Solve for s -4|s+10|<-16

$- 4 | s + 10 | < - 16$

Write $- 4 | s + 10 | < - 16$ as a piecewise.

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

$s + 10 \geq 0$

Subtract $10$ from both sides of the inequality.

$s \geq - 10$

In the piece where $s + 10$ is non-negative, remove the absolute value.

$- 4 (s + 10) < - 16$

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

$s + 10 < 0$

Subtract $10$ from both sides of the inequality.

$s < - 10$

In the piece where $s + 10$ is negative, remove the absolute value and multiply by $- 1$ .

$- 4 (- (s + 10)) < - 16$

Write as a piecewise.

${\begin{cases} - 4 (s + 10) < - 16 & s \geq - 10 \\ - 4 (- (s + 10)) < - 16 & s < - 10 \end{cases}$

Simplify $- 4 (s + 10) < - 16$ .

Apply the distributive property.

${\begin{cases} - 4 s - 4 \cdot 10 < - 16 & s \geq - 10 \\ - 4 (- (s + 10)) < - 16 & s < - 10 \end{cases}$

Multiply $- 4$ by $10$ .

${\begin{cases} - 4 s - 40 < - 16 & s \geq - 10 \\ - 4 (- (s + 10)) < - 16 & s < - 10 \end{cases}$

Simplify $- 4 (- (s + 10)) < - 16$ .

Apply the distributive property.

${\begin{cases} - 4 s - 40 < - 16 & s \geq - 10 \\ - 4 (- s - 1 \cdot 10) < - 16 & s < - 10 \end{cases}$

Multiply $- 1$ by $10$ .

${\begin{cases} - 4 s - 40 < - 16 & s \geq - 10 \\ - 4 (- s - 10) < - 16 & s < - 10 \end{cases}$

Apply the distributive property.

${\begin{cases} - 4 s - 40 < - 16 & s \geq - 10 \\ - 4 (- s) - 4 \cdot - 10 < - 16 & s < - 10 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} - 4 s - 40 < - 16 & s \geq - 10 \\ 4 s - 4 \cdot - 10 < - 16 & s < - 10 \end{cases}$

Multiply $- 4$ by $- 10$ .

${\begin{cases} - 4 s - 40 < - 16 & s \geq - 10 \\ 4 s + 40 < - 16 & s < - 10 \end{cases}$

Solve $- 4 s - 40 < - 16$ for $s$ .

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

Add $40$ to both sides of the inequality.

$- 4 s < - 16 + 40$

Add $- 16$ and $40$ .

$- 4 s < 24$

Divide each term in $- 4 s < 24$ by $- 4$ and simplify.

Divide each term in $- 4 s < 24$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 s}{- 4} > \frac{24}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 s}{- 4} > \frac{24}{- 4}$

Divide $s$ by $1$ .

$s > \frac{24}{- 4}$

Simplify the right side.

Divide $24$ by $- 4$ .

$s > - 6$

Solve $4 s + 40 < - 16$ for $s$ .

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

Subtract $40$ from both sides of the inequality.

$4 s < - 16 - 40$

Subtract $40$ from $- 16$ .

$4 s < - 56$

Divide each term in $4 s < - 56$ by $4$ and simplify.

Divide each term in $4 s < - 56$ by $4$ .

$\frac{4 s}{4} < \frac{- 56}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 s}{4} < \frac{- 56}{4}$

Divide $s$ by $1$ .

$s < \frac{- 56}{4}$

Simplify the right side.

Divide $- 56$ by $4$ .

$s < - 14$

Find the union of the solutions.

$s < - 14$ or $s > - 6$

The result can be shown in multiple forms.

Inequality Form:

$s < - 14 or s > - 6$

Interval Notation:

$(- \infty, - 14) \cup (- 6, \infty)$

Solve for s -4|s+10|<-16

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