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

$\frac{(x + 4) (x - 3)}{(x - 2) (x^{2} + 2)} \geq 0$

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x + 4 = 0$

$x - 3 = 0$

$x - 2 = 0$

$x^{2} + 2 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Add $3$ to both sides of the equation.

$x = 3$

Add $2$ to both sides of the equation.

$x = 2$

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

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

$x = \pm \sqrt{- 2}$

Simplify $\pm \sqrt{- 2}$ .

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

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

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

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.

$x = i \sqrt{2}$

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

$x = - i \sqrt{2}$

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

$x = i \sqrt{2}, - i \sqrt{2}$

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = - 4$

$x = 3$

$x = 2$

$x = i \sqrt{2}, - i \sqrt{2}$

Consolidate the solutions.

$x = - 4, 3, 2$

Find the domain of $\frac{(x + 4) (x - 3)}{(x - 2) (x^{2} + 2)}$ .

Set the denominator in $\frac{(x + 4) (x - 3)}{(x - 2) (x^{2} + 2)}$ equal to $0$ to find where the expression is undefined.

$(x - 2) (x^{2} + 2) = 0$

Solve for $x$ .

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

$x - 2 = 0$

$x^{2} + 2 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

$x^{2} + 2 = 0$

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

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

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

$x = \pm \sqrt{- 2}$

Simplify $\pm \sqrt{- 2}$ .

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

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

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

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.

$x = i \sqrt{2}$

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

$x = - i \sqrt{2}$

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

$x = i \sqrt{2}, - i \sqrt{2}$

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

$x = 2, i \sqrt{2}, - i \sqrt{2}$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 2) \cup (2, \infty)$

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 2$

$2 < x < 3$

$x > 3$

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.

$\frac{((- 6) + 4) ((- 6) - 3)}{((- 6) - 2) ({(- 6)}^{2} + 2)} \geq 0$

The left side $- 0.05921052$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{((0) + 4) ((0) - 3)}{((0) - 2) ({(0)}^{2} + 2)} \geq 0$

The left side $3$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 3$ and see if this value makes the original inequality true.

$x = 2.5$

Replace $x$ with $2.5$ in the original inequality.

$\frac{((2.5) + 4) ((2.5) - 3)}{((2.5) - 2) ({(2.5)}^{2} + 2)} \geq 0$

The left side $- 0.\overline{78}$ is less 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.

$\frac{((6) + 4) ((6) - 3)}{((6) - 2) ({(6)}^{2} + 2)} \geq 0$

The left side $0.19736842$ 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.

$x < - 4$ False

$- 4 < x < 2$ True

$2 < x < 3$ False

$x > 3$ True

$x < - 4$ False

$- 4 < x < 2$ True

$2 < x < 3$ False

$x > 3$ True

The solution consists of all of the true intervals.

$- 4 \leq x < 2$ or $x \geq 3$

The result can be shown in multiple forms.

Inequality Form:

$- 4 \leq x < 2 or x \geq 3$

Interval Notation:

$[- 4, 2) \cup [3, \infty)$

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

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