Solve the Absolute Value Inequality for v 5+|3v|>20

$5 + | 3 v | > 20$

Step 1

Write $5 + | 3 v | > 20$ as a piecewise.

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

$3 v \geq 0$

Divide each term in $3 v \geq 0$ by $3$ and simplify.

Divide each term in $3 v \geq 0$ by $3$ .

$\frac{3 v}{3} \geq \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 v}{3} \geq \frac{0}{3}$

Divide $v$ by $1$ .

$v \geq \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$v \geq 0$

In the piece where $3 v$ is non-negative, remove the absolute value.

$5 + 3 v > 20$

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

$3 v < 0$

Divide each term in $3 v < 0$ by $3$ and simplify.

Divide each term in $3 v < 0$ by $3$ .

$\frac{3 v}{3} < \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 v}{3} < \frac{0}{3}$

Divide $v$ by $1$ .

$v < \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$v < 0$

In the piece where $3 v$ is negative, remove the absolute value and multiply by $- 1$ .

$5 - 3 v > 20$

Write as a piecewise.

${\begin{cases} 5 + 3 v > 20 & v \geq 0 \\ 5 - 3 v > 20 & v < 0 \end{cases}$

Step 2

Solve $5 + 3 v > 20$ for $v$ .

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

Subtract $5$ from both sides of the inequality.

$3 v > 20 - 5$

Subtract $5$ from $20$ .

$3 v > 15$

Divide each term in $3 v > 15$ by $3$ and simplify.

Divide each term in $3 v > 15$ by $3$ .

$\frac{3 v}{3} > \frac{15}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 v}{3} > \frac{15}{3}$

Divide $v$ by $1$ .

$v > \frac{15}{3}$

Simplify the right side.

Divide $15$ by $3$ .

$v > 5$

Step 3

Solve $5 - 3 v > 20$ for $v$ .

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

Subtract $5$ from both sides of the inequality.

$- 3 v > 20 - 5$

Subtract $5$ from $20$ .

$- 3 v > 15$

Divide each term in $- 3 v > 15$ by $- 3$ and simplify.

Divide each term in $- 3 v > 15$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 v}{- 3} < \frac{15}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 v}{- 3} < \frac{15}{- 3}$

Divide $v$ by $1$ .

$v < \frac{15}{- 3}$

Simplify the right side.

Divide $15$ by $- 3$ .

$v < - 5$

Step 4

Find the union of the solutions.

$v < - 5$ or $v > 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$v < - 5 or v > 5$

Interval Notation:

$(- \infty, - 5) \cup (5, \infty)$

Step 6

Solve the Absolute Value Inequality for v 5+|3v|>20

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