Solve for s -2|s+6|<-16

$- 2 | s + 6 | < - 16$

Step 1

Write $- 2 | s + 6 | < - 16$ as a piecewise.

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

$s + 6 \geq 0$

Subtract $6$ from both sides of the inequality.

$s \geq - 6$

In the piece where $s + 6$ is non-negative, remove the absolute value.

$- 2 (s + 6) < - 16$

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

$s + 6 < 0$

Subtract $6$ from both sides of the inequality.

$s < - 6$

In the piece where $s + 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- 2 (- (s + 6)) < - 16$

Write as a piecewise.

${\begin{cases} - 2 (s + 6) < - 16 & s \geq - 6 \\ - 2 (- (s + 6)) < - 16 & s < - 6 \end{cases}$

Simplify $- 2 (s + 6) < - 16$ .

Apply the distributive property.

${\begin{cases} - 2 s - 2 \cdot 6 < - 16 & s \geq - 6 \\ - 2 (- (s + 6)) < - 16 & s < - 6 \end{cases}$

Multiply $- 2$ by $6$ .

${\begin{cases} - 2 s - 12 < - 16 & s \geq - 6 \\ - 2 (- (s + 6)) < - 16 & s < - 6 \end{cases}$

Simplify $- 2 (- (s + 6)) < - 16$ .

Apply the distributive property.

${\begin{cases} - 2 s - 12 < - 16 & s \geq - 6 \\ - 2 (- s - 1 \cdot 6) < - 16 & s < - 6 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} - 2 s - 12 < - 16 & s \geq - 6 \\ - 2 (- s - 6) < - 16 & s < - 6 \end{cases}$

Apply the distributive property.

${\begin{cases} - 2 s - 12 < - 16 & s \geq - 6 \\ - 2 (- s) - 2 \cdot - 6 < - 16 & s < - 6 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} - 2 s - 12 < - 16 & s \geq - 6 \\ 2 s - 2 \cdot - 6 < - 16 & s < - 6 \end{cases}$

Multiply $- 2$ by $- 6$ .

${\begin{cases} - 2 s - 12 < - 16 & s \geq - 6 \\ 2 s + 12 < - 16 & s < - 6 \end{cases}$

Step 2

Solve $- 2 s - 12 < - 16$ for $s$ .

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

Add $12$ to both sides of the inequality.

$- 2 s < - 16 + 12$

Add $- 16$ and $12$ .

$- 2 s < - 4$

Divide each term in $- 2 s < - 4$ by $- 2$ and simplify.

Divide each term in $- 2 s < - 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 s}{- 2} > \frac{- 4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 s}{- 2} > \frac{- 4}{- 2}$

Divide $s$ by $1$ .

$s > \frac{- 4}{- 2}$

Simplify the right side.

Divide $- 4$ by $- 2$ .

$s > 2$

Step 3

Solve $2 s + 12 < - 16$ for $s$ .

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

Subtract $12$ from both sides of the inequality.

$2 s < - 16 - 12$

Subtract $12$ from $- 16$ .

$2 s < - 28$

Divide each term in $2 s < - 28$ by $2$ and simplify.

Divide each term in $2 s < - 28$ by $2$ .

$\frac{2 s}{2} < \frac{- 28}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 s}{2} < \frac{- 28}{2}$

Divide $s$ by $1$ .

$s < \frac{- 28}{2}$

Simplify the right side.

Divide $- 28$ by $2$ .

$s < - 14$

Step 4

Find the union of the solutions.

$s < - 14$ or $s > 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$s < - 14 or s > 2$

Interval Notation:

$(- \infty, - 14) \cup (2, \infty)$

Step 6

Solve for s -2|s+6|<-16

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