Question 647364
<pre>
Absolute value bars either make negatives into positives or keep positives
positive.

Rule for getting rid of the absolute value bars | | around a number:

1. If the number inside the | | is positive, simply erase the | |.
2. If the number inside the | | is negative, change the number to 
   positive and erase the absolute value bars.
3. The absolute value of zero is zero, |0| = 0

Examples:

 |3| = 3 because 3 is already positive so you just erase the | |.

|-3| = 3 because -3 is negative and so you change the sign to positive
       and erase the | |.

|57| = 57 because 57 is already positive and you just erase the | |

|-234| = 234 because -234 is negative and so you change the sign to positive
       and erase the | |.

That's all there is to the rule.

The rule your teacher gave you amounts to the same thing.  Here is why:

First part:

|K|= K if K>(or equal to)0

"If K is greater than or equal 0" is the same as saying 
"K is positive or zero", and so you just erase the absolute value 
bars around K

The second part is what seems different but isn't.  It's confusing because 
often a positive letter stands for a negative number. That's what K is 
in this part of the rule. K is a positive letter standing for a negative
number.

|K| = -K if K < 0

That says if POSITIVE K stands for a NEGATIVE number (less than 0), then
to make it positive you will need to put another negative sign before that
negative number that K stands for already has so that the negative sign that 
K already has will multiply by the negative sign that you put in front of
whatever negative number K stands for and that negative sign will cause the
result to be a positive number.

I agree that's very confusing!  So just remember that the absolute value bars
just say "If it's positive, leave it positive, and if it's negative,
make it positive".

-------------------------------

Let's look at some examples with letters:

Ex: Simplify |x-3| if possible

No absolute values are ever negative. You cannot remove the absolute value 
bars because -- since you cannot know what x is, you cannot tell whether x-3
is positive or negative.  For instance if x were 1, x-3 would be -2, a
negative number.  So you cannot remove the absolute value bars. 
So it is not possible to simplify |x-3|. You cannot remove the bars.

Ex: Simplify |3y+12| if possible.

You can simplify by taking out a positive 3

       3|y+4|

but you have to leave the absolute value bars because if y were equal
to, say, -9, then y+4 would be negative.  So you couldn't simplify it 
further.

Now let's look at the example in your book:

Simplify |(x-2)²| if possible.

Notice that (x-2) is SQUARED.  When you square a POSITIVE number you
get a POSITIVE number.  But also when you SQUARE a NEGATIVE number you
ALSO get a POSITIVE number!   Therefore in the case of (x-2)² you
will NEVER get a negative number, regardless of what x is. That's because
even if (x-2) is negative, it will then get squared and the answer will be positive.  Therefore you can simplify by removing the absolute value bars
because (x-2)² will be positive regardless of whether (x-2) is positive or negative.

That's why the absolute value bars are not necessary around something
SQUARED.

That's why |(x-2)²| can be simplified by removing the absolute values,
getting just (x-2)² without absolute value bars.

To recap:

You cannot remove that absolute values for |x-2| because what's inside might
be negative.  However you CAN remove the absolute value bars for |(x-2)²|
because what's inside cannot be negative.

I hope this clears it up a little for you.  Don't worry about the "K".  Just
think of absolute value bars as "positive makers or keepers".

Edwin</pre>