Is x^4 + y^4 > z^4?
(1) x^2 + y^2 > z^2
(2) x+y > z
I am terrible at inequalities. Please explain why E is the correct answer. Thank you!
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- divya
Is x^4 + y^4 > z^4?
(1) x^2 + y^2 > z^2
(2) x+y > z
1. x^2 + y^2 > z^2 -- > Eq 1
we can square both sides:
(x^2 + y^2) ^ 2 > ( z^2 ) 2
-- > x^4 + y^4 + 2x^2y^2 > z^4
however we do not know the value of x and y, thus we cannot determine whether x^4 + y^4 > z^4? -- > Insufficient
2. x + y > z , again this doesn't tell us anything much, if we square and then square again ( i.e raise (( x+ y) ^ 2) ^ 2 ) it will give us the combinations/variables of xy as well. Thus insufficient
Adding both 1, & 2 together -- > doesn't give any additional info.
Therefore answer E.
(1) x^2 + y^2 > z^2
(2) x+y > z
1. x^2 + y^2 > z^2 -- > Eq 1
we can square both sides:
(x^2 + y^2) ^ 2 > ( z^2 ) 2
-- > x^4 + y^4 + 2x^2y^2 > z^4
however we do not know the value of x and y, thus we cannot determine whether x^4 + y^4 > z^4? -- > Insufficient
2. x + y > z , again this doesn't tell us anything much, if we square and then square again ( i.e raise (( x+ y) ^ 2) ^ 2 ) it will give us the combinations/variables of xy as well. Thus insufficient
Adding both 1, & 2 together -- > doesn't give any additional info.
Therefore answer E.
- RonPurewal
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in my learned opinion, this is one of the most difficult inequality problems that gmatprep has put out there in some time, so you definitely shouldn't feel bad about tanking it.
along the way, we're going to learn 2 VERY important takeaways about data sufficiency number plugging. in fact, the first takeaway is so important that i'll state it 3 times.
here it is for the first time:
takeaway #1: when you plug numbers on a DS problem, YOUR GOAL IS TO PROVE THAT THE STATEMENT IS INSUFFICIENT.
therefore, as soon as you get a 'yes' answer, you should be TRYING to get a 'no' answer to go along with it; and, as soon as you get a 'no' answer, you should be TRYING to get a 'yes' answer to go along with it.
--
statement (2)
you need to pick numbers such that x + y > z, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0.
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x + y > z.
fortunately, this is somewhat simple to do: just make z a big negative number.
try x = 1, y = 1, z = -100
in this case, x + y > z (satisfying statement two), but x^4 + y^4 is clearly less than z^4, so, NO to the prompt question.
insufficient.
--
statement (1)
you need to pick numbers such that x^2 + y^2 > z^2, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0 (the same set of numbers we picked last time).
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x^2 + y^2 > z^2.
unfortunately, this isn't as easy to do as it was last time; we can't just make z a huge negative number, because z^2 would then still be a giant positive number (thwarting our efforts at obeying the criterion).
so, we have to finesse this one a bit, but the deal is still to make z as big as possible while still obeying the criterion.
let's let x and y randomly be 3 and 3.
then x^2 + y^2 = 18. we need z^2 to be less than this, but still as big as possible. so let's let z = 4 (so that z^2 = 16, which is pretty close).**
with these numbers, x^4 + y^4 = 162, which is much less than z^4 = 256. therefore, NO to the prompt question, so, insufficient.
answer = e.
--
by the way, you may have noticed that divya didn't get the algebra to work, so she just tossed her electronic hands in the air and said 'i give up'.
now, clearly, NOT FIGURING OUT the algebra doesn't PROVE that a statement is insufficient, but, whether intentionally or not, divya is onto something here. specifically:
takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.
since the algebra just doesn't work out - especially for a student as strong as divya (she has posted some pretty amazing stuff on other threads) - you should have a strong inclination to think that the statements are insufficient.
and you'd be right.
along the way, we're going to learn 2 VERY important takeaways about data sufficiency number plugging. in fact, the first takeaway is so important that i'll state it 3 times.
here it is for the first time:
takeaway #1: when you plug numbers on a DS problem, YOUR GOAL IS TO PROVE THAT THE STATEMENT IS INSUFFICIENT.
therefore, as soon as you get a 'yes' answer, you should be TRYING to get a 'no' answer to go along with it; and, as soon as you get a 'no' answer, you should be TRYING to get a 'yes' answer to go along with it.
--
statement (2)
you need to pick numbers such that x + y > z, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0.
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x + y > z.
fortunately, this is somewhat simple to do: just make z a big negative number.
try x = 1, y = 1, z = -100
in this case, x + y > z (satisfying statement two), but x^4 + y^4 is clearly less than z^4, so, NO to the prompt question.
insufficient.
--
statement (1)
you need to pick numbers such that x^2 + y^2 > z^2, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0 (the same set of numbers we picked last time).
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x^2 + y^2 > z^2.
unfortunately, this isn't as easy to do as it was last time; we can't just make z a huge negative number, because z^2 would then still be a giant positive number (thwarting our efforts at obeying the criterion).
so, we have to finesse this one a bit, but the deal is still to make z as big as possible while still obeying the criterion.
let's let x and y randomly be 3 and 3.
then x^2 + y^2 = 18. we need z^2 to be less than this, but still as big as possible. so let's let z = 4 (so that z^2 = 16, which is pretty close).**
with these numbers, x^4 + y^4 = 162, which is much less than z^4 = 256. therefore, NO to the prompt question, so, insufficient.
answer = e.
