Trigonometry Examples

$x^{2} - 11 x + 24 > 0$

Step 1

Convert the inequality to an equation.

$x^{2} - 11 x + 24 = 0$

Step 2

Factor $x^{2} - 11 x + 24$ using the AC method.

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Step 2.1

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 $- 11$ .

$- 8, - 3$

Step 2.2

Write the factored form using these integers.

$(x - 8) (x - 3) = 0$

Step 3

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

$x - 8 = 0$

$x - 3 = 0$

Step 4

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.2

Add $8$ to both sides of the equation.

$x = 8$

Step 5

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

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Step 5.1

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

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

The final solution is all the values that make $(x - 8) (x - 3) = 0$ true.

$x = 8, 3$

Step 7

Use each root to create test intervals.

$x < 3$

$3 < x < 8$

$x > 8$

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 $x < 3$ to see if it makes the inequality true.

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Step 8.1.1

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

$x = 0$

Step 8.1.2

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

${(0)}^{2} - 11 \cdot 0 + 24 > 0$

Step 8.1.3

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

True

Step 8.2

Test a value on the interval $3 < x < 8$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $3 < x < 8$ and see if this value makes the original inequality true.

$x = 6$

Step 8.2.2

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

${(6)}^{2} - 11 \cdot 6 + 24 > 0$

Step 8.2.3

The left side $- 6$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval $x > 8$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Step 8.3.2

Replace $x$ with $10$ in the original inequality.

${(10)}^{2} - 11 \cdot 10 + 24 > 0$

Step 8.3.3

The left side $14$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.4

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

$x < 3$ True

$3 < x < 8$ False

$x > 8$ True

$x < 3$ True

$3 < x < 8$ False

$x > 8$ True

Step 9

The solution consists of all of the true intervals.

$x < 3$ or $x > 8$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < 3 or x > 8$

Interval Notation:

$(- \infty, 3) \cup (8, \infty)$

Step 11