

The left side $27$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.4

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

$x < - 3$ True

$- 3 < x < 3$ False

$x > 3$ True

$x < - 3$ True

$- 3 < x < 3$ False

$x > 3$ True

Step 7

The solution consists of all of the true intervals.

$x < - 3$ or $x > 3$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - 3 or x > 3$

Interval Notation:

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

Step 9