Question 647364: Hi, I'm in a hybrid math class at my college, so i don't get to see my teacher for another week and i have no idea how to do this problem. Can you please help me?
I don't understand the "K" rule:
K={K if K>(or equal to)0
{-K if K< 0
I have an example from my book here: (I am just having a hard time with this because my book does not explain anything.)
absol. (x-2)^2 absol.
I thought you would do this: (x-2)(x-2) and multiply: x^2+4
The answer in the book says (x-2)^2 is the answer along with: absol.K absol.=K if K> (or equal to) 0
I dont understand how to find out what K rule to use if there is no real number value.
Thank you,
Sydney
Found 2 solutions by ewatrrr, Edwin McCravy:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website!
Hi,
the idea is by definition: absolute value |number| ALWAYS has an 'absolutely' ≥ 0 result
regardless if the number is positive or negative
for ex: |4| = 4 and |-4| = 4
As to:
|K| = K, that would only be true if K itself were ≥ 0
|(x-2)^2| = (x-2)^2 | quanity (x-2) 'squared' is ALWAYS ≥ 0
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
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
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