Algebra Examples

$x^{2} \leq 36$

Step 1

Solve $- 6 \leq x \leq 6$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify the equation.

Tap for more steps...

Step 1.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{36}$

Step 1.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.2.1

Simplify $\sqrt{36}$ .

Tap for more steps...

Step 1.2.2.1.1

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

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

Step 1.2.2.1.2

Pull terms out from under the radical.

$| x | \leq | 6 |$

Step 1.2.2.1.3

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

$| x | \leq 6$

Step 1.3

Write $| x | \leq 6$ as a piecewise.

Tap for more steps...

Step 1.3.1

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

$x \geq 0$

Step 1.3.2

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

$x \leq 6$

Step 1.3.3

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

$x < 0$

Step 1.3.4

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

$- x \leq 6$

Step 1.3.5

Write as a piecewise.

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

Step 1.4

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

$0 \leq x \leq 6$

Step 1.5

Solve $- x \leq 6$ when $x < 0$ .

Tap for more steps...

Step 1.5.1

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

Tap for more steps...

Step 1.5.1.1

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

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

Step 1.5.1.2

Simplify the left side.

Tap for more steps...

Step 1.5.1.2.1

Dividing two negative values results in a positive value.

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

Step 1.5.1.2.2

Divide $x$ by $1$ .

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

Step 1.5.1.3

Simplify the right side.

Tap for more steps...

Step 1.5.1.3.1

Divide $6$ by $- 1$ .

$x \geq - 6$

Step 1.5.2

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

$- 6 \leq x < 0$

Step 1.6

Find the union of the solutions.

$- 6 \leq x \leq 6$

Step 2

Use the inequality $- 6 \leq x \leq 6$ to build the set notation.

${x | - 6 \leq x \leq 6}$

Step 3