Solve the Absolute Value Inequality for v |-2v-4|<8

$| - 2 v - 4 | < 8$

Step 1

Write $| - 2 v - 4 | < 8$ as a piecewise.

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

$- 2 v - 4 \geq 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$- 2 v \geq 4$

Divide each term in $- 2 v \geq 4$ by $- 2$ and simplify.

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

$\frac{- 2 v}{- 2} \leq \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 v}{- 2} \leq \frac{4}{- 2}$

Divide $v$ by $1$ .

$v \leq \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$v \leq - 2$

In the piece where $- 2 v - 4$ is non-negative, remove the absolute value.

$- 2 v - 4 < 8$

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

$- 2 v - 4 < 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$- 2 v < 4$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $v$ by $1$ .

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

Simplify the right side.

Divide $4$ by $- 2$ .

$v > - 2$

In the piece where $- 2 v - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 2 v - 4) < 8$

Write as a piecewise.

${\begin{cases} - 2 v - 4 < 8 & v \leq - 2 \\ - (- 2 v - 4) < 8 & v > - 2 \end{cases}$

Simplify $- (- 2 v - 4) < 8$ .

Apply the distributive property.

${\begin{cases} - 2 v - 4 < 8 & v \leq - 2 \\ - (- 2 v) - - 4 < 8 & v > - 2 \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 2 v - 4 < 8 & v \leq - 2 \\ 2 v - - 4 < 8 & v > - 2 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} - 2 v - 4 < 8 & v \leq - 2 \\ 2 v + 4 < 8 & v > - 2 \end{cases}$

Step 2

Solve $- 2 v - 4 < 8$ when $v \leq - 2$ .

Solve $- 2 v - 4 < 8$ for $v$ .

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

Add $4$ to both sides of the inequality.

$- 2 v < 8 + 4$

Add $8$ and $4$ .

$- 2 v < 12$

Divide each term in $- 2 v < 12$ by $- 2$ and simplify.

Divide each term in $- 2 v < 12$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 v}{- 2} > \frac{12}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 v}{- 2} > \frac{12}{- 2}$

Divide $v$ by $1$ .

$v > \frac{12}{- 2}$

Simplify the right side.

Divide $12$ by $- 2$ .

$v > - 6$

Find the intersection of $v > - 6$ and $v \leq - 2$ .

$- 6 < v \leq - 2$

Step 3

Solve $2 v + 4 < 8$ when $v > - 2$ .

Solve $2 v + 4 < 8$ for $v$ .

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

Subtract $4$ from both sides of the inequality.

$2 v < 8 - 4$

Subtract $4$ from $8$ .

$2 v < 4$

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

Divide each term in $2 v < 4$ by $2$ .

$\frac{2 v}{2} < \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 v}{2} < \frac{4}{2}$

Divide $v$ by $1$ .

$v < \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$v < 2$

Find the intersection of $v < 2$ and $v > - 2$ .

$- 2 < v < 2$

Step 4

Find the union of the solutions.

$- 6 < v < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 6 < v < 2$

Interval Notation:

$(- 6, 2)$

Step 6

Solve the Absolute Value Inequality for v |-2v-4|<8

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