Solve the Inequality for x (x+5)(x^2+4)(x-6)<=0

$(x + 5) (x^{2} + 4) (x - 6) \leq 0$

Step 1

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

$x + 5 = 0$

$x^{2} + 4 = 0$

$x - 6 = 0$

Step 2

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Solve $x^{2} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

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

$x = \pm i \cdot \sqrt{2^{2}}$

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

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

Step 4

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.

$x = 6$

Step 5

The final solution is all the values that make $(x + 5) (x^{2} + 4) (x - 6) \leq 0$ true.

$x = - 5, 2 i, - 2 i, 6$

Step 6

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 6$

$x > 6$

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 $x < - 5$ 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.

$((- 8) + 5) ({(- 8)}^{2} + 4) ((- 8) - 6) \leq 0$

The left side $2856$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) + 5) ({(0)}^{2} + 4) ((0) - 6) \leq 0$

The left side $- 120$ 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.

$((8) + 5) ({(8)}^{2} + 4) ((8) - 6) \leq 0$

The left side $1768$ 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.

$x < - 5$ False

$- 5 < x < 6$ True

$x > 6$ False

$x < - 5$ False

$- 5 < x < 6$ True

$x > 6$ False

Step 8

The solution consists of all of the true intervals.

$- 5 \leq x \leq 6$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq 6$

Interval Notation:

$[- 5, 6]$

Step 10

Solve the Inequality for x (x+5)(x^2+4)(x-6)<=0

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