Is a>0? Need help with this inequality?

[abhi.skyisthelimit]

Hello Ron,

Here goes my first post to you. First of all, I am sorry in case I have violated, any guidelines for posting question in general math questions category. Finally, I have realized that inequalities with absolute values give me jitters and I am still trying my best to get better at solving inequalities. I saw this question on a forum other day, and tried to solve it. I got it wrong, but the authors reply was not convincing either so I thought of coming to you for a better explanation. Some people commented that it is a OG question, but I did not really find it in OG12. So, Here it goes:

Is a > 0 ?

(1) a^3 - a < 0
(2) 1 - a^2 > 0

This is how I approached this problem. Please explain where I made mistake and where I am lacking in my concept of inequalities related to this problem.

Stat 1:

(a^3-a)<0
a(a^2-1)<0
a<0 or (a^2-1)<0
a<0 or a^2<1
a<0; -1<a<1
Since a can be both -ve and +ve : Not sufficient.

Stat 2:

1 - a^2 > 0
multiply both sides with -1.
(a^2-1)<0
a^2<1
-1<a<1
Since a can be both -ve and +ve : Not sufficient.
Combining 1 and 2
-1<a<1
Since a can still be both -ve and +ve : Not sufficient.
Answer E!

The author closed in on C as the correct answer, but couple of people comments as D to be correct answer. I know I am lacking few nuances about the inequalities. Please recommend how to get strong on them?

[RonPurewal]

hi, welcome to the forum.

sorry, but you must post the ORIGINAL SOURCE of the problem when you post it here. (this means the source of original authorship; it's not good enough to report that you saw it on another forum or other such secondhand source.)
if you don't know the original source of the problem, then we can't host it here; copyright issues.

please read the forum rules -- first thread in this folder, and in every other folder on the forum. all these things are explained in there.

please post the original source of the problem within the next week, or we'll have to kill the thread.
thanks.

[abhi.skyisthelimit]

Hello Ron,

Much thanks for your prompt reply. I found this problem yesterday on http://www.beatthegmat.com. Following is the link to the problem page:

http://www.beatthegmat.com/mba/2013/07/21/understanding-powers-the-theoretical-advantage

I am sorry for not providing the exact source of the problem earlier.

This is all I know about this problem, and I hope I have not violated any copyright issues.

Please let me know if now it is OK, and if you would be able to help me with this problem now!

Regards,
Abhishek

[RonPurewal]

well, technically, we should still kill the thread, because that's still not an original source. in fact, the author makes it explicit that it's not an original source ("i saw this problem on a forum").

but, since there is nothing particularly unique about the inequalities here, i suppose we can make an exception.

by the way, please write out the words "positive" and "negative".
i personally find "+ve" and "-ve" not only annoying, but also much more difficult to read. i'm sure that others feel the same. please take the extra second or two to type these words out in full; thank you.

Stat 1:

(a^3-a)<0
a(a^2-1)<0
a<0 or (a^2-1)<0
a<0 or a^2<1
a<0; -1<a<1
Since a can be both -ve and +ve : Not sufficient.

this doesn't work.
first, i'm curious why you didn't break x^2 - 1 into (x - 1)(x + 1); that may have helped.

the reason why this doesn't work is that a product is only negative if EXACTLY ONE of the two things is negative. so, (a)(a^2 - 1) is only negative if EXACTLY ONE of (a) and (a^2 - 1) is negative.

if 0 < a < 1, then that happens, since (a) is positive but (a^2 - 1) is negative.
on the other hand, if -1 < a < 0, then BOTH of these expressions are negative, and so the product is positive.
if a < -1, though, (a^2 - 1) becomes positive while (a) remains negative, and so the inequality works again.

so, the actual solution here is either a < -1 or 0 < a < 1. that's insufficient, but it doesn't work the way you have here.

Stat 2:

1 - a^2 > 0
multiply both sides with -1.
(a^2-1)<0
a^2<1
-1<a<1

this is fine.

when you put these together, though, you'll notice that the only common values are 0 < x < 1. so, (c).

more to the point --
you shouldn't be so thoroughly dependent on mucking around with theoretical algebra stuff. if you think that negative values will work, TRY those values, and see whether they really do work!

try any negative number in the whole world. no matter what, one of the two statements (or both of them, if you happen to pick -1) will be false.

[RonPurewal]

also, when you get to using the 2 statements together, you can just take these factorings:
statement 1: (a)(a^2 - 1) is negative
statement 2: (a^2 - 1) is negative.
(you know the latter of these because a^2 - 1 is just the opposite of 1 - a^2.)

that's enough to settle the issue: if "a" times a negative number is still a negative number, then "a" must be positive. done.

[abhi.skyisthelimit]

Hello Ron,

Thanks a lot for such a valuable answer. I can clearly see what a silly assumption I made while solving this question. I solved it just like an equality, without realizing the fact that when product of two values is Negative, then they should have opposite sign. What I liked the most, is your second reply. That one could have solved in seconds, rather than doing all that math..

