Teaching Absolute Value Inequalities – Another Idea

I am still searching for a good way to teach the basic structure for solving absolute value inequalities.  Here are two  models I posted earlier – with lots of arrows and with a weird formula pattern.  This time I went for total simplification – just two associations to understand and memorize.

I started as usual with my “join the club” exercise where everyone lined up and stepped forward if their number satisfied the given inequality.  Also as usual I emphasized that either the middle or the tails stepped forward.  Then I drew a number line like this.

Number Line with Blanks

And we filled in the blanks with either “<” or “>”.  They soon figured out that “<” was used in all cases – smallest on left, largest on right.  I then wrote |x|< 4, threw out numbers and they said “yes” if they were solutions, “no” otherwise Then I drew

Number Line Solution to Less Than Ineguality

to get

Number Line Solution-Use Less Than or Egual Sign

Thus |x|\leq a or |x|< a indicates that  the unknown in the middle.  So

Number Line Solution-Use Less Than or EgualIn the Middle

Similarly with |x|> 4.  Now the unknown is in the tails (extremes).

Number Line Solution-Use Tails with Greater Than


How to Set up the Solutions toAbsolute Value Inequalities

Easy to explain though it took some coaching in the group work before they got comfortable with the middle versus tails idea.


About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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