Pre-Algebra Examples

$| x | < 3$

Step 1

Write

$| x | < 3$ 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.

$x \geq 0$

Step 1.2

In the piece where

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

$x < 3$

Step 1.3

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

$x < 0$

Step 1.4

In the piece where

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

$- 1$ .

$- x < 3$

Step 1.5

Write as a piecewise.

${\begin{cases} x < 3 & x \geq 0 \\ - x < 3 & x < 0 \end{cases}$

Step 2

Find the intersection of

$x < 3$ and

$x \geq 0$ .

$0 \leq x < 3$

Step 3

Solve

$- x < 3$ when

$x < 0$ .

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Step 3.1

Divide each term in

$- x < 3$ by

$- 1$ and simplify.

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Step 3.1.1

Divide each term in

$- x < 3$ by

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

$\frac{- x}{- 1} > \frac{3}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{3}{- 1}$

Step 3.1.2.2

Divide

$x$ by

$1$ .

$x > \frac{3}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide

$3$ by

$- 1$ .

$x > - 3$

Step 3.2

Find the intersection of

$x > - 3$ and

$x < 0$ .

$- 3 < x < 0$

Step 4

Find the union of the solutions.

$- 3 < x < 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 3$

Interval Notation:

$(- 3, 3)$

Step 6