1.6 - Absolute Value Equations and Inequalities

1.6 - Absolute Value Equations and Inequalities

Absolute Value

The absolute value of a number (written

 $\left|x\right|$  ) is the distance from x to zero on the number line. Because absolute value is a distance it can only ever be zero or positive. Absolute value cannot be negative.

Solving Absolute Value Equations and Inequalities

• The solutions of

   $\left|x\right|=3$  are the two points that are 3 units away from zero. The solution is a disjunction:  $x=-3\ or\ x=3$  .

• 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:  $x<-3\ or\ x>3$  

Let's practice. Solve

 $\left|x-7\right|=5$  

• 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.

Multiple Choice

Solve

 $\left|3x\right|+5=14$  
(Hint: isolate the absolute value term first)

•  $x=-3$  

•  $x=3\ or\ x=-3$  

•  $x=\frac{19}{3}or\ x=-\frac{19}{3}$  

• No solution

Multiple Choice

Solve

 $\left|4x-8\right|>12$  

•  $x>-1$  

•  $x<5$  

•  $x<-1\ or\ x>5$  

•  $-1<x<5$  

Open Ended

What questions do you have for me?