Solve the Absolute Value Inequality for v |v+5|-6<-5

$| v + 5 | - 6 < - 5$

Write $| v + 5 | - 6 < - 5$ as a piecewise.

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

$v + 5 \geq 0$

Subtract $5$ from both sides of the inequality.

$v \geq - 5$

In the piece where $v + 5$ is non-negative, remove the absolute value.

$v + 5 - 6 < - 5$

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

$v + 5 < 0$

Subtract $5$ from both sides of the inequality.

$v < - 5$

In the piece where $v + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (v + 5) - 6 < - 5$

Write as a piecewise.

${\begin{cases} v + 5 - 6 < - 5 & v \geq - 5 \\ - (v + 5) - 6 < - 5 & v < - 5 \end{cases}$

Subtract $6$ from $5$ .

${\begin{cases} v - 1 < - 5 & v \geq - 5 \\ - (v + 5) - 6 < - 5 & v < - 5 \end{cases}$

Simplify $- (v + 5) - 6 < - 5$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} v - 1 < - 5 & v \geq - 5 \\ - v - 1 \cdot 5 - 6 < - 5 & v < - 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} v - 1 < - 5 & v \geq - 5 \\ - v - 5 - 6 < - 5 & v < - 5 \end{cases}$

Subtract $6$ from $- 5$ .

${\begin{cases} v - 1 < - 5 & v \geq - 5 \\ - v - 11 < - 5 & v < - 5 \end{cases}$

Solve $v - 1 < - 5$ when $v \geq - 5$ .

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

Add $1$ to both sides of the inequality.

$v < - 5 + 1$

Add $- 5$ and $1$ .

$v < - 4$

Find the intersection of $v < - 4$ and $v \geq - 5$ .

$- 5 \leq v < - 4$

Solve $- v - 11 < - 5$ when $v < - 5$ .

Solve $- v - 11 < - 5$ for $v$ .

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

Add $11$ to both sides of the inequality.

$- v < - 5 + 11$

Add $- 5$ and $11$ .

$- v < 6$

Divide each term in $- v < 6$ by $- 1$ and simplify.

Divide each term in $- v < 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- v}{- 1} > \frac{6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{v}{1} > \frac{6}{- 1}$

Divide $v$ by $1$ .

$v > \frac{6}{- 1}$

Simplify the right side.

Divide $6$ by $- 1$ .

$v > - 6$

Find the intersection of $v > - 6$ and $v < - 5$ .

$- 6 < v < - 5$

Find the union of the solutions.

$- 6 < v < - 4$

The result can be shown in multiple forms.

Inequality Form:

$- 6 < v < - 4$

Interval Notation:

$(- 6, - 4)$

Solve the Absolute Value Inequality for v |v+5|-6<-5

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