Solve for n |-8n|<32

$| - 8 n | < 32$

Step 1

Write $| - 8 n | < 32$ as a piecewise.

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

$- 8 n \geq 0$

Divide each term in $- 8 n \geq 0$ by $- 8$ and simplify.

Divide each term in $- 8 n \geq 0$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 n}{- 8} \leq \frac{0}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 n}{- 8} \leq \frac{0}{- 8}$

Divide $n$ by $1$ .

$n \leq \frac{0}{- 8}$

Simplify the right side.

Divide $0$ by $- 8$ .

$n \leq 0$

In the piece where $- 8 n$ is non-negative, remove the absolute value.

$- 8 n < 32$

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

$- 8 n < 0$

Divide each term in $- 8 n < 0$ by $- 8$ and simplify.

Divide each term in $- 8 n < 0$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 n}{- 8} > \frac{0}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 n}{- 8} > \frac{0}{- 8}$

Divide $n$ by $1$ .

$n > \frac{0}{- 8}$

Simplify the right side.

Divide $0$ by $- 8$ .

$n > 0$

In the piece where $- 8 n$ is negative, remove the absolute value and multiply by $- 1$ .

$8 n < 32$

Write as a piecewise.

${\begin{cases} - 8 n < 32 & n \leq 0 \\ 8 n < 32 & n > 0 \end{cases}$

Step 2

Solve $- 8 n < 32$ when $n \leq 0$ .

Divide each term in $- 8 n < 32$ by $- 8$ and simplify.

Divide each term in $- 8 n < 32$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 n}{- 8} > \frac{32}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 n}{- 8} > \frac{32}{- 8}$

Divide $n$ by $1$ .

$n > \frac{32}{- 8}$

Simplify the right side.

Divide $32$ by $- 8$ .

$n > - 4$

Find the intersection of $n > - 4$ and $n \leq 0$ .

$- 4 < n \leq 0$

Step 3

Solve $8 n < 32$ when $n > 0$ .

Divide each term in $8 n < 32$ by $8$ and simplify.

Divide each term in $8 n < 32$ by $8$ .

$\frac{8 n}{8} < \frac{32}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 n}{8} < \frac{32}{8}$

Divide $n$ by $1$ .

$n < \frac{32}{8}$

Simplify the right side.

Divide $32$ by $8$ .

$n < 4$

Find the intersection of $n < 4$ and $n > 0$ .

$0 < n < 4$

Step 4

Find the union of the solutions.

$- 4 < n < 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 4 < n < 4$

Interval Notation:

$(- 4, 4)$

Step 6

Solve for n |-8n|<32

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