Trigonometry Examples

$x^{2} + 3 x - 10 > 8$

Step 1

Convert the inequality to an equation.

$x^{2} + 3 x - 10 = 8$

Step 2

Subtract $8$ from both sides of the equation.

$x^{2} + 3 x - 10 - 8 = 0$

Step 3

Subtract $8$ from $- 10$ .

$x^{2} + 3 x - 18 = 0$

Step 4

Factor $x^{2} + 3 x - 18$ 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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 4.2

Write the factored form using these integers.

$(x - 3) (x + 6) = 0$

Step 5

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

$x - 3 = 0$

$x + 6 = 0$

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

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Step 7.1

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

$x + 6 = 0$

Step 7.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 8

The final solution is all the values that make $(x - 3) (x + 6) = 0$ true.

$x = 3, - 6$

Step 9

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 3$

$x > 3$

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 $x < - 6$ to see if it makes the inequality true.

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Step 10.1.1

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

$x = - 8$

Step 10.1.2

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

${(- 8)}^{2} + 3 (- 8) - 10 > 8$

Step 10.1.3

The left side $30$ is greater than the right side $8$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 6 < x < 3$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 6 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

Step 10.2.2

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

${(0)}^{2} + 3 (0) - 10 > 8$

Step 10.2.3

The left side $- 10$ is not greater than the right side $8$ , which means that the given statement is false.

False

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.

${(6)}^{2} + 3 (6) - 10 > 8$

Step 10.3.3

The left side $44$ is greater than the right side $8$ , which means that the given statement is always true.

True

Step 10.4

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

$x < - 6$ True

$- 6 < x < 3$ False

$x > 3$ True

$x < - 6$ True

$- 6 < x < 3$ False

$x > 3$ True

Step 11

The solution consists of all of the true intervals.

$x < - 6$ or $x > 3$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < - 6 or x > 3$

Interval Notation:

$(- \infty, - 6) \cup (3, \infty)$

Step 13