Solve the Inequality for x x^4>81x^2

$x^{4} > 81 x^{2}$

Step 1

Subtract $81 x^{2}$ from both sides of the inequality.

$x^{4} - 81 x^{2} > 0$

Step 2

Convert the inequality to an equation.

$x^{4} - 81 x^{2} = 0$

Step 3

Factor the left side of the equation.

Factor $x^{2}$ out of $x^{4} - 81 x^{2}$ .

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

$x^{2} x^{2} - 81 x^{2} = 0$

Factor $x^{2}$ out of $- 81 x^{2}$ .

$x^{2} x^{2} + x^{2} \cdot - 81 = 0$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 81$ .

$x^{2} (x^{2} - 81) = 0$

Rewrite $81$ as $9^{2}$ .

$x^{2} (x^{2} - 9^{2}) = 0$

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 $b = 9$ .

$x^{2} ((x + 9) (x - 9)) = 0$

Remove unnecessary parentheses.

$x^{2} (x + 9) (x - 9) = 0$

Step 4

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

$x^{2} = 0$

$x + 9 = 0$

$x - 9 = 0$

Step 5

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 $0$ .

$x = 0$

Step 6

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 7

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Step 8

The final solution is all the values that make $x^{2} (x + 9) (x - 9) = 0$ true.

$x = 0, - 9, 9$

Step 9

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 0$

$0 < x < 9$

$x > 9$

Step 10

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 $x < - 9$ to see if it makes the inequality true.

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Replace $x$ with $- 12$ in the original inequality.

${(- 12)}^{4} > 81 {(- 12)}^{2}$

The left side $20736$ is greater than the right side $11664$ , which means that the given statement is always true.

True

Test a value on the interval $- 9 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < 0$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

${(- 4)}^{4} > 81 {(- 4)}^{2}$

The left side $256$ is not greater than the right side $1296$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 9$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

${(4)}^{4} > 81 {(4)}^{2}$

The left side $256$ is not greater than the right side $1296$ , which means that the given statement is false.

False

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

${(12)}^{4} > 81 {(12)}^{2}$

The left side $20736$ is greater than the right side $11664$ , which means that the given statement is always true.

True

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

$x < - 9$ True

$- 9 < x < 0$ False

$0 < x < 9$ False

$x > 9$ True

$x < - 9$ True

$- 9 < x < 0$ False

$0 < x < 9$ False

$x > 9$ True

Step 11

The solution consists of all of the true intervals.

$x < - 9$ or $x > 9$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < - 9 or x > 9$

Interval Notation:

$(- \infty, - 9) \cup (9, \infty)$

Step 13

Solve the Inequality for x x^4>81x^2

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