--
by the way, you may have noticed that divya didn't get the algebra to work, so she just tossed her electronic hands in the air and said 'i give up'.
now, clearly, NOT FIGURING OUT the algebra doesn't PROVE that a statement is insufficient, but, whether intentionally or not, divya is onto something here. specifically:
takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.
since the algebra just doesn't work out - especially for a student as strong as divya (she has posted some pretty amazing stuff on other threads) - you should have a strong inclination to think that the statements are insufficient.
and you'd be right.
- RonPurewal
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mbarshaik Wrote:My approach to these kind of questions is to jump to the Values between 0 and 1.
Whenever we see a Exponent with a inequality and nothing is mentioned about teh variables then put a value between 0 and 1.
Thats what i did for this question. Take x = 1/2 and y = 1/2 this makes life easy.
that's not a bad approach. numbers between 0 and 1 have qualitatively different behavior when raised to powers, so you may discover exceptions to patterns more readily with this method.
however, i'm interested in your approach here - "take x = 1/2 and y = 1/2." does this generate some sort of obvious solution to the problem? for this particular problem, it seems as though you still need to choose numbers somewhat cleverly to realize that statement #1 is insufficient.
when you let x and y = 1/2, what are your choice(s) for z?
- RonPurewal
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let me say the following:
this problem should serve as a nice wake-up call to any and all students who don't like "plug-in methods", or who abjure such methods so that they can keep searching ... and searching ... and searching for the elusive "textbook" method.
this problem is pretty much ONLY soluble with plug-in methods. therefore, you MUST make plug-in methods part of your arsenal if you want a fighting chance at all quant problems you'll see.
this is the case for a great many difficult inequality problems, by the way: the most difficult among those problems will often require some sort of plug-ins, or, at the very least, they will be hell on earth if you try to use theory.
this problem should serve as a nice wake-up call to any and all students who don't like "plug-in methods", or who abjure such methods so that they can keep searching ... and searching ... and searching for the elusive "textbook" method.
this problem is pretty much ONLY soluble with plug-in methods. therefore, you MUST make plug-in methods part of your arsenal if you want a fighting chance at all quant problems you'll see.
this is the case for a great many difficult inequality problems, by the way: the most difficult among those problems will often require some sort of plug-ins, or, at the very least, they will be hell on earth if you try to use theory.
- kylo
i think there is a much simpler approach for this problem -
stat1 - x^2 + y^2 = z^2
squaring & solving we get -
x^4 + y^4 > z^4 - 2*x^2*y^2
now there are 2 possibities -
1) x^4 + y^4 > z^4 & this condition satisfies the above equation.
2) z^4 - 2*x^2*y^2 < x^4 + y^4 < z^4 & this condition satisfies the above equation.
similarly we can derive the same logic from stat2.
hence IMO E.
Thanks!
stat1 - x^2 + y^2 = z^2
squaring & solving we get -
x^4 + y^4 > z^4 - 2*x^2*y^2
now there are 2 possibities -
1) x^4 + y^4 > z^4 & this condition satisfies the above equation.
2) z^4 - 2*x^2*y^2 < x^4 + y^4 < z^4 & this condition satisfies the above equation.
similarly we can derive the same logic from stat2.
hence IMO E.
Thanks!
- RonPurewal
- Students
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interesting solution.
um, don't you mean that the second one doesn't satisfy the equation? that, after all, would be the genesis of your decision that the statement is insufficient.
this solution does work, although i think you're severely underrating the difficulty of coming up with it (or something like it).
--
that would require you to square the equation twice, resulting in an absolute mess of terms.
you can probably still write cases the same way, though.
--
this is one of those problems for which i really wish i could see an official solution. i'd wager that the official solution would merely proffer numbers that prove the statements insufficient, without providing any sort of explanation or justification for those choices of numbers.
in other words, numbers like the ones in my post above, but absent the derivation of those numbers. (there are several problems in the official guide that do this - they just pull numbers out of thin air to prove a statement insufficient).
kylo Wrote:1) x^4 + y^4 > z^4 & this condition satisfies the above equation.
2) z^4 - 2*x^2*y^2 < x^4 + y^4 < z^4 & this condition satisfies the above equation.
um, don't you mean that the second one doesn't satisfy the equation? that, after all, would be the genesis of your decision that the statement is insufficient.
this solution does work, although i think you're severely underrating the difficulty of coming up with it (or something like it).
--
similarly we can derive the same logic from stat2.
that would require you to square the equation twice, resulting in an absolute mess of terms.
you can probably still write cases the same way, though.
--
this is one of those problems for which i really wish i could see an official solution. i'd wager that the official solution would merely proffer numbers that prove the statements insufficient, without providing any sort of explanation or justification for those choices of numbers.
in other words, numbers like the ones in my post above, but absent the derivation of those numbers. (there are several problems in the official guide that do this - they just pull numbers out of thin air to prove a statement insufficient).
- RonPurewal
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kylo Wrote:the above mentioned solution might be useful for such type of questions when u have less time to spent in actual GMAT.
I suppose sometimes