Absolute Value Equations

Date: 10/30/2002 at 16:53:48
From: Abel Dallas
Subject: Absolute value equations

What is the answer to 'the absolute value of x-3 plus the absolute 
value of 3-x equals 12' ?

Date: 11/03/2002 at 16:56:07
From: Doctor Ian
Subject: Re: Absolute value equations

Hi Abel,

Let's look at that:

  |x-3| + |x+3| = 12

The first term tells you how far x is away from 3. For example, when
x is 4, 

  |4 - 3| = |1| = 1

And when x is 2, 

  |2 - 3| = |-1| = 1

Similarly, the second term tells you how far x is away from -3.  

So the in words, the equation is saying:

  What is x, if the distance from 3, plus the distance from -3, 
  is 12? 

Let's think about how that would look on a number line.  Suppose we
want to test the value x = -7:


            |------------10---------------|
            x
            |----4------|

   <--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->
     -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

For this value of x, the total distance is 14.  Let's substitute that
into the expression, and see what it says:

  |-7 - 3| + |-7 + 3| = |-10| + |-4|

                      = 10 + 4

                      = 14

So the picture and the equation seem to be saying the same thing. 

The symmetry of the situation suggests that there will be two values
of x such that the sum of the distances to -3 and 3 will be 12. Try
to find them, and let me know what you come up with. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/