Solve the Absolute Value Inequality for r -51=3-|6-12r|

$- 51 = 3 - | 6 - 12 r |$

Rewrite the equation as $3 - | 6 - 12 r | = - 51$ .

$3 - | 6 - 12 r | = - 51$

Move all terms not containing $| 6 - 12 r |$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- | 6 - 12 r | = - 51 - 3$

Subtract $3$ from $- 51$ .

$- | 6 - 12 r | = - 54$

Divide each term in $- | 6 - 12 r | = - 54$ by $- 1$ and simplify.

Divide each term in $- | 6 - 12 r | = - 54$ by $- 1$ .

$\frac{- | 6 - 12 r |}{- 1} = \frac{- 54}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{| 6 - 12 r |}{1} = \frac{- 54}{- 1}$

Divide $| 6 - 12 r |$ by $1$ .

$| 6 - 12 r | = \frac{- 54}{- 1}$

Simplify the right side.

Divide $- 54$ by $- 1$ .

$| 6 - 12 r | = 54$

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

$6 - 12 r = \pm 54$

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.

$6 - 12 r = 54$

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

Subtract $6$ from both sides of the equation.

$- 12 r = 54 - 6$

Subtract $6$ from $54$ .

$- 12 r = 48$

Divide each term in $- 12 r = 48$ by $- 12$ and simplify.

Divide each term in $- 12 r = 48$ by $- 12$ .

$\frac{- 12 r}{- 12} = \frac{48}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 r}{- 12} = \frac{48}{- 12}$

Divide $r$ by $1$ .

$r = \frac{48}{- 12}$

Simplify the right side.

Divide $48$ by $- 12$ .

$r = - 4$

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

$6 - 12 r = - 54$

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

Subtract $6$ from both sides of the equation.

$- 12 r = - 54 - 6$

Subtract $6$ from $- 54$ .

$- 12 r = - 60$

Divide each term in $- 12 r = - 60$ by $- 12$ and simplify.

Divide each term in $- 12 r = - 60$ by $- 12$ .

$\frac{- 12 r}{- 12} = \frac{- 60}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 r}{- 12} = \frac{- 60}{- 12}$

Divide $r$ by $1$ .

$r = \frac{- 60}{- 12}$

Simplify the right side.

Divide $- 60$ by $- 12$ .

$r = 5$

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

$r = - 4, 5$

Solve the Absolute Value Inequality for r -51=3-|6-12r|

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