Pre-Algebra Examples

$| n | + 4 < 12$

Step 1

Write $| n | + 4 < 12$ 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.

$n \geq 0$

Step 1.2

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

$n + 4 < 12$

Step 1.3

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

$n < 0$

Step 1.4

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

$- n + 4 < 12$

Step 1.5

Write as a piecewise.

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

Step 2

Solve $n + 4 < 12$ when $n \geq 0$ .

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Step 2.1

Move all terms not containing $n$ to the right side of the inequality.

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Step 2.1.1

Subtract $4$ from both sides of the inequality.

$n < 12 - 4$

Step 2.1.2

Subtract $4$ from $12$ .

$n < 8$

Step 2.2

Find the intersection of $n < 8$ and $n \geq 0$ .

$0 \leq n < 8$

Step 3

Solve $- n + 4 < 12$ when $n < 0$ .

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Step 3.1

Solve $- n + 4 < 12$ for $n$ .

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Step 3.1.1

Move all terms not containing $n$ to the right side of the inequality.

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Step 3.1.1.1

Subtract $4$ from both sides of the inequality.

$- n < 12 - 4$

Step 3.1.1.2

Subtract $4$ from $12$ .

$- n < 8$

Step 3.1.2

Divide each term in $- n < 8$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- n < 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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.

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

Step 3.1.2.2.2

Divide $n$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $8$ by $- 1$ .

$n > - 8$

Step 3.2

Find the intersection of $n > - 8$ and $n < 0$ .

$- 8 < n < 0$

Step 4

Find the union of the solutions.

$- 8 < n < 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 8 < n < 8$

Interval Notation:

$(- 8, 8)$

Step 6