Solve the Inequality for x (x+1)(x-5)(x^2+9)<=0

$(x + 1) (x - 5) (x^{2} + 9) \leq 0$

Step 1

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

$x + 1 = 0$

$x - 5 = 0$

$x^{2} + 9 = 0$

Step 2

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 3

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 4

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Solve $x^{2} + 9 = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

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

$x = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$x = \pm i \cdot \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

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

$x = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$x = \pm 3 i$

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.

$x = 3 i$

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

$x = - 3 i$

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

$x = 3 i, - 3 i$

Step 5

The final solution is all the values that make $(x + 1) (x - 5) (x^{2} + 9) \leq 0$ true.

$x = - 1, 5, 3 i, - 3 i$

Step 6

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 5$

$x > 5$

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 $x < - 1$ to see if it makes the inequality true.

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

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$((- 4) + 1) ((- 4) - 5) ({(- 4)}^{2} + 9) \leq 0$

The left side $675$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) + 1) ((0) - 5) ({(0)}^{2} + 9) \leq 0$

The left side $- 45$ is less 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.

$((8) + 1) ((8) - 5) ({(8)}^{2} + 9) \leq 0$

The left side $1971$ 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.

$x < - 1$ False

$- 1 < x < 5$ True

$x > 5$ False

$x < - 1$ False

$- 1 < x < 5$ True

$x > 5$ False

Step 8

The solution consists of all of the true intervals.

$- 1 \leq x \leq 5$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x \leq 5$

Interval Notation:

$[- 1, 5]$

Step 10

Solve the Inequality for x (x+1)(x-5)(x^2+9)<=0

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