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GMATPrep Question

by student22 » Mon Mar 22, 2010 1:45 pm
I'm having a lot of trouble for some reason wrapping my mind around this question. Here is my reasoning and steps on solving it

if x < 0 Let x = -5 --> |-5| * -(-5) = 5 * 5 = 25. The square root of 25 is 5.

To prove it, I also entered the same thing into Excel:

=SQRT(-(-5)*ABS(-5)) --> Answer = 5.

Why is the official answer -x? I thought that it was IMPOSSIBLE for a square root to have a negative answer except with imaginary numbers and similar voodoo math.

If anyone can explain this for me step by step, I'd really appreciate it.



Also, I know that it's been discussed in this thread before. Sorry for reposting it, but I can't grasp any of the solutions:
https://www.beatthegmat.com/if-x-0-then- ... 44325.html


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by stephen@knewton » Mon Mar 22, 2010 2:47 pm
Hi. Let me see if I can shed some light. This is a really interesting problem ...

First and foremost, you are SO close! Ironically, your strategy of plugging in -5 has produced/confirmed the correct answer. And you are also correct in stating that (in the universe of GMAT problem solving) radicals / square roots are to be considered positive.

Now, simply recall that if X = -5, then 5 in fact DOES equal -X!

(And by that I mean to display my enthusiasm for math, not X factorial, by the way ...)

So yes, on the GMAT we should indeed consider radicals to be positive, but -X in this case DOES represent a postive number! Just make sure you don't get thrown off by the negative sign attached to the variable, and you are home free. In other words, a variable with a negative sign is simply the number equally far from zero with the opposite sign. It does NOT tell you the actual sign of X.

Hope that helps!

And while I must HIGHLY endorse the plugging-in approach that the OP has introduced below, for anyone else looking for a purely number properties based approach, you CAN tackle the problem this way:

1) When X is negative, and ONLY when X is negative, -X is equal to |X|
2) Thus the problem is really asking for the square root of |X|*|X|
3) In the GMAT universe, that will have a positive solution of |X|
4) Just as we saw in the first step, that means that the answer is -X

Lastly, while we shouldn't dwell on imaginary numbers and other "voodoo" (I love that by the way, might use it in class) let's clarify exactly what we know about positive and negative roots:

Anything squared will be positive, so we cannot have a negative SQUARE. Thus the square root of a negative number is imaginary, and that's where the voodoo comes in. But remember, a positive number DOES have a negative square root (whether it's an integer or not), it's just that on the GMAT we treat "radical 25" only as the positive root, ie 5. That makes problem solving easier, and it's a good rule to know on PS, but remember: if you see X^2 = 25 in DATA SUFFICIENCY, there are positive AND negative solutions for X.

I think that about covers it ... hope this helps ... cool question :)

Cheers,
Steve P.
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by kstv » Mon Mar 22, 2010 4:57 pm
If x < 0, √-x |x| is
|x| = - x , + x
let take + x Step A
√-x |x| = √-x * + x = √-x²
In GMAT only real no. are to be considered
Step A should not have been considered in the first place as it is given x < 0 in the Q stem
so √-x |x| for -x = √-x * -x = √x²
the values are + x or - x
again only x < 0 is to be considered
the answer is -x or Option 1

I hope this is correct interpretation of stephen@knewton's advice. Also, the Q specifies to consider only the cases where x < 0.
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by student22 » Mon Mar 22, 2010 8:40 pm
stephen@knewton thanks for the very thorough response, especially by explaining it through number properties. Honestly, I tried this problem with the plugging in approach because I couldn't think of a more abstract way to solve it. Especially since on data sufficiency sufficiency problems you can sometimes get into trouble by relying on plugging in numbers.

I'll bookmark this thread to go over this problem a couple more times before the test just be sure I remember it.
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by stephen@knewton » Tue Mar 23, 2010 7:08 am
I'm very glad this was helpful!

Just don't shy away TOO much from the plugging-in strategy, even on DS. It worked on this problem, and can be very useful in cases where the conceptual approach is just out of reach. And remember, the GMAT is designed to make sure that you spend most of your time answering questions JUST outside of your comfort zone! So use number properties when you can, but definitely practice plugging-in, so you'll be comfortable using it when you need it.

To your comment about DS, let me say this: plugging in numbers is a great way to show INSUFFICIENCY, and it can also give you that 90% confidence of sufficiency that you need to answer and move on.

Good luck!

Steve
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by rockeyb » Tue Mar 23, 2010 7:59 am
An alternate approach :

Given : x< 0 and we need to find the value of SQRT( -x |x|)

since x < 0 that is x is - ve .

| x |will always be +ve .

So we get SQRT ( -x * x)

we can also write this as (-x ^2)^1/2

we are left with -x .
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by stephen@knewton » Tue Mar 23, 2010 9:14 am
Hi Rockeyb,

Thanks for adding your alternate approach. It's great to see so many people thinking through this question in different ways. Maybe just a couple of words of caution in response:

To recap, we are told that x is negative and asked to find the value of SQRT(-x |x|). The correct answer is -x.

One of the steps you've suggested goes as follows:

| x | will always be +ve .
So we get SQRT ( -x * x)

You are correct that |x| will always be a positive value. But that does NOT mean that you should drop the negative sign from X! Remember, when X is negative, -X is itself a positive value! In other words, when x is negative (and only then), |x| = -x.

So what we're actually going to see here is SQRT(-x * -x)

This seems to be the point that is causing the most trouble on this question, so try to remember that -x is simply the number as far away from zero on the number line as x is, but in the other direction. It's NOT necessarily a negative, and in this case it specifically is not!

(I'm also wondering if you simply forgot a negative sign there, because you return to the correct expression in your next step)

we can also write this as (-x ^2)^1/2

From that point, just be careful about what happens when you square something and then take the square root of that. This process tends to HIDE negative/positiveness. Because we assume radicals to be positive in problem solving, for negative values of X the expression above DOES yield -X. But for positive values, it's X. This problem only works because we DO know the sign of X, and that's why I really like the plugging in approach for this one.

Cheers, Steve
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by rockeyb » Tue Mar 23, 2010 9:38 am
Hey Steve ,

I was expecting this post from some one , certainly did not expect it from an Instructor :) .

The moment I wrote this post I was aware that I needed to consider the -ve case too .

Since |x| can either be +x OR -x .

But played along since it was given x<0 so was not going to effect my answer .

You are right this may cause trouble and we should always consider the -ve case when variables are used as you rightly said they HIDE THE SIGN .

Thanks again for correcting me :)

Regards ,

Pranab Banerjee.
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by student22 » Tue Mar 23, 2010 3:01 pm
stephen@knewton wrote: This seems to be the point that is causing the most trouble on this question, so try to remember that -x is simply the number as far away from zero on the number line as x is, but in the other direction. It's NOT necessarily a negative, and in this case it specifically is not!
Exactly, this is the part that I keep reminding myself about whenever I see an absolute value problem. That it's not really giving you a value, but rather the distance it is from 0.

So, |-8| is NOT really 8. Instead, it's distance from 0 is 8.

So, yeah, you're right that this is what confuses people about problems like this.
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