Trigonometry Examples

$x^{2} - 18 x + 81 < 0$

Step 1

Convert the inequality to an equation.

$x^{2} - 18 x + 81 = 0$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$x^{2} - 18 x + 9^{2} = 0$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 x = 2 \cdot x \cdot 9$

Step 2.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 9 + 9^{2} = 0$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

${(x - 9)}^{2} = 0$

Step 3

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 4

Add $9$ to both sides of the equation.

$x = 9$

Step 5

Use each root to create test intervals.

$x < 9$

$x > 9$

Step 6

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 6.1

Test a value on the interval $x < 9$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $x < 9$ and see if this value makes the original inequality true.

$x = 0$

Step 6.1.2

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

${(0)}^{2} - 18 \cdot 0 + 81 < 0$

Step 6.1.3

The left side $81$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.2

Test a value on the interval $x > 9$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Step 6.2.2

Replace $x$ with $12$ in the original inequality.

${(12)}^{2} - 18 \cdot 12 + 81 < 0$

Step 6.2.3

The left side $9$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.3

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

$x < 9$ False

$x > 9$ False

$x < 9$ False

$x > 9$ False

Step 7

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 8