Solving Absolute Value Equations

Date: 01/12/2004 at 20:04:39
From: Baffled Dad
Subject: Absolute value

My child was recently given this homework problem in absolute values:
ABS(x + 1) = ABS(x - 1), solve for x.  I thought I knew how to solve
these problems, but now I'm not sure.  It seems x = 0 is the only
solution since ABS(+1) = ABS(-1).  But I do not know how to begin to
set this problem up in algebra form. 

If the problem were ABS(x+1) = 1, then I would setup the problem as 
follows:

(x + 1) = 1  or  -(x + 1) = 1   
      x = 0        x + 1  = -1 
                       x  = -2

But having variables on both sides is more confusing.  Can you help?

Date: 01/12/2004 at 20:29:14
From: Doctor Schwa
Subject: Re: Absolute value

Hi Baffled Dad -

You have the right idea!  To extend your thinking, if abs(some stuff)
= abs(other stuff), there are two possibilities:

  (some stuff) = (other stuff)   OR   (some stuff) = -(other stuff)

So, for abs(x + 1) = abs(x - 1), either

  x + 1 = x - 1    OR    x + 1 = -(x - 1)
      1 = -1             x + 1 = -x + 1
    Impossible              2x = 0
    No solution              x = 0

This confirms your thought that x = 0 is the only possible solution. 
Note that there will not always be an impossible side--usually each
statement will yield distinct solutions.

Finally, what happens if we start the other way?

-(x + 1) = x - 1   OR   x + 1 = x - 1
  -x - 1 = x - 1            1 = -1
       0 = 2x              Impossible
       0 = x               No solution

As you can see, the overall result is the same.  Thus, you can choose
whichever initial statement you like and you don't have to consider
the other way around.

Enjoy,

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