Solve for v 4v-3=|2v+9|

$4 v - 3 = | 2 v + 9 |$

Step 1

Rewrite the equation as $| 2 v + 9 | = 4 v - 3$ .

$| 2 v + 9 | = 4 v - 3$

Step 2

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

$2 v + 9 = \pm (4 v - 3)$

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.

$2 v + 9 = 4 v - 3$

Move all terms containing $v$ to the left side of the equation.

Subtract $4 v$ from both sides of the equation.

$2 v + 9 - 4 v = - 3$

Subtract $4 v$ from $2 v$ .

$- 2 v + 9 = - 3$

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

Subtract $9$ from both sides of the equation.

$- 2 v = - 3 - 9$

Subtract $9$ from $- 3$ .

$- 2 v = - 12$

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

Divide each term in $- 2 v = - 12$ by $- 2$ .

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

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 $- 2$ .

$v = 6$

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

$2 v + 9 = - (4 v - 3)$

Simplify $- (4 v - 3)$ .

Rewrite.

$2 v + 9 = 0 + 0 - (4 v - 3)$

Simplify by adding zeros.

$2 v + 9 = - (4 v - 3)$

Apply the distributive property.

$2 v + 9 = - (4 v) - - 3$

Multiply.

Multiply $4$ by $- 1$ .

$2 v + 9 = - 4 v - - 3$

Multiply $- 1$ by $- 3$ .

$2 v + 9 = - 4 v + 3$

Move all terms containing $v$ to the left side of the equation.

Add $4 v$ to both sides of the equation.

$2 v + 9 + 4 v = 3$

Add $2 v$ and $4 v$ .

$6 v + 9 = 3$

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

Subtract $9$ from both sides of the equation.

$6 v = 3 - 9$

Subtract $9$ from $3$ .

$6 v = - 6$

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

Divide each term in $6 v = - 6$ by $6$ .

$\frac{6 v}{6} = \frac{- 6}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 v}{6} = \frac{- 6}{6}$

Divide $v$ by $1$ .

$v = \frac{- 6}{6}$

Simplify the right side.

Divide $- 6$ by $6$ .

$v = - 1$

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

$v = 6, - 1$

Step 4

Exclude the solutions that do not make $4 v - 3 = | 2 v + 9 |$ true.

$v = 6$

Solve for v 4v-3=|2v+9|

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