Solve the Inequality for x x^4>16x^2

$x^{4} > 16 x^{2}$

Subtract $16 x^{2}$ from both sides of the inequality.

$x^{4} - 16 x^{2} > 0$

Convert the inequality to an equation.

$x^{4} - 16 x^{2} = 0$

Factor the left side of the equation.

Factor $x^{2}$ out of $x^{4} - 16 x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} - 16 x^{2} = 0$

Factor $x^{2}$ out of $- 16 x^{2}$ .

$x^{2} x^{2} + x^{2} \cdot - 16 = 0$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 16$ .

$x^{2} (x^{2} - 16) = 0$

Rewrite $16$ as $4^{2}$ .

$x^{2} (x^{2} - 4^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$x^{2} ((x + 4) (x - 4)) = 0$

Remove unnecessary parentheses.

$x^{2} (x + 4) (x - 4) = 0$

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

$x^{2} = 0$

$x + 4 = 0$

$x - 4 = 0$

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

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.

$x = - 4$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

$x = 0, - 4, 4$

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 0$

$0 < x < 4$

$x > 4$

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.

${(- 6)}^{4} > 16 {(- 6)}^{2}$

The left side $1296$ is greater than the right side $576$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

${(- 2)}^{4} > 16 {(- 2)}^{2}$

The left side $16$ is not greater than the right side $64$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

${(2)}^{4} > 16 {(2)}^{2}$

The left side $16$ is not greater than the right side $64$ , which means that the given statement is false.

False

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.

${(6)}^{4} > 16 {(6)}^{2}$

The left side $1296$ is greater than the right side $576$ , which means that the given statement is always true.

True

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

$x < - 4$ True

$- 4 < x < 0$ False

$0 < x < 4$ False

$x > 4$ True

$x < - 4$ True

$- 4 < x < 0$ False

$0 < x < 4$ False

$x > 4$ True

The solution consists of all of the true intervals.

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

The result can be shown in multiple forms.

Inequality Form:

$x < - 4 or x > 4$

Interval Notation:

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

Solve the Inequality for x x^4>16x^2

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