Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.3

Simplify the right side.

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Step 1.3.1

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.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

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

Step 3.2.2

Subtract $1$ from $3$ .

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

Step 3.3

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

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

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Cancel the common factor of $3$ .

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Step 3.4.1.1.1.1

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

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

Step 3.4.1.1.1.2

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

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

Step 3.4.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 3.4.1.1.1.4

Cancel the common factor.

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

Step 3.4.1.1.1.5

Rewrite the expression.

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

Step 3.4.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.2.1

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

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

Step 3.4.1.1.2.2

Cancel the common factor.

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

Step 3.4.1.1.2.3

Rewrite the expression.

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

Step 3.4.1.1.3

Multiply.

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Step 3.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.1.1.3.2

Multiply $v$ by $1$ .

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

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

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

Step 3.4.2.1.2

Cancel the common factor.

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

Step 3.4.2.1.3

Rewrite the expression.

$v = - 3$

Step 3.5

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

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

Step 3.6

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

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Step 3.6.1

Subtract $1$ from both sides of the equation.

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

Step 3.6.2

Subtract $1$ from $- 3$ .

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

Step 3.7

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

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

Step 3.8

Simplify both sides of the equation.

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Step 3.8.1

Simplify the left side.

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Step 3.8.1.1

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

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Step 3.8.1.1.1

Cancel the common factor of $3$ .

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Step 3.8.1.1.1.1

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

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

Step 3.8.1.1.1.2

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

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

Step 3.8.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 3.8.1.1.1.4

Cancel the common factor.

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

Step 3.8.1.1.1.5

Rewrite the expression.

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

Step 3.8.1.1.2

Cancel the common factor of $2$ .

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Step 3.8.1.1.2.1

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

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

Step 3.8.1.1.2.2

Cancel the common factor.

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

Step 3.8.1.1.2.3

Rewrite the expression.

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

Step 3.8.1.1.3

Multiply.

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Step 3.8.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.8.1.1.3.2

Multiply $v$ by $1$ .

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

Step 3.8.2

Simplify the right side.

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Step 3.8.2.1

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

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Step 3.8.2.1.1

Cancel the common factor of $2$ .

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Step 3.8.2.1.1.1

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

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

Step 3.8.2.1.1.2

Factor $2$ out of $- 4$ .

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

Step 3.8.2.1.1.3

Cancel the common factor.

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

Step 3.8.2.1.1.4

Rewrite the expression.

$v = - 3 \cdot - 2$

Step 3.8.2.1.2

Multiply $- 3$ by $- 2$ .

$v = 6$

Step 3.9

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

$v = - 3, 6$