Great Insight!

Another thing, what I would like to ask specifically about data sufficiency questions is that almost everywhere, people say that you should first tackle both the statement individually, and then take them together, but I see in most of the problem, if you look at both the statement together, then you can get an idea about the fact, where GMAT is trying to trick you, or at least where the trap lies. I mean or the second statement can slightly indicate, what you should really consider, to make the first statement sufficient.

I do not know if I have correctly conveyed, what I want to say. But, in one of your Thursday with Ron, video you also recommended to sometimes look at both statements together, to see if you can find some insight about what exactly that question it trying to test.

Please Advise!

Regards

[tim]

You have the right idea in that while investigating one statement, looking at the other statement will often give you a clue as to what the GMAT is after. This is particularly useful when trying to decide how to create two scenarios that will give you different answers to the question at the top of the page. This does not mean that you are combining the statements though, and you should go to great lengths to avoid combining the statements until you have dealt with each one on its own.

[RonPurewal]

I do not know if I have correctly conveyed, what I want to say. But, in one of your Thursday with Ron, video you also recommended to sometimes look at both statements together, to see if you can find some insight about what exactly that question it trying to test.

Please Advise!

Regards

i always look at both statements, for a reason that's a bit less sophisticated: i want to know if either one of them is super easy to settle.
for instance, on OG12 DS problem #154 (can't reproduce here), i would most certainly kill statement 2 first -- quick elimination of (b) and (d) -- and then move on to the more beastly statement 1.

there's also the idea of just "putting your head up and looking around your environment".
e.g., if it's a problem where choice C is clearly a trap, then that's also something you can only notice by looking at both statements. (conveniently, the aforementioned #154 is also one of these problems.)
or, if you notice that one of the statements says something like "x is an integer" ... well, then, you'd darn well better think about non-integer values in the other statement.

in sum, i'd say this:
if you are still struggling with the fundaments of DS -- like, you don't remember which answer choice means what, or you don't have a firm handle on what "sufficient" and/or "not sufficient" means on every single problem -- then it's best to hold off on more advanced / more strategic modes of thinking, like these.
but, if you have the fundaments down cold, then adding another layer of strategy to your overall skill set can help you "level up", as it were.

[abhi.skyisthelimit]

Thanks Tim and Ron!

I am starting to look at both statements together now, but only to have an idea about what the question is trying to ask, or which statement is easy to handle first, so that its easy to make educated guess.

I think I am getting better, but inequality and absolute values still stays my weakness. Also, It happens a lot of time that every question can be easily done in 2 min or less as long as you can apply the right technique in first go. But, in many question (at least with me), it happens that I work on a question, and then later realize that this is not the best way. Then I change my approach and solve it some other way and can easily solve it in 2 min or less.

In real test, how to make sure you hit the question right in first go, and not waste time switching techniques. For example, Say I have spend 1.5 -2 min solving a hard question, and them realize that I did a mistake and figures out the right way, but by that time I have already lost my 2 min, and working on that problem will take more time. What is best at that time, to guess even if in know i can solve it, or solve it with another technique.

-Abhishek

[RonPurewal]

so, yeah, this whole "I have to solve EVERY PROBLEM in under two minutes" thing -- well... no.

don't forget, ladies and gentlemen, that "2 minutes" is an AVERAGE time!
not a time limit!
just think about how averages work: basically, about half of the problems should take 2 minutes or less ... and the other half should take more than 2 minutes.

so, the question you're asking here is probably a non-issue in the first place. it's perfectly ok to take over 2 minutes on problems, provided you're actually getting somewhere.
if you are genuinely, truly stuck on a problem, then you should quit right away, even if it has only been thirty seconds. ("quit" doesn't necessarily mean "immediately guess and leave the problem" -- first, you should think about whether there are any other approaches you could take.)

[kouranjelika]

Hey Ron,

I am also trying to sharpen my inequality with absolute value approaches and skills. It's a huge problem area for me. Not only do I often find myself deep in a rabbit hole, even when I first look at these questions, it's right away an "oh-oh" moment, which I def can't have on the test.
The sad thing is that I actually had no problem with the material in the Stategy Guide (sometimes the concepts there are a bit easy in comparison to what we face in the actual test) and generally algebra is my favorite subject. I really need to get to a place where I'll see these questions and actually get excited becuase I totally get it. As it stands now, I either spend way too long testing each case or try to solve it algebraicly and always get it wrong.

I've been watching your archival videos. Just finished one from Feb 18, 2010. You mention that these problems sometimes cannot be solved algebraicly at all. Do you recommend just thinking of it conceptually? Can you give me some insight on how you look at it when you first see a new one like this. May be you can also recommend other Archived study halls that you've done on these monster problems?

If I can explore the problem in this particular thread further as well (I tried solving it):

Is a>0?
(1) a^3-a<0
(2) 1-a^2>0

Stat 1:
yields a<-1 or 0>a>1 is this correct?
Stat 2:
yields a is either a positive or a negative fraction yes?

