Trigonometry Examples

$x^{2} - x + 5 < 11$

Step 1

Convert the inequality to an equation.

$x^{2} - x + 5 = 11$

Step 2

Subtract $11$ from both sides of the equation.

$x^{2} - x + 5 - 11 = 0$

Step 3

Subtract $11$ from $5$ .

$x^{2} - x - 6 = 0$

Step 4

Factor $x^{2} - x - 6$ using the AC method.

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Step 4.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 4.2

Write the factored form using these integers.

$(x - 3) (x + 2) = 0$

Step 5

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

$x - 3 = 0$

$x + 2 = 0$

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

Add $3$ to both sides of the equation.

$x = 3$

Step 7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 7.1

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

$x + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8

The final solution is all the values that make $(x - 3) (x + 2) = 0$ true.

$x = 3, - 2$

Step 9

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 3$

$x > 3$

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 $x < - 2$ to see if it makes the inequality true.

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Step 10.1.1

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

$x = - 4$

Step 10.1.2

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

${(- 4)}^{2} - (- 4) + 5 < 11$

Step 10.1.3

The left side $25$ is not less than the right side $11$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $- 2 < x < 3$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 2 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

Step 10.2.2

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

${(0)}^{2} - (0) + 5 < 11$

Step 10.2.3

The left side $5$ is less than the right side $11$ , which means that the given statement is always true.

True

Step 10.3

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

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Step 10.3.1

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

$x = 6$

Step 10.3.2

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

${(6)}^{2} - (6) + 5 < 11$

Step 10.3.3

The left side $35$ is not less than the right side $11$ , which means that the given statement is false.

False

Step 10.4

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

$x < - 2$ False

$- 2 < x < 3$ True

$x > 3$ False

$x < - 2$ False

$- 2 < x < 3$ True

$x > 3$ False

Step 11

The solution consists of all of the true intervals.

$- 2 < x < 3$

Step 12

Convert the inequality to interval notation.

$(- 2, 3)$

Step 13