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Absolute Caution

Date: 09/28/2012 at 13:43:35
From: Mark
Subject: Distributing into an absolute value exoression.

In your response to a math teacher about whether absolute value symbols
should be treated as grouping symbols, you said that it is possible to
distribute into them.

I've always advised my students not to, since if the number they
distribute is negative, that will surely change the sign of the answer.

For example, this should have no solution at all:

   -2|x + 5| = 12

But if you distribute the -2 into the absolute value expression, you get 

   |-2x - 10| = 12 

This leads to the solutions of 

   -2x - 10 = 12,    x = -11
   -2x - 10 = -12,   x = 1
But neither of these is a solution to the original equation, so the
mistake was distributing the -2.

Of course if you distribute only the 2 and leave the "opposite sign" in
front of the absolute values symbols, all is well.

In a basic algebra class I prefer to tell them not to mix any part of the
absolute value expression with any part of the equation outside of it
until they've put the equation into the form |expression| = C. (We don't
try to solve any with the variable both inside and outside the absolute
value expression.)

I think I will stop telling them that they may not distribute into
absolute value symbols though, as a result of reading your discussion.


Date: 09/28/2012 at 15:48:49
From: Doctor Peterson
Subject: Re: Distributing into an absolute value exoression.

Hi, Mark.

I think you're referring to this page:

   Absolute Value as a Grouping Symbol? 

I didn't go into detail there about distributing into an absolute value; I
had said a little more about it here:

   Distribution and Absolute Values 

It's important to give those details:

   If a, b, and c are real numbers, and a > 0, then
     a |b + c| = |ab + ac|

This is really a combination of the distributive property of
multiplication over addition, and the multiplication property of absolute
value, which I mentioned on the latter page. To wit:

   If a and b are real numbers, then

     |a| |b| = |ab|

This lets us say

   a |b + c| = |a(b + c)|     (because a = |a|, since a>0)
             = |ab + ac|

So in teaching, I would probably not teach simply that you can distribute
into an absolute value, but rather that you can introduce a POSITIVE
multiplier inside an absolute value, and distribute there.

Keeping the ideas separate makes it less likely that a student would make
the mistake of "distributing" a negative number.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Basic Algebra

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