Algebra Examples

| x | > 13

$| x | > 13$

Step 1

Write

| x | > 13

$| x | > 13$ as a piecewise.

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Step 1.1

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

x \geq 0

$x \geq 0$

Step 1.2

In the piece where

x

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

x > 13

$x > 13$

Step 1.3

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

x < 0

$x < 0$

Step 1.4

In the piece where

x

$x$ is negative, remove the absolute value and multiply by

- 1

$- 1$ .

- x > 13

$- x > 13$

Step 1.5

Write as a piecewise.

{\begin{cases} x > 13 & x \geq 0 \\ - x > 13 & x < 0 \end{cases}

${\begin{cases} x > 13 & x \geq 0 \\ - x > 13 & x < 0 \end{cases}$

{\begin{cases} x > 13 & x \geq 0 \\ - x > 13 & x < 0 \end{cases}

${\begin{cases} x > 13 & x \geq 0 \\ - x > 13 & x < 0 \end{cases}$

Step 2

Find the intersection of

x > 13

$x > 13$ and

x \geq 0

$x \geq 0$ .

x > 13

$x > 13$

Step 3

Divide each term in

- x > 13

$- x > 13$ by

- 1

$- 1$ and simplify.

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Step 3.1

Divide each term in

- x > 13

$- x > 13$ by

- 1

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

\frac{- x}{- 1} < \frac{13}{- 1}

$\frac{- x}{- 1} < \frac{13}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

\frac{x}{1} < \frac{13}{- 1}

$\frac{x}{1} < \frac{13}{- 1}$

Step 3.2.2

Divide

x

$x$ by

1

$1$ .

x < \frac{13}{- 1}

$x < \frac{13}{- 1}$

x < \frac{13}{- 1}

$x < \frac{13}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide

13

$13$ by

- 1

$- 1$ .

x < - 13

$x < - 13$

x < - 13

$x < - 13$

x < - 13

$x < - 13$

Step 4

Find the union of the solutions.

x < - 13

$x < - 13$ or

x > 13

$x > 13$

Step 5

Convert the inequality to interval notation.

(- \infty, - 13) \cup (13, \infty)

$(- \infty, - 13) \cup (13, \infty)$

Step 6