Lesson: Solve absolute value inequalties

Do Now: (5 minutes)

1) Solve for x:

|4x + 1| = 15

2) Solve and graph:

3x + 2 < 8 or –x + 3 < -2

3) 8 ¼ - 2 â…

Problem Solving: (25 minutes)

Today we are going to take a look at absolute value inequalities.  How were absolute value equations different than other equations that we solved? (They became two equations)  How do you think an absolute value inequality is going to be different from other inequalities?  (There will be two inequalities?

Where else have we seen two inequalities in the same problem? (Compound inequalities). 

Note - for the purposes of formatting this lesson to post, ( ) are used to indicate absolute value.

(x) > 6

Absolute value is a distance.  What do you think this inequality is telling us about distance?

The distance between x and zero is greater than 6.

x < -6 or x > 6

Graph on a number line.  Since we are looking at distances that are greater than 6, we end up with the union of two inequalities.

·         When an absolute value inequality has the greater than sign, the is the union of two inequalities

Ex. (x) <0.5

The distance between x and zero is less than or equal to 0.5

Graph on a number line.  We are looking at distances that are less than 0.5 from 0, so we have the intersection of two graphs.

·         When an absolute value inequality has the less than sign, the graphs of the solution intersect.

Ex.(x - 5) > 7

Ask students how they would solve if this was an equation instead of an inequality.  What do we know about equations and inequalities? (They follow the same rules for solving)

Ex.(-4x - 5) < 9

Since it is a less than equation, make it into an “and” inequality with the absolute value expression in the middle and the negative in the front.

Solving absolute value inequalities:

1) Isolate the absolute value

2) Rewrite as a compound inequality (or inequality if there is a > sign, and inequality if there is a < sign.

3) Solve the inequality