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Num Prop - concept explanation through question required

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by kaushikkumar1987 » Sat Jun 18, 2011 6:02 am
Q. When is |x-4| equal to 4-x?

It is very obvious that x-4 is 4-x when we multiply the first with -1. But correct me if I am wrong |x-4| will be equal to (x-4) isn't it?

The explanation to this question states - 'Absolute brackets can only do one of two things to the expression inside of them: (a) leave the expression unchanged, whenever the expression is 0 or positive or (b) change the sign of whole expression whenever the expression is 0 or negative.'

Can someone please explain the above in detail.

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Kaushik K
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by Frankenstein » Sat Jun 18, 2011 6:09 am
Hi,
Rule: |x| = x if x>=0
|x| = -x if x<=0
When x=0, |x| = 0. So, it doesn't really matter if it is x or -x.
Coming to this problem: |x-4| = 4-x
This is of the form |x| = -x.
So, (x-4)<=0
So, x<=4
Cheers!

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by PencilNerd » Sat Jun 25, 2011 11:54 am
Absolute value always yields a positive result. Therefore if the expression in the brackets actually yields a negative result, absolute value forces you to multiply by -1 to get a positive answer. In such a case, (x - 4) inside the brackets becomes (4 - x) when you take the absolute value, because you have to multiply by -1.

In your example,
|x - 4| let's say x is actually 1.
thus, |x - 4| = |1 - 4| = |-3| = 3.
so when x = 1, |x - 4| = 3. As you can see, this is an example where
|x - 4| was equal to 4 - x.

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