Why Some Solutions to Absolute Value Equations are False

Date: 10/24/2004 at 19:09:23
From: Dan
Subject: Why are some values rejected when solve absolute value eq's?

Hi there,

I'm curious as to why some values are rejected when solving absolute
value equations.  I understand that this only occurs when you have
checked your solution and one of the values turns out to be false, but
my question is WHY the equation can yield false values.

An example of this would be the following question:

  |4x+3| + 21 = 5x

which yields the possible x-values of 24 and 2.  When you check those
values, 2 turns out to not work.  WHY?  Any help as to why this can
happen would be greatly appreciated.

Date: 10/24/2004 at 21:30:04
From: Doctor Rick
Subject: Re: Why are some values rejected when solve absolute value eq's?

Hi, Dan.

Let's trace through the solution and then trace through what happens 
when we plug in the spurious solution.  First, my solution:

  |4x+3| + 21 = 5x
  |4x+3| = 5x-21
    4x+3 = 5x-21   OR  -(4x+3) = 5x-21
    21+3 = 5x-4x         -4x-3 = 5x-21
      24 = x             -3+21 = 5x+4x
                            18 = 9x
                             2 = x

So we have two (tentative) solutions: x = 24 or 2.  Now we'll plug 
each solution into the original equation and evaluate:

  |4x+3| + 21 = 5x         |4x+3| + 21 = 5x
  |4*24+3| + 21 = 5*24     |4*2+3| + 21 = 5*2
  |99| + 21 = 120          |11| + 21 = 10
  99 + 21 = 120            11 + 21 = 10
  120 = 120                32 = 10
  CORRECT                  WRONG

Why did the second solution fail?  It would have worked if the 
absolute value of 11 were -11; then the third line would read

  -11 + 21 = 10

which is correct.  The problem arose because the absolute value of a 
quantity is the negative of that quantity ONLY IF the quantity is 
negative.  In other words, a more formal definition of absolute value is:

        { x if x > 0
  |x| = { 0 if x = 0
        {-x if x < 0

This was not taken into account in my solution method.  In the third
line of the solution, I said that the absolute value could be either
the quantity itself, 4x+3, or its negative, -(4x+3).  I didn't check 
at that stage whether the quantity (4x+3) was negative or positive.  I
hold off this check until after I have found the candidate solutions.
This is why the testing of the candidate solutions is an integral part
of the solution process, and not just a double-check on my work.

Does this help?

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