Trigonometry Examples

$x^{2} + 8 x > - 12$

Step 1

Convert the inequality to an equation.

$x^{2} + 8 x = - 12$

Step 2

Add $12$ to both sides of the equation.

$x^{2} + 8 x + 12 = 0$

Step 3

Factor $x^{2} + 8 x + 12$ using the AC method.

Tap for more steps...

Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $8$ .

$2, 6$

Step 3.2

Write the factored form using these integers.

$(x + 2) (x + 6) = 0$

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 $x$ .

Tap for more steps...

Step 5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

Tap for more steps...

Step 6.1

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

$x + 6 = 0$

Step 6.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 7

The final solution is all the values that make $(x + 2) (x + 6) = 0$ true.

$x = - 2, - 6$

Step 8

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - 2$

$x > - 2$

Step 9

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

Tap for more steps...

Step 9.1

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

Tap for more steps...

Step 9.1.1

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

$x = - 8$

Step 9.1.2

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

${(- 8)}^{2} + 8 (- 8) > - 12$

Step 9.1.3

The left side $0$ is greater than the right side $- 12$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $- 6 < x < - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 9.2.1

Choose a value on the interval $- 6 < x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 9.2.2

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

${(- 4)}^{2} + 8 (- 4) > - 12$

Step 9.2.3

The left side $- 16$ is not greater than the right side $- 12$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 9.3.1

Choose a value on the interval $x > - 2$ and see if this value makes the original inequality true.

$x = 0$

Step 9.3.2

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

${(0)}^{2} + 8 (0) > - 12$

Step 9.3.3

The left side $0$ is greater than the right side $- 12$ , which means that the given statement is always true.

True

Step 9.4

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

$x < - 6$ True

$- 6 < x < - 2$ False

$x > - 2$ True

$x < - 6$ True

$- 6 < x < - 2$ False

$x > - 2$ True

Step 10

The solution consists of all of the true intervals.

$x < - 6$ or $x > - 2$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - 6 or x > - 2$

Interval Notation:

$(- \infty, - 6) \cup (- 2, \infty)$

Step 12