The relation between compound inequalities and absolute value inequalities

 I finally ended up using a textbook that I like.  It is older but it groups many topics together so they flow well from one to another.  One such flow is the book teaches how to solve compound inequalities first and in the very next section, students are asked to use what they learned with compound inequalities to solve absolute value inequalities.  

I think this made it so much easier for my students because they could "see" why these inequalities had to be solved as two problems rather than one due to the definition of an absolute value. I was able to relate the AND's and OR's to the appropriate absolute value inequality even to the point of showing how can be solved the same exact way as you do for a compound AND statement.

It was nice to have the book actually teach absolute value inequalities by having students rewrite them into compound inequality equations after reducing any into the absolute value being greater than or less than to a value. When using the idea of rewriting the absolute value inequality into a compound inequality, most of my Algebra I students were nodding and it was like a light bulb going off in their heads.

The biggest issue they had was to remember that the absolute value inequality with a less than sign was solved using the AND while the greater than required the OR situation to solve.  In addition, I had to remind them about the special cases of no solution or all possible solutions because they got in a rhythm solving the inequalities.  It's the same as when students first solve inequalities and just whiz through one that equals a negative number.

Once I finish this section, I should take time to have students convert compound inequality problems into absolute value inequalities so students better understand it is a two way relationship.  The previous section did not take time to show how to do that so they only do the conversion one way.  I think I'll find a worksheet of mixed compound and absolute value inequality problems so they can practice going from one form to the other.  I don't know if I'll actually have them solve the problem but I want to have them comfortable going from one to the other.

The longer I teach, the more I realize how important it is for students to make connections between one thing to the next.  I've expanded how I teach things so I am introducing ideas before we actually get to them or add depth to a topic as I teach it.  This is one topic, I need to show to students.  Let me know what you think, I'd love to hear.  have a good day.