Trigonometry Examples

$n^{2} + 2 n < 24$

Step 1

Convert the inequality to an equation.

$n^{2} + 2 n = 24$

Step 2

Subtract $24$ from both sides of the equation.

$n^{2} + 2 n - 24 = 0$

Step 3

Factor $n^{2} + 2 n - 24$ using the AC method.

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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 $- 24$ and whose sum is $2$ .

$- 4, 6$

Step 3.2

Write the factored form using these integers.

$(n - 4) (n + 6) = 0$

Step 4

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

$n - 4 = 0$

$n + 6 = 0$

Step 5

Set $n - 4$ equal to $0$ and solve for $n$ .

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Step 5.1

Set $n - 4$ equal to $0$ .

$n - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$n = 4$

Step 6

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

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Step 6.1

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

$n + 6 = 0$

Step 6.2

Subtract $6$ from both sides of the equation.

$n = - 6$

Step 7

The final solution is all the values that make $(n - 4) (n + 6) = 0$ true.

$n = 4, - 6$

Step 8

Use each root to create test intervals.

$n < - 6$

$- 6 < n < 4$

$n > 4$

Step 9

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 9.1

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

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Step 9.1.1

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

$n = - 8$

Step 9.1.2

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

${(- 8)}^{2} + 2 (- 8) < 24$

Step 9.1.3

The left side $48$ is not less than the right side $24$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- 6 < n < 4$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 6 < n < 4$ and see if this value makes the original inequality true.

$n = 0$

Step 9.2.2

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

${(0)}^{2} + 2 (0) < 24$

Step 9.2.3

The left side $0$ is less than the right side $24$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $n > 4$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $n > 4$ and see if this value makes the original inequality true.

$n = 6$

Step 9.3.2

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

${(6)}^{2} + 2 (6) < 24$

Step 9.3.3

The left side $48$ is not less than the right side $24$ , which means that the given statement is false.

False

Step 9.4

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

$n < - 6$ False

$- 6 < n < 4$ True

$n > 4$ False

$n < - 6$ False

$- 6 < n < 4$ True

$n > 4$ False

Step 10

The solution consists of all of the true intervals.

$- 6 < n < 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- 6 < n < 4$

Interval Notation:

$(- 6, 4)$

Step 12