Solve for v -3|1-2/3v|=-9

$- 3 | 1 - \frac{2}{3} v | = - 9$

Step 1

Divide each term in $- 3 | 1 - \frac{2}{3} v | = - 9$ by $- 3$ and simplify.

Divide each term in $- 3 | 1 - \frac{2}{3} v | = - 9$ by $- 3$ .

$\frac{- 3 | 1 - \frac{2}{3} v |}{- 3} = \frac{- 9}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 | 1 - \frac{2}{3} v |}{- 3} = \frac{- 9}{- 3}$

Divide $| 1 - \frac{2}{3} v |$ by $1$ .

$| 1 - \frac{2}{3} v | = \frac{- 9}{- 3}$

Simplify each term.

Combine $v$ and $\frac{2}{3}$ .

$| 1 - \frac{v \cdot 2}{3} | = \frac{- 9}{- 3}$

Move $2$ to the left of $v$ .

$| 1 - \frac{2 v}{3} | = \frac{- 9}{- 3}$

Simplify the right side.

Divide $- 9$ by $- 3$ .

$| 1 - \frac{2 v}{3} | = 3$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$1 - \frac{2 v}{3} = \pm 3$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$1 - \frac{2 v}{3} = 3$

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

Subtract $1$ from both sides of the equation.

$- \frac{2 v}{3} = 3 - 1$

Subtract $1$ from $3$ .

$- \frac{2 v}{3} = 2$

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 v}{3}) = - \frac{3}{2} \cdot 2$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{3}{2} (- \frac{2 v}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 v}{3}) = - \frac{3}{2} \cdot 2$

Move the leading negative in $- \frac{2 v}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 v}{3} = - \frac{3}{2} \cdot 2$

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 v}{3} = - \frac{3}{2} \cdot 2$

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 v}{3} = - \frac{3}{2} \cdot 2$

Rewrite the expression.

$\frac{- 1}{2} (- 2 v) = - \frac{3}{2} \cdot 2$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2 v$ .

$\frac{- 1}{2} (2 (- v)) = - \frac{3}{2} \cdot 2$

Cancel the common factor.

$\frac{- 1}{2} (2 (- v)) = - \frac{3}{2} \cdot 2$

Rewrite the expression.

$- - v = - \frac{3}{2} \cdot 2$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 v = - \frac{3}{2} \cdot 2$

Multiply $v$ by $1$ .

$v = - \frac{3}{2} \cdot 2$

Simplify the right side.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$v = \frac{- 3}{2} \cdot 2$

Cancel the common factor.

$v = \frac{- 3}{2} \cdot 2$

Rewrite the expression.

$v = - 3$

Next, use the negative value of the $\pm$ to find the second solution.

$1 - \frac{2 v}{3} = - 3$

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

Subtract $1$ from both sides of the equation.

$- \frac{2 v}{3} = - 3 - 1$

Subtract $1$ from $- 3$ .

$- \frac{2 v}{3} = - 4$

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 v}{3}) = - \frac{3}{2} \cdot - 4$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{3}{2} (- \frac{2 v}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 v}{3}) = - \frac{3}{2} \cdot - 4$

Move the leading negative in $- \frac{2 v}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 v}{3} = - \frac{3}{2} \cdot - 4$

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 v}{3} = - \frac{3}{2} \cdot - 4$

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 v}{3} = - \frac{3}{2} \cdot - 4$

Rewrite the expression.

$\frac{- 1}{2} (- 2 v) = - \frac{3}{2} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2 v$ .

$\frac{- 1}{2} (2 (- v)) = - \frac{3}{2} \cdot - 4$

Cancel the common factor.

$\frac{- 1}{2} (2 (- v)) = - \frac{3}{2} \cdot - 4$

Rewrite the expression.

$- - v = - \frac{3}{2} \cdot - 4$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 v = - \frac{3}{2} \cdot - 4$

Multiply $v$ by $1$ .

$v = - \frac{3}{2} \cdot - 4$

Simplify the right side.

Simplify $- \frac{3}{2} \cdot - 4$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$v = \frac{- 3}{2} \cdot - 4$

Factor $2$ out of $- 4$ .

$v = \frac{- 3}{2} \cdot (2 (- 2))$

Cancel the common factor.

$v = \frac{- 3}{2} \cdot (2 \cdot - 2)$

Rewrite the expression.

$v = - 3 \cdot - 2$

Multiply $- 3$ by $- 2$ .

$v = 6$

The complete solution is the result of both the positive and negative portions of the solution.

$v = - 3, 6$

Solve for v -3|1-2/3v|=-9

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