Trigonometry Examples

$(x - 4) (x - 5) > 0$

Step 1

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

$x - 4 = 0$

$x - 5 = 0$

Step 2

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

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Step 2.1

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

$x - 4 = 0$

Step 2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4

The final solution is all the values that make $(x - 4) (x - 5) > 0$ true.

$x = 4, 5$

Step 5

Use each root to create test intervals.

$x < 4$

$4 < x < 5$

$x > 5$

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 $x < 4$ to see if it makes the inequality true.

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Step 6.1.1

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

$x = 0$

Step 6.1.2

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

$((0) - 4) ((0) - 5) > 0$

Step 6.1.3

The left side $20$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.2

Test a value on the interval $4 < x < 5$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $4 < x < 5$ and see if this value makes the original inequality true.

$x = 4.5$

Step 6.2.2

Replace $x$ with $4.5$ in the original inequality.

$((4.5) - 4) ((4.5) - 5) > 0$

Step 6.2.3

The left side $- 0.25$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 6.3

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 6.3.2

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

$((8) - 4) ((8) - 5) > 0$

Step 6.3.3

The left side $12$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.4

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

$x < 4$ True

$4 < x < 5$ False

$x > 5$ True

$x < 4$ True

$4 < x < 5$ False

$x > 5$ True

Step 7

The solution consists of all of the true intervals.

$x < 4$ or $x > 5$

Step 8

Convert the inequality to interval notation.

$(- \infty, 4) \cup (5, \infty)$

Step 9