Solve the Inequality for x (x-5)(x-9)>0

$(x - 5) (x - 9) > 0$

Step 1

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

$x - 5 = 0$

$x - 9 = 0$

Step 2

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 3

Set $x - 9$ equal to $0$ and solve for $x$ .

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

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Step 4

The final solution is all the values that make $(x - 5) (x - 9) > 0$ true.

$x = 5, 9$

Step 5

Use each root to create test intervals.

$x < 5$

$5 < x < 9$

$x > 9$

Step 6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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

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

$x = 0$

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

$((0) - 5) ((0) - 9) > 0$

The left side $45$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $5 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $5 < x < 9$ and see if this value makes the original inequality true.

$x = 7$

Replace $x$ with $7$ in the original inequality.

$((7) - 5) ((7) - 9) > 0$

The left side $- 4$ is not greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 12$

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

$((12) - 5) ((12) - 9) > 0$

The left side $21$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < 5$ True

$5 < x < 9$ False

$x > 9$ True

$x < 5$ True

$5 < x < 9$ False

$x > 9$ True

Step 7

The solution consists of all of the true intervals.

$x < 5$ or $x > 9$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < 5 or x > 9$

Interval Notation:

$(- \infty, 5) \cup (9, \infty)$

Step 9

Solve the Inequality for x (x-5)(x-9)>0

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