Solve the Inequality for x (x+2)^2(x+8)(x-6)<0

${(x + 2)}^{2} (x + 8) (x - 6) < 0$

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

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

$x + 8 = 0$

$x - 6 = 0$

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.

$x = - 2$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make ${(x + 2)}^{2} (x + 8) (x - 6) < 0$ true.

$x = - 2, - 8, 6$

Use each root to create test intervals.

$x < - 8$

$- 8 < x < - 2$

$- 2 < x < 6$

$x > 6$

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 $x < - 8$ to see if it makes the inequality true.

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

${((- 10) + 2)}^{2} ((- 10) + 8) ((- 10) - 6) < 0$

The left side $2048$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 8 < x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < - 2$ and see if this value makes the original inequality true.

$x = - 5$

Replace $x$ with $- 5$ in the original inequality.

${((- 5) + 2)}^{2} ((- 5) + 8) ((- 5) - 6) < 0$

The left side $- 297$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 2 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

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

${((0) + 2)}^{2} ((0) + 8) ((0) - 6) < 0$

The left side $- 192$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

${((8) + 2)}^{2} ((8) + 8) ((8) - 6) < 0$

The left side $3200$ is not less 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.

$x < - 8$ False

$- 8 < x < - 2$ True

$- 2 < x < 6$ True

$x > 6$ False

$x < - 8$ False

$- 8 < x < - 2$ True

$- 2 < x < 6$ True

$x > 6$ False

The solution consists of all of the true intervals.

$- 8 < x < - 2$ or $- 2 < x < 6$

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < - 2 or - 2 < x < 6$

Interval Notation:

$(- 8, - 2) \cup (- 2, 6)$

Solve the Inequality for x (x+2)^2(x+8)(x-6)<0

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