Absolute Value Equations

Absolute Value Equations

What We Know

• absolute value is the distance from 0

• distance always has to be positive

• there are almost always 2 solutions

• if an absolute value = 0, there's only 1 solutions

• if an absolute value = a negative number, then there is no solution

Multiple Choice

 $\left|x\right|=\ -\ 5$  

• 5

• -5

• 5 or -5

• no solution

Multiple Choice

 $\left|x\right|\ =\ 6$  

• 6

• -6

• 6 or -6

• no solution

Solving Absolute Value Equations

• Isolate the absolute value by following the rules for solving equations

• Create 2 equations - one equal to a positive and one equal to a negative
  

• Solve for the variables as "normal"

Open Ended

What are the steps to solve this: $2\ +\ \left|x-4\right|\ =\ 5$  

Open Ended

What are the steps to solve this:  $3\left|x+1\right|\ +2=4$  

Multiple Choice

 $\left|x+5\right|\ =\ -3$  

•  $-2\ or\ -8$  

• no solution

•  $0$  

Multiple Choice

 $\left|x+5\right|=0$  

• no solution

•  $0$  

•  $-5$  

•  $-5\ or\ 5$  

Multiple Choice

Solve:
 $\left|\frac{3}{4}a-3\right|=9$  

•  $\left\{16,\ -8\right\}$  

•  $\left\{16,\ -16\right\}$  

•  $\left\{-8,\ 8\right\}$  

• no solution