Inequalities and Absolute Value

Date: 8/28/96 at 20:50:45
From: Brian Hicks
Subject: Inequalities and Absolute Value

Hi Dr Math! 
 
I have a few problems with inequalities and absolute values... 
What would the solution set be to something like this... 
 
  | X | > X
                
For this inequality, I had  X > X  or X < -X  How could this result be 
possible?

And... 
 
  | X + 2 | - X >= 0     On this problem, my solution set was 
                             { X : 2>_0 or X < -X } 
 
  48 | 7K-30 |           Where K = 14.  My solution to this one      
                         was 48 | 68 | But, I cannot distribute      
                         regularly right?  How would I solve this? 
 
Thanks!  I appreciate your time and advice. 
 
Brian 

Date: 8/29/96 at 14:21:57
From: Doctor Jodi
Subject: Re: Inequalities and Absolute Value

Hi there! Let me go through each of your problems and try to give you 
some suggestions.

> | X | > X          

Since x > x is NEVER true, you can eliminate that as a "solution":  no 
matter what the value of x, it will never be greater than itself.  So 
this problem could also be written x < -x.  For any value less than 
zero, x will be less than its opposite.  Does that make sense?
 
> | X + 2 | - X >= 0     On this problem, my solution set was 
>                             { X : 2>_0 or X < -X } 

When you're doing problems like this one, you'll want to remember when 
the solutions apply.  Here's how I'd do this problem:

|x + 2| - x >= 0

|x + 2| >= x

Now you need to divide this problem into two parts:

|x + 2| is positive or zero:
----------------------------

Since |x + 2| is the same as x + 2, you can remove the abs. value sign 
without changing the rest of the problem:

x + 2 >= x
2 >= 0

Since 2 is ALWAYS greater than zero, this means that if |x + 2| is 
non-negative the statement 

|x + 2| - x >= 0

is true.  So the final answer set must include all numbers greater 
than or equal to -2.

Now there's the second part of the problem:
|x + 2| is negative:
--------------------
This happens when x is less than -2.

The absolute value sign will change |x + 2| to its opposite, - (x + 2)
(aka -x -2 ), so

 |x + 2| >= x
-(x + 2) >= x
 -x - 2  >= x
     -2  >= 2x
     -1  >= x

Thus, this statement is true for any x that's less than or equal to -1 
and less than -2.  This means it's true for all numbers less than -2.

Combining the first and the second parts of the problem, we see that 
the equation holds for all real numbers.

-------------------------------
> 48 | 7K-30 |                        

The absolute value of 68 is 68, so now multiply 48 * 68 to get the 
answer.

-Doctor Jodi,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/