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### 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)

would be a name for it.

Anyway, I'm glad I could help!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra

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