Solve the Inequality for x x(x+2)(x-2)<=0

$x (x + 2) (x - 2) \leq 0$

Step 1

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

$x = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

Set $x - 2$ equal to $0$ and solve for $x$ .

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 5

The final solution is all the values that make $x (x + 2) (x - 2) \leq 0$ true.

$x = 0, - 2, 2$

Step 6

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 2$

$x > 2$

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.

$(- 4) ((- 4) + 2) ((- 4) - 2) \leq 0$

The left side $- 48$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$(- 1) ((- 1) + 2) ((- 1) - 2) \leq 0$

The left side $3$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 1$

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

$(1) ((1) + 2) ((1) - 2) \leq 0$

The left side $- 3$ is less 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.

$(4) ((4) + 2) ((4) - 2) \leq 0$

The left side $48$ is greater than the right side $0$ , which means that the given statement is false.

False

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

$x < - 2$ True

$- 2 < x < 0$ False

$0 < x < 2$ True

$x > 2$ False

$x < - 2$ True

$- 2 < x < 0$ False

$0 < x < 2$ True

$x > 2$ False

Step 8

The solution consists of all of the true intervals.

$x \leq - 2$ or $0 \leq x \leq 2$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 2 or 0 \leq x \leq 2$

Interval Notation:

$(- \infty, - 2] \cup [0, 2]$

Step 10

Solve the Inequality for x x(x+2)(x-2)<=0

Related Questions

Support LCMGCF.com by disabling your adblocker.