Precalculus Examples

$x^{2} > 36$

Step 1

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

$\sqrt{x^{2}} > \sqrt{36}$

Step 2

Simplify the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Pull terms out from under the radical.

$| x | > \sqrt{36}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\sqrt{36}$ .

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Step 2.2.1.1

Rewrite $36$ as $6^{2}$ .

$| x | > \sqrt{6^{2}}$

Step 2.2.1.2

Pull terms out from under the radical.

$| x | > | 6 |$

Step 2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$| x | > 6$

Step 3

Write $| x | > 6$ as a piecewise.

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Step 3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 3.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 6$

Step 3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 3.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 6$

Step 3.5

Write as a piecewise.

${\begin{cases} x > 6 & x \geq 0 \\ - x > 6 & x < 0 \end{cases}$

Step 4

Find the intersection of $x > 6$ and $x \geq 0$ .

$x > 6$

Step 5

Divide each term in $- x > 6$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x > 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{6}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{6}{- 1}$

Step 5.2.2

Divide $x$ by $1$ .

$x < \frac{6}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $6$ by $- 1$ .

$x < - 6$

Step 6

Find the union of the solutions.

$x < - 6$ or $x > 6$

Step 7

Convert the inequality to interval notation.

$(- \infty, - 6) \cup (6, \infty)$

Step 8