Solve the Inequality for x (2-x)(x+1)(x^2-16)>0

$(2 - x) (x + 1) (x^{2} - 16) > 0$

Step 1

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

$2 - x = 0$

$x + 1 = 0$

$x^{2} - 16 = 0$

Step 2

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

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

$2 - x = 0$

Solve $2 - x = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

Step 3

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $x^{2} - 16$ equal to $0$ and solve for $x$ .

Set $x^{2} - 16$ equal to $0$ .

$x^{2} - 16 = 0$

Solve $x^{2} - 16 = 0$ for $x$ .

Add $16$ to both sides of the equation.

$x^{2} = 16$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{16}$

Simplify $\pm \sqrt{16}$ .

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

Step 5

The final solution is all the values that make $(2 - x) (x + 1) (x^{2} - 16) > 0$ true.

$x = 2, - 1, 4, - 4$

Step 6

Use each root to create test intervals.

$x < - 4$

$- 4 < x < - 1$

$- 1 < x < 2$

$2 < x < 4$

$x > 4$

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 $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$(2 - (- 6)) ((- 6) + 1) ({(- 6)}^{2} - 16) > 0$

The left side $- 800$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < - 1$ and see if this value makes the original inequality true.

$x = - 2$

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

$(2 - (- 2)) ((- 2) + 1) ({(- 2)}^{2} - 16) > 0$

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

True

Test a value on the interval $- 1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$(2 - (0)) ((0) + 1) ({(0)}^{2} - 16) > 0$

The left side $- 32$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $2 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 4$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$(2 - (3)) ((3) + 1) ({(3)}^{2} - 16) > 0$

The left side $28$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$(2 - (6)) ((6) + 1) ({(6)}^{2} - 16) > 0$

The left side $- 560$ is not 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.

$x < - 4$ False

$- 4 < x < - 1$ True

$- 1 < x < 2$ False

$2 < x < 4$ True

$x > 4$ False

$x < - 4$ False

$- 4 < x < - 1$ True

$- 1 < x < 2$ False

$2 < x < 4$ True

$x > 4$ False

Step 8

The solution consists of all of the true intervals.

$- 4 < x < - 1$ or $2 < x < 4$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < - 1 or 2 < x < 4$

Interval Notation:

$(- 4, - 1) \cup (2, 4)$

Step 10

Solve the Inequality for x (2-x)(x+1)(x^2-16)>0

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