Algebra Examples

- 4 | s + 10 | < - 16

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

Step 1

Write

- 4 | s + 10 | < - 16

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

s + 10 \geq 0

$s + 10 \geq 0$

Step 1.2

Subtract

10

$10$ from both sides of the inequality.

s \geq - 10

$s \geq - 10$

Step 1.3

In the piece where

s + 10

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

- 4 (s + 10) < - 16

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

Step 1.4

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

s + 10 < 0

$s + 10 < 0$

Step 1.5

Subtract

10

$10$ from both sides of the inequality.

s < - 10

$s < - 10$

Step 1.6

In the piece where

s + 10

$s + 10$ is negative, remove the absolute value and multiply by

- 1

$- 1$ .

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

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

Step 1.7

Write as a piecewise.

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

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

Step 1.8

Simplify

- 4 (s + 10) < - 16

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

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Step 1.8.1

Apply the distributive property.

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

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

Step 1.8.2

Multiply

- 4

$- 4$ by

10

$10$ .

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

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

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

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

Step 1.9

Simplify

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

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

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Step 1.9.1

Apply the distributive property.

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

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

Step 1.9.2

Multiply

- 1

$- 1$ by

10

$10$ .

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

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

Step 1.9.3

Apply the distributive property.

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

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

Step 1.9.4

Multiply

$- 1$ by

$- 4$ .

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

Step 1.9.5

Multiply

$- 4$ by

$- 10$ .

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

Step 2

Solve

$- 4 s - 40 < - 16$ for

$s$ .

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Step 2.1

Move all terms not containing

$s$ to the right side of the inequality.

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Step 2.1.1

Add

$40$ to both sides of the inequality.

$- 4 s < - 16 + 40$

Step 2.1.2

Add

$- 16$ and

$40$ .

$- 4 s < 24$

Step 2.2

Divide each term in

$- 4 s < 24$ by

$- 4$ and simplify.

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Step 2.2.1

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.

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of

$- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide

$s$ by

$1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide

$24$ by

$- 4$ .

$s > - 6$

Step 3

Solve

$4 s + 40 < - 16$ for

$s$ .

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Step 3.1

Move all terms not containing

$s$ to the right side of the inequality.

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Step 3.1.1

Subtract

$40$ from both sides of the inequality.

$4 s < - 16 - 40$

Step 3.1.2

Subtract

$40$ from

$- 16$ .

$4 s < - 56$

Step 3.2

Divide each term in

$4 s < - 56$ by

$4$ and simplify.

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Step 3.2.1

Divide each term in

$4 s < - 56$ by

$4$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

$4$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide

$s$ by

$1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide

$- 56$ by

$4$ .

$s < - 14$

Step 4

Find the union of the solutions.

$s < - 14$ or

$s > - 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$s < - 14 or s > - 6$

Interval Notation:

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

Step 6