Precalculus Examples

$x^{3} + x^{2} + 36 x + 36 < 0$

Step 1

Convert the inequality to an equation.

$x^{3} + x^{2} + 36 x + 36 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$(x^{3} + x^{2}) + 36 x + 36 = 0$

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x + 1) + 36 (x + 1) = 0$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$(x + 1) (x^{2} + 36) = 0$

Step 3

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

$x + 1 = 0$

$x^{2} + 36 = 0$

Step 4

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

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x^{2} + 36$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 36$ equal to $0$ .

$x^{2} + 36 = 0$

Step 5.2

Solve $x^{2} + 36 = 0$ for $x$ .

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Step 5.2.1

Subtract $36$ from both sides of the equation.

$x^{2} = - 36$

Step 5.2.2

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

$x = \pm \sqrt{- 36}$

Step 5.2.3

Simplify $\pm \sqrt{- 36}$ .

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Step 5.2.3.1

Rewrite $- 36$ as $- 1 (36)$ .

$x = \pm \sqrt{- 1 (36)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{36}$

Step 5.2.3.3

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

$x = \pm i \cdot \sqrt{36}$

Step 5.2.3.4

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2}}$

Step 5.2.3.5

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

$x = \pm i \cdot 6$

Step 5.2.3.6

Move $6$ to the left of $i$ .

$x = \pm 6 i$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = 6 i$

Step 5.2.4.2

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

$x = - 6 i$

Step 5.2.4.3

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

$x = 6 i, - 6 i$

Step 6

The final solution is all the values that make $(x + 1) (x^{2} + 36) = 0$ true.

$x = - 1, 6 i, - 6 i$

Step 7

The solution consists of all of the true intervals.

$x < - 1$

Step 8

Convert the inequality to interval notation.

$(- \infty, - 1)$

Step 9