Absolute Values : if x<0

I have replied on the other thread

https://www.beatthegmat.com/gmat-prep-qu ... 36977.html

Thanks for the response dumb.doofus, I appreciate it but I can't follow your logic. You are making some 'logic jumps' that I don't understand.

[quote="dumb.doofus"]
Let's say x = -2
so for x < 0, |x| = -x

therefore root(-x|x|) = root(-(-2)(-(-2)) = root(4) = 2

So root(-x|x|) = 2

but x = -2, so therefore root(-x|x|) = 2 = -x and that's the answer..

[/quote="dumb.doofus"]

1) for x <0, |x| = -x ?
Aren't absolute values always greater than 0?
Therefore how is |x| = -x ?
I understand |-x| = x and |x| = x but not |x| = -x.

2) All of sudden there seems to be an additional sign appearing in your root equation.

[quote="dumb.doofus"]root(-(-2)(-(-2))[/quote="dumb.doofus"]

I think the usage of brackets is causing confusion.

How do you get |-2| = (-(-2))?
Do you mean |-2| = (-|-2|)?
Or do you mean |-2| = |-2| = 2?

3) Can you please elaborate on the step:

[quote="dumb.doofus"]
but x = -2, so therefore root(-x|x|) = 2 = -x and that's the answer.[/quote="dumb.doofus"]

You've just proved that when x = -2 the expression sqrt(-x|x|) = 2.
Then you say, BUT x=-2 (which we've just evaluated), so therefore its something else? Can you please explicitly state the logic that has lead you to this conclusion.

Thanks.

Alrighty Wally.. I'll explain to you everything one by one...

1. The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?", not "in which direction?". This means not only that | 3 | = 3, because 3 is three units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.
When x > 0, |x| = x, because x is positive.. and therefore that is the distance from 0
When x < 0, |x| = -x.. because x is negative and we want a positive distance... let's say x is -2, so by the above equation.. |x| = -x, since x = -2, the equation becomes, |x| = -(-2) = 2 which is the distance from 0.
You can read more about it: https://en.wikipedia.org/wiki/Absolute_value

2. Here x = -2 so |x| = -x and so -(-2)..

3. After solving the equation, we found that sqrt(-x|x|) = 2, but we already know that x = -2 or I can write this equation after multiplying by a minus sign as -x = 2..
and that's what I have written.. since the answer is 2.. that is equal to -x

Hope this clarifies..