Solving Linear Inequalities with Absolute Values

Date: 05/31/98 at 18:09:29
From: Carolyn
Subject: Linear inequalities

I have two questions that I am having difficulty solving. 

1) 3 <= |x + 2| < 8
   I have broken it into two parts:

     (a) |x + 2| < 8   and 
     (b) |x + 2| >= 3 

   From that, I have gotten for (a) |x| < 6 or |x| < -10 and for 
   (b) |x| >= 1 or |x| >= -5. Now, how do I put that on the line, or
   is there more I have to do?

2) 3 < |2x + 1| or |x + 1| - 3 > 2    
   I have done this so far: 2 + 3 < |x + 1|, which means 5 < |x + 1|. 
   I'm lost at this point.

Date: 05/31/98 at 23:27:48
From: Doctor Pat
Subject: Re: Linear inequalities

Carolyn, 

I'm going to try to help you see a new way to approach these, rather 
than just correct the mistakes above. It seems like you're trying to 
plow through this with some memorized algebra techniques without 
really understanding the idea, which for me is the most important 
part, so let's try to find a way to "see" the answer, then work 
from there. 

Let's look at the whole first problem: 3 <= |x + 2| < 8. We are 
looking for some numbers on the number line so that when we put them 
into the equation, the whole thing is true. Draw a number line and 
mark off x values in increments of, say, five or so, like this

      -10  -5  0   5  10  15   20  
   ____|___|___|___|___|___|___|___|___

Now, let's get an idea about which neighborhood would be the one we 
want. We take all these numbers, add two, then write the absolute 
value like this:

      -10  -5  0   5   10  15  20  
   ____|___|___|___|___|___|___|___|___
       8   3   2   7   12  17  22  
   
Which numbers are between three and eight? Well, by luck we see that 
numbers from -10 to -5 seem to work. It also looks like we should find 
another set of numbers just a little to the right of the origin (0). 
We could put more values in the area from 0 to ten and repeat, but I 
think you see where this is going. Let's look back at the values you 
have already found: -10, -5, 1, and 6. You found endpoints at all the 
right places; now, all we have to do is write them out as intervals 
joined by the "or" statement. Can you look at the number line again 
and write that out? You want a statement that says "x is between -10 
and -5 or else x is between 1 and 6." You need to put in the correct 
endpoint values to match up the < verses <=, but that is nothing 
compared to all the work you've already done. 

I will let you try to draw the picture of the next one. I hope this 
helps you make sense of absolute values and inequalities. Math is 
really difficult if you are not even sure when you are doing it right, 
so try to keep understanding the big ideas, and the rest sort of works 
out.  

Good luck and write back if you need more help.
 
-Doctor Pat, The Math Forum
Check out our web site! http://mathforum.org/dr.math/