Convert to Interval Notation x(x-5)^2(x+2)^2>0

$x {(x - 5)}^{2} {(x + 2)}^{2} > 0$

Step 1

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

$x = 0$

${(x - 5)}^{2} = 0$

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

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{2}$ equal to $0$ .

${(x - 5)}^{2} = 0$

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 4

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 $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

The final solution is all the values that make $x {(x - 5)}^{2} {(x + 2)}^{2} > 0$ true.

$x = 0, 5, - 2$

Step 6

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 5$

$x > 5$

Step 7

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 $x < - 2$ 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.

$(- 4) {((- 4) - 5)}^{2} {((- 4) + 2)}^{2} > 0$

The left side $- 1296$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Replace $x$ with $- 1$ in the original inequality.

$(- 1) {((- 1) - 5)}^{2} {((- 1) + 2)}^{2} > 0$

The left side $- 36$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 5$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$(2) {((2) - 5)}^{2} {((2) + 2)}^{2} > 0$

The left side $288$ is greater than the right side $0$ , which means that the given statement is always true.

True

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.

$x = 8$

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

$(8) {((8) - 5)}^{2} {((8) + 2)}^{2} > 0$

The left side $7200$ 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.

$x < - 2$ False

$- 2 < x < 0$ False

$0 < x < 5$ True

$x > 5$ True

$x < - 2$ False

$- 2 < x < 0$ False

$0 < x < 5$ True

$x > 5$ True

Step 8

The solution consists of all of the true intervals.

$0 < x < 5$ or $x > 5$

Step 9

Convert the inequality to interval notation.

$(0, 5) \cup (5, \infty)$

Step 10

Convert to Interval Notation x(x-5)^2(x+2)^2>0

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