1.6 - Absolute Value Equations and Inequalities

The solutions of

The solutions of  $\left|x\right|<3$  are the points that are less than 3 units away from zero. The solution is a conjunction:  $-3<x<3$  

The solutions of  $\left|x\right|>3$  are the points that are more than 3 units away from zero. The solution is a disjunction:  $x<-3\ or\ x>3$  

Important: any time you are solving an absolute value equation, you must write and solve two equations.

 $x-7=5\ or\ x-7=-5$  

     $+7\ \ +7\ \ \ \ \ \ +7\ +7$  

 $x=12\ or\ x=2$  

And this checks out. 12-7 is 5, which has an absolute value of 5. And 2-7 is -5, which has an absolute value of 5.