Therefore, putting the statements together, we know a must be a fraction (from statement 2), in which case it could only be a positive fraction (from statement 1). Hence the answer to the question would be satisfactory (Yes, a is indeed greater than 0 or simply is positive). So answer C.

If you don't mind please comment on my reasoning above. It is super awkward for me to do it like this (verbally), not through a clean algebra proof method (I know I can do that here too by plugging numbers..I think that would be the only way).

But as I am getting used to using this conceptual approach, which sort of is starting to click in and make sense, the other inequality stuff (the simpler kind) starts to totally not make sense anymore. Ie (source: Thursdays with Ron - Feb 18, 2010 - Slide 4; but really any made up example would do to showcase this):
Is |x|=y-z?
(1) x+y=z
(2) x<0

Ok so, mechanically I just jump into setting the two cases:
if x>0
is x=y-z?

if x<0
is -x=y-z

Stat 1:
y-z=-x; but we don't know if x<0; so Not Sufficient
Stat 2:
obv not sufficient, no info on y & z.
Together: we got all that we need, x<0 (stat 2) & if x<0 then y-z=-x (stat 1). And the answer is C, sufficent when we have both pieces of info together.

BUT THEN! I start to conceptually think about this mechanism that we apply. And I realize, I do it (and I did every drill on this and got practically all correct. I do the both cases, test them, etc), but I DON'T get how it is in fact true.
Look, if y-z=|x|; how could y-z EVER be equal to a negative x, even if x was negative. why would we have such a situation in the first place, when x will be absolute value and no matter it's sign the subtraction of the two other variables MUST be greater or equal to zero (zero only if x turns out to be zero). I mean how does that make any sense? And using this flawed reasoning one could conclude that, "hey, Stat 1 is sufficient." Dammit!
PLEASE find the flaw in my reasoning...it's literally driving me insane! I feel like I am getting worse at the GMAT...

[RonPurewal]

The sad thing is that I actually had no problem with the material in the Stategy Guide (sometimes the concepts there are a bit easy in comparison to what we face in the actual test) and generally algebra is my favorite subject.

In itself, this isn't surprising; the strategy guides aren't meant to operate like the official GMAT problems. The strategy guides are prerequisite material; once you've mastered that material, then you're ready to tackle the (more unusual) problems that show up on the test.

Here's the best analogy I have here:
The strategy-guide material is like practicing scales on the piano; the GMAT problems are like playing actual pieces of music.
You can see the point: Once you've mastered the scales"”i.e., once the scales are relatively easy for you, andy ou can consistently do them almost perfectly"”then you're ready to start playing actual pieces of music.

[RonPurewal]

I've been watching your archival videos. Just finished one from Feb 18, 2010. You mention that these problems sometimes cannot be solved algebraicly at all. Do you recommend just thinking of it conceptually?

My recommendation is "do whatever comes to mind".
If you see something that has a conceptual meaning for you, then think conceptually. If not, don't. (For instance, "|x| = y - z", from the problem you posted above, has no immediate conceptual meaning to me.)
More importantly, you should NEVER try to have one "primary" method for approaching these problems. The entire point of this test is that no method is going to work all the time; you're going to need flexibility.

May be you can also recommend other Archived study halls that you've done on these monster problems?

To answer this question, I would have to search the study-hall page in exactly the same way you would. (I don't remember the contents of previous study halls, especially if they are more than a few months old.)

[RonPurewal]

Is a>0?
(1) a^3-a<0
(2) 1-a^2>0

Stat 1:
yields a<-1 or 0>a>1 is this correct?
Stat 2:
yields a is either a positive or a negative fraction yes?

OK except for two things:
* Should say 0 < a < 1. (You wrote the signs backward; "0 > a > 1" is impossible.) I'm sure you understand that a is supposed to be between 0 and 1 here; you just didn't write it correctly. No big deal.
* In statement 2, in addition to the possibilities you've listed (which are correct), a = 0 is also a possibility. This turns out to have no impact on the problem at hand, but it could be disastrous on another problem.

Therefore, putting the statements together, we know a must be a fraction (from statement 2), in which case it could only be a positive fraction (from statement 1). Hence the answer to the question would be satisfactory (Yes, a is indeed greater than 0 or simply is positive). So answer C.

Yes.

[RonPurewal]

Look, if y-z=|x|; how could y-z EVER be equal to a negative x, even if x was negative. why would we have such a situation in the first place, when x will be absolute value and no matter it's sign the subtraction of the two other variables MUST be greater or equal to zero (zero only if x turns out to be zero). I mean how does that make any sense? And using this flawed reasoning one could conclude that, "hey, Stat 1 is sufficient." Dammit!

I think you're assuming that "-x" is never positive. That's incorrect; if x is negative, then -x is positive. (For instance, if x is -14, then -x is 14.)
That seems to be the only issue here.
(Incidentally, if you spend too much time studying "advanced" math, you'll be more likely to make mistakes like this one. The GMAT tests math that's pretty close to the ground.)