Distribution and Absolute Values

Date: 09/22/1999 at 20:39:45
From: Lea Ann Smith
Subject: Distribution through an Absolute Value Sign

I am a first year teacher in an Algebra 2 class. I am trying to 
determine if it is okay to distribute through an absolute value sign. 
I was told that I need to isolate the absolute value sign prior to 
solving the problem, but I am having a hard time convincing myself 
(and my class) that I can't distribute through the absolute value 
sign. For example, the following problem gives the same answer two 
different ways:

Method 1

     2|3x+1| = 10
      |3x+1| =  5

        3x+1 = 5     or   -(3x+1) =  5
             :               3x+1 = -5
          3x = 4               3x = -6
           x = 4/3              x = -2

Method 2

     2|3x+1| = 10
      |6x+2| = 10

        6x+2 = 10    or   -(6x+2) =  10
             :               6x+2 = -10
          6x =  8              6x = -12
           x =  4/3             x = -2

Is distributing through the absolute value sign a generally applicable 
method or will it only work in special cases?

Thank you,
Lea Ann Smith

Date: 09/22/1999 at 23:21:41
From: Doctor Peterson
Subject: Re: Distribution through an Absolute Value Sign

Hi, Lea Ann.

>Is distributing through the absolute value sign a generally 
>applicable method or will it only work in special cases?

Neither. It won't always work, but it's not only for special cases: it 
will work half the time.

Specifically, the case you have to watch out for is distributing a 
negative number. The correct "distributive" property for absolute 
values is

     |a|*|b| = |a*b|
not
       a*|b| = |a*b|

because in the latter if a were negative, you would have a negative on 
the left and a positive on the right. You can say that the latter 
statement is true, however, as long as a is positive, and that is the 
rule you were using.

In your example (and probably in all examples you would have thought 
of, because you're aware of the rule subconsciously), the multiplier 
is positive and you can distribute.

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

Date: 09/23/1999 at 15:31:13
From: Lea Ann Smith
Subject: Re: Distribution through an Absolute Value Sign

Thank you so much for your prompt and logical response to my question. 
I showed your response in class today and it was a huge help, both 
from a math standpoint and from a technology/Internet standpoint.

Regards from a happy teacher,
Lea Ann

Date: 09/23/1999 at 17:08:10
From: Doctor Peterson
Subject: Re: Distribution through an Absolute Value Sign

Hi, Lea Ann.

I hope you caught my little mistake.

I used your term to describe the property I explained, but it's really 
not a distributive property at all; that's just how you were using it, 
with an addition inside the absolute value. A distributive property 
involves two binary operations like multiplication and addition, 
whereas absolute value is a unary operator or function. I'm not sure 
if there's a name for this property, but I'd just call it the 
multiplicative property of absolute value. If we write absolute value 
as a function f, it would say:

     f(a * b) = f(a) * f(b)

I suspect we don't talk about this property enough; if we did, there 
would be a name for it.

Anyway, I'm glad I could help!

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