Solve the Absolute Value Inequality for n -5|4+n|<-15

$- 5 | 4 + n | < - 15$

Write $- 5 | 4 + n | < - 15$ as a piecewise.

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

$4 + n \geq 0$

Subtract $4$ from both sides of the inequality.

$n \geq - 4$

In the piece where $4 + n$ is non-negative, remove the absolute value.

$- 5 (4 + n) < - 15$

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

$4 + n < 0$

Subtract $4$ from both sides of the inequality.

$n < - 4$

In the piece where $4 + n$ is negative, remove the absolute value and multiply by $- 1$ .

$- 5 (- (4 + n)) < - 15$

Write as a piecewise.

${\begin{cases} - 5 (4 + n) < - 15 & n \geq - 4 \\ - 5 (- (4 + n)) < - 15 & n < - 4 \end{cases}$

Simplify $- 5 (4 + n) < - 15$ .

Apply the distributive property.

${\begin{cases} - 5 \cdot 4 - 5 n < - 15 & n \geq - 4 \\ - 5 (- (4 + n)) < - 15 & n < - 4 \end{cases}$

Multiply $- 5$ by $4$ .

${\begin{cases} - 20 - 5 n < - 15 & n \geq - 4 \\ - 5 (- (4 + n)) < - 15 & n < - 4 \end{cases}$

Simplify $- 5 (- (4 + n)) < - 15$ .

Apply the distributive property.

${\begin{cases} - 20 - 5 n < - 15 & n \geq - 4 \\ - 5 (- 1 \cdot 4 - n) < - 15 & n < - 4 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} - 20 - 5 n < - 15 & n \geq - 4 \\ - 5 (- 4 - n) < - 15 & n < - 4 \end{cases}$

Apply the distributive property.

${\begin{cases} - 20 - 5 n < - 15 & n \geq - 4 \\ - 5 \cdot - 4 - 5 (- n) < - 15 & n < - 4 \end{cases}$

Multiply $- 5$ by $- 4$ .

${\begin{cases} - 20 - 5 n < - 15 & n \geq - 4 \\ 20 - 5 (- n) < - 15 & n < - 4 \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} - 20 - 5 n < - 15 & n \geq - 4 \\ 20 + 5 n < - 15 & n < - 4 \end{cases}$

Solve $- 20 - 5 n < - 15$ for $n$ .

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

Add $20$ to both sides of the inequality.

$- 5 n < - 15 + 20$

Add $- 15$ and $20$ .

$- 5 n < 5$

Divide each term in $- 5 n < 5$ by $- 5$ and simplify.

Divide each term in $- 5 n < 5$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 n}{- 5} > \frac{5}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 n}{- 5} > \frac{5}{- 5}$

Divide $n$ by $1$ .

$n > \frac{5}{- 5}$

Simplify the right side.

Divide $5$ by $- 5$ .

$n > - 1$

Solve $20 + 5 n < - 15$ for $n$ .

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

Subtract $20$ from both sides of the inequality.

$5 n < - 15 - 20$

Subtract $20$ from $- 15$ .

$5 n < - 35$

Divide each term in $5 n < - 35$ by $5$ and simplify.

Divide each term in $5 n < - 35$ by $5$ .

$\frac{5 n}{5} < \frac{- 35}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 n}{5} < \frac{- 35}{5}$

Divide $n$ by $1$ .

$n < \frac{- 35}{5}$

Simplify the right side.

Divide $- 35$ by $5$ .

$n < - 7$

Find the union of the solutions.

$n < - 7$ or $n > - 1$

The result can be shown in multiple forms.

Inequality Form:

$n < - 7 or n > - 1$

Interval Notation:

$(- \infty, - 7) \cup (- 1, \infty)$

Solve the Absolute Value Inequality for n -5|4+n|<-15

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