## A Year of Nights with Wondering – III

Solving Absolute Value Equalities and Inequalities

We were working on solving absolute value equalities and inequalities.  A few days ago we went through the exercise I described in this post and now I was consistently and emphatically making connections to the number line using words like “middle” and “tail.”   When my students started working problems at their desks however, some of them became vocal.  They couldn’t figure out how to or when to “switch” the inequality signs.  It dawned on me that some of them were looking at the problems as just symbols to push around on the paper and they were frustrated because they couldn’t find a pattern.  As part of one problem one student had written $-2>3x+7>2$.  My jaw dropped.  This was wonderfully correct as far as the problem went but wonderfully incorrect since the transitivity of the $<$ sign was violated.  I tried to convince the student of that fact but he didn’t go for it.  I woke up last night with a new pattern in my head.  Look at this potential crib sheet.

Absolute Values Equalities and Inequalities - Summary

The pattern can be applied to work these problems.

Absolute Values Equalities and Inequalities - Examples

I think I will show my students this post next class session. I know that the steps marked with an ‘*” have a notation error and thus reinforce a bad habit.  I will dutifully warn them with appropriate histrionics.

Will I have committed a great sin?  I tried all the methods I know to help them conceptualize the solving process. Is it okay to provide the students who still don’t get it with an easy though incorrect pattern to memorize?  I am not sure but they really wanted something and said so.  I am going to go with their instincts.