Solve the Absolute Value Inequality for r 3-1/2|r+9|=-11

$3 - \frac{1}{2} | r + 9 | = - 11$

Combine $| r + 9 |$ and $\frac{1}{2}$ .

$3 - \frac{| r + 9 |}{2} = - 11$

Move all terms not containing $| r + 9 |$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- \frac{| r + 9 |}{2} = - 11 - 3$

Subtract $3$ from $- 11$ .

$- \frac{| r + 9 |}{2} = - 14$

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{| r + 9 |}{2}) = - 2 \cdot - 14$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{| r + 9 |}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{| r + 9 |}{2}$ into the numerator.

$- 2 \frac{- | r + 9 |}{2} = - 2 \cdot - 14$

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- | r + 9 |}{2} = - 2 \cdot - 14$

Cancel the common factor.

$2 \cdot - 1 \frac{- | r + 9 |}{2} = - 2 \cdot - 14$

Rewrite the expression.

$- - | r + 9 | = - 2 \cdot - 14$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 | r + 9 | = - 2 \cdot - 14$

Multiply $| r + 9 |$ by $1$ .

$| r + 9 | = - 2 \cdot - 14$

Simplify the right side.

Multiply $- 2$ by $- 14$ .

$| r + 9 | = 28$

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

$r + 9 = \pm 28$

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.

$r + 9 = 28$

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

Subtract $9$ from both sides of the equation.

$r = 28 - 9$

Subtract $9$ from $28$ .

$r = 19$

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

$r + 9 = - 28$

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

Subtract $9$ from both sides of the equation.

$r = - 28 - 9$

Subtract $9$ from $- 28$ .

$r = - 37$

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

$r = 19, - 37$

Solve the Absolute Value Inequality for r 3-1/2|r+9|=-11

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