Absolute Value in Open Sentences

Our homework for Tuesday started a new topic, which was absolute values.  All of the problems given for homework involved absolute values, either in equations or inequalities.  By themselves, absolute values are very easy.  The absolute value of a number is how far away a number is from zero.  A few examples:  
| –3 | = 3
| 28| = 28
|-8.4 | = 8.4

However, in the homework absolute value signs generally involved splitting a problem into two different problems, and solving the problem through these, as shown in the example below.  As for what happened during class on Tuesday, the majority of class involved going over the homework, but after this we tried more problems, and for these problems we paired up and worked on the board, allowing each other to compare are work to others, and find where we may have made mistakes.  Here is an example of a problem from the board from Tuesday in class:
 

The first step in this problem was splitting it into two different inequalities, and knowing that there would be an "and" in between, because of the less than sign.  Each of these two inequalities now splits into two more, leaving four inequalities which need to be solved.  The way to find this second set of two inequalities is to write the problem as is but without absolute value for the first one, and then switch the sign and multiply the side without the variable by negative one. After solving for the variable in these four inequalites, we know that the line we graph has to be less than or equal to five AND greater than or equal to negative four, as well as greater than or equal two one OR less than or equal to zero.  At the end of the problem, the line is graphed and appropriately shaded.

The homework for Tuesday night was similar, involving solving for variables by splitting absolute value problems into two equations or two inequalities.  Most of class Wednesday was spent putting the homework problems on the board, and then discussing them.  While most of the homework given was very similar to the above problem we did in class, the last problem was an extension of what we did in class.  The way the last problem was different was that there was an absolute value on each side of the inequality.  Here is the problem from when it was put on the board in class, after we discussed it.

 Although this problem looked different, solving it involved the same general steps.  First, the problem was split into two inequalities, with an and in between, and then each of these two inequalities became two more, the first pair having an and in between, the second pair with an or in between.  The way to solve this kind of problem, as we learned in class, was to find the two pairs by pretending the absolute value sign was only on one side, and then only on the other.  After using that trick to find the inequalities, all that was left was solving for the variable for each one, and after doing this we found that this line has to be greater than or equal to five AND less than or equal to fifteen, as well as less than or equal to fifteen OR greater than or equal to fifteen.  Although we got less than or equal to fifteen twice, it is still important to always find all the inequalities, because generally they are all different.  At the end of the problem you can see what this looks like graphed on a line.

One part of these problems that kept confusing me is whether to put "and" or "or" in between two inequalities.  While this may seem simple, it was actually one of the most tricky parts of the problem for me, and often where I made mistakes.  Usually, after finding the inequalities, I would draw a quick sketch of the line to try to figure it out, but Wednesday in class I had an aha moment when I realized it was as simple as this:

≤ - and
≥ - or

Here is a video I found on absolute value in inequalities.  We have not covered some things he talks about near the end of the video, but up until that point there is lots of useful information.  This video would be a great way to review this topic before a test or exams.  Also, problems exactly like the ones above can be found in the math book on page 73.  

- Ben Freedman