Precalculus Examples

$2 x^{2} - x^{4} \leq 0$

Step 1

Convert the inequality to an equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

$x^{2} = 0$

$2 - x^{2} = 0$

Step 4

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

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Step 4.1

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

$x^{2} = 0$

Step 4.2

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

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Step 4.2.1

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

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

$x = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

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

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Step 5.1

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

$2 - x^{2} = 0$

Step 5.2

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

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Step 5.2.1

Subtract $2$ from both sides of the equation.

$- x^{2} = - 2$

Step 5.2.2

Divide each term in $- x^{2} = - 2$ by $- 1$ and simplify.

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Step 5.2.2.1

Divide each term in $- x^{2} = - 2$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 2}{- 1}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 2}{- 1}$

Step 5.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 2}{- 1}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide $- 2$ by $- 1$ .

$x^{2} = 2$

Step 5.2.3

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

$x = \pm \sqrt{2}$

Step 5.2.4

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

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Step 5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 5.2.4.2

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

$x = - \sqrt{2}$

Step 5.2.4.3

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

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

Step 6

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

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 7

Use each root to create test intervals.

$x < - \sqrt{2}$

$- \sqrt{2} < x < 0$

$0 < x < \sqrt{2}$

$x > \sqrt{2}$

Step 8

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 8.1

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

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Step 8.1.1

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

$x = - 4$

Step 8.1.2

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

$2 {(- 4)}^{2} - {(- 4)}^{4} \leq 0$

Step 8.1.3

The left side $- 224$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.2

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

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Step 8.2.1

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

$x = - 1$

Step 8.2.2

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

$2 {(- 1)}^{2} - {(- 1)}^{4} \leq 0$

Step 8.2.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 8.3

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

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Step 8.3.1

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

$x = 1$

Step 8.3.2

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

$2 {(1)}^{2} - {(1)}^{4} \leq 0$

Step 8.3.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

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Step 8.4.1

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

$x = 4$

Step 8.4.2

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

$2 {(4)}^{2} - {(4)}^{4} \leq 0$

Step 8.4.3

The left side $- 224$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.5

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

$x < - \sqrt{2}$ True

$- \sqrt{2} < x < 0$ False

$0 < x < \sqrt{2}$ False

$x > \sqrt{2}$ True

$x < - \sqrt{2}$ True

$- \sqrt{2} < x < 0$ False

$0 < x < \sqrt{2}$ False

$x > \sqrt{2}$ True

Step 9

The solution consists of all of the true intervals.

$x \leq - \sqrt{2}$ or $x \geq \sqrt{2}$ or $x = 0$

Step 10

Combine the intervals.

$x \leq - \sqrt{2} or x = 0 or x \geq \sqrt{2}$

Step 11

Convert the inequality to interval notation.

$(- \infty, - \sqrt{2}] \cup [0, 0] \cup [\sqrt{2}, \infty)$

Step 12