Precalculus Examples

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

Step 1

Subtract $4 x^{2}$ from both sides of the inequality.

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

Step 2

Convert the inequality to an equation.

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

Step 3

Factor the left side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

$x^{2} x^{2} + x^{2} \cdot - 4 = 0$

Step 3.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 4$ .

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

Step 3.2

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

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

Step 3.3

Factor.

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Step 3.3.1

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 $b = 2$ .

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

Step 3.3.2

Remove unnecessary parentheses.

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

Step 4

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

$x^{2} = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 5

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

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Step 5.1

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

$x^{2} = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \pm \sqrt{0}$

Step 5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.2.2.1

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

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

Step 5.2.2.2

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

$x = \pm 0$

Step 5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

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

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Step 6.1

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

$x + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7

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

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Step 7.1

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

$x - 2 = 0$

Step 7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 8

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

$x = 0, - 2, 2$

Step 9

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 2$

$x > 2$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 10.1

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

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Step 10.1.1

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

$x = - 4$

Step 10.1.2

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

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

Step 10.1.3

The left side $256$ is greater than the right side $64$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 10.2.2

Replace $x$ with $- 1$ in the original inequality.

${(- 1)}^{4} > 4 {(- 1)}^{2}$

Step 10.2.3

The left side $1$ is not greater than the right side $4$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 10.3.2

Replace $x$ with $1$ in the original inequality.

${(1)}^{4} > 4 {(1)}^{2}$

Step 10.3.3

The left side $1$ is not greater than the right side $4$ , which means that the given statement is false.

False

Step 10.4

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

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Step 10.4.1

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

$x = 4$

Step 10.4.2

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

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

Step 10.4.3

The left side $256$ is greater than the right side $64$ , which means that the given statement is always true.

True

Step 10.5

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

$x < - 2$ True

$- 2 < x < 0$ False

$0 < x < 2$ False

$x > 2$ True

$x < - 2$ True

$- 2 < x < 0$ False

$0 < x < 2$ False

$x > 2$ True

Step 11

The solution consists of all of the true intervals.

$x < - 2$ or $x > 2$

Step 12

Convert the inequality to interval notation.

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

Step 13