Solve the Inequality for x (x-2)^2(x+9)>0

${(x - 2)}^{2} (x + 9) > 0$

Step 1

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

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

$x + 9 = 0$

Step 2

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Solve ${(x - 2)}^{2} = 0$ for $x$ .

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 3

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

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

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 4

The final solution is all the values that make ${(x - 2)}^{2} (x + 9) > 0$ true.

$x = 2, - 9$

Step 5

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 2$

$x > 2$

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 $x < - 9$ 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.

${((- 12) - 2)}^{2} ((- 12) + 9) > 0$

The left side $- 588$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 9 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

${((0) - 2)}^{2} ((0) + 9) > 0$

The left side $36$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 4$

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

${((4) - 2)}^{2} ((4) + 9) > 0$

The left side $52$ 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.

$x < - 9$ False

$- 9 < x < 2$ True

$x > 2$ True

$x < - 9$ False

$- 9 < x < 2$ True

$x > 2$ True

Step 7

The solution consists of all of the true intervals.

$- 9 < x < 2$ or $x > 2$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < 2 or x > 2$

Interval Notation:

$(- 9, 2) \cup (2, \infty)$

Step 9

Solve the Inequality for x (x-2)^2(x+9)>0

Related Questions

Support LCMGCF.com by disabling your adblocker.