Solve the Inequality for x x^3+x^2+49x+49<0

$x^{3} + x^{2} + 49 x + 49 < 0$

Step 1

Convert the inequality to an equation.

$x^{3} + x^{2} + 49 x + 49 = 0$

Step 2

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} + x^{2}) + 49 x + 49 = 0$

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

$x^{2} (x + 1) + 49 (x + 1) = 0$

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

$(x + 1) (x^{2} + 49) = 0$

Step 3

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

$x + 1 = 0$

$x^{2} + 49 = 0$

Step 4

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 5

Set $x^{2} + 49$ equal to $0$ and solve for $x$ .

Set $x^{2} + 49$ equal to $0$ .

$x^{2} + 49 = 0$

Solve $x^{2} + 49 = 0$ for $x$ .

Subtract $49$ from both sides of the equation.

$x^{2} = - 49$

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

$x = \pm \sqrt{- 49}$

Simplify $\pm \sqrt{- 49}$ .

Rewrite $- 49$ as $- 1 (49)$ .

$x = \pm \sqrt{- 1 (49)}$

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{49}$

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

$x = \pm i \cdot \sqrt{49}$

Rewrite $49$ as $7^{2}$ .

$x = \pm i \cdot \sqrt{7^{2}}$

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

$x = \pm i \cdot 7$

Move $7$ to the left of $i$ .

$x = \pm 7 i$

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.

$x = 7 i$

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

$x = - 7 i$

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

$x = 7 i, - 7 i$

Step 6

The final solution is all the values that make $(x + 1) (x^{2} + 49) = 0$ true.

$x = - 1, 7 i, - 7 i$

Step 7

The solution consists of all of the true intervals.

$x < - 1$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - 1$

Interval Notation:

$(- \infty, - 1)$

Step 9

Solve the Inequality for x x^3+x^2+49x+49<0

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