Pre-Algebra Examples

| n | + 4 < 12

$| n | + 4 < 12$

Step 1

Write

| n | + 4 < 12

$| n | + 4 < 12$ 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.

n \geq 0

$n \geq 0$

Step 1.2

In the piece where

n

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

n + 4 < 12

$n + 4 < 12$

Step 1.3

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

n < 0

$n < 0$

Step 1.4

In the piece where

n

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

- 1

$- 1$ .

- n + 4 < 12

$- n + 4 < 12$

Step 1.5

Write as a piecewise.

{\begin{cases} n + 4 < 12 & n \geq 0 \\ - n + 4 < 12 & n < 0 \end{cases}

${\begin{cases} n + 4 < 12 & n \geq 0 \\ - n + 4 < 12 & n < 0 \end{cases}$

{\begin{cases} n + 4 < 12 & n \geq 0 \\ - n + 4 < 12 & n < 0 \end{cases}

${\begin{cases} n + 4 < 12 & n \geq 0 \\ - n + 4 < 12 & n < 0 \end{cases}$

Step 2

Solve

n + 4 < 12

$n + 4 < 12$ when

n \geq 0

$n \geq 0$ .

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Step 2.1

Move all terms not containing

n

$n$ to the right side of the inequality.

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Step 2.1.1

Subtract

4

$4$ from both sides of the inequality.

n < 12 - 4

$n < 12 - 4$

Step 2.1.2

Subtract

4

$4$ from

12

$12$ .

n < 8

$n < 8$

n < 8

$n < 8$

Step 2.2

Find the intersection of

n < 8

$n < 8$ and

n \geq 0

$n \geq 0$ .

0 \leq n < 8

$0 \leq n < 8$

0 \leq n < 8

$0 \leq n < 8$

Step 3

Solve

- n + 4 < 12

$- n + 4 < 12$ when

n < 0

$n < 0$ .

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Step 3.1

Solve

- n + 4 < 12

$- n + 4 < 12$ for

n

$n$ .

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Step 3.1.1

Move all terms not containing

n

$n$ to the right side of the inequality.

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Step 3.1.1.1

Subtract

4

$4$ from both sides of the inequality.

- n < 12 - 4

$- n < 12 - 4$

Step 3.1.1.2

Subtract

4

$4$ from

12

$12$ .

- n < 8

$- n < 8$

- n < 8

$- n < 8$

Step 3.1.2

Divide each term in

- n < 8

$- n < 8$ by

- 1

$- 1$ and simplify.

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Step 3.1.2.1

Divide each term in

- n < 8

$- n < 8$ by

- 1

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

\frac{- n}{- 1} > \frac{8}{- 1}

$\frac{- n}{- 1} > \frac{8}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

\frac{n}{1} > \frac{8}{- 1}

$\frac{n}{1} > \frac{8}{- 1}$

Step 3.1.2.2.2

Divide

n

$n$ by

1

$1$ .

n > \frac{8}{- 1}

$n > \frac{8}{- 1}$

n > \frac{8}{- 1}

$n > \frac{8}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide

8

$8$ by

- 1

$- 1$ .

n > - 8

$n > - 8$

n > - 8

$n > - 8$

n > - 8

$n > - 8$

n > - 8

$n > - 8$

Step 3.2

Find the intersection of

n > - 8

$n > - 8$ and

n < 0

$n < 0$ .

- 8 < n < 0

$- 8 < n < 0$

- 8 < n < 0

$- 8 < n < 0$

Step 4

Find the union of the solutions.

- 8 < n < 8

$- 8 < n < 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

- 8 < n < 8

$- 8 < n < 8$

Interval Notation:

(- 8, 8)

$(- 8, 8)$

Step 6