Solve the Inequality for x (x-3)(x^2+1)(x+4)>0

$(x - 3) (x^{2} + 1) (x + 4) > 0$

Step 1

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

$x - 3 = 0$

$x^{2} + 1 = 0$

$x + 4 = 0$

Step 2

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 3

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Solve $x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

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

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

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

$x = \pm i$

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.

$x = i$

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

$x = - i$

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

$x = i, - i$

Step 4

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

The final solution is all the values that make $(x - 3) (x^{2} + 1) (x + 4) > 0$ true.

$x = 3, i, - i, - 4$

Step 6

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 3$

$x > 3$

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.

$((- 6) - 3) ({(- 6)}^{2} + 1) ((- 6) + 4) > 0$

The left side $666$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) - 3) ({(0)}^{2} + 1) ((0) + 4) > 0$

The left side $- 12$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

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

$((6) - 3) ({(6)}^{2} + 1) ((6) + 4) > 0$

The left side $1110$ is greater than the right side $0$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 4$ True

$- 4 < x < 3$ False

$x > 3$ True

$x < - 4$ True

$- 4 < x < 3$ False

$x > 3$ True

Step 8

The solution consists of all of the true intervals.

$x < - 4$ or $x > 3$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x < - 4 or x > 3$

Interval Notation:

$(- \infty, - 4) \cup (3, \infty)$

Step 10

Solve the Inequality for x (x-3)(x^2+1)(x+4)>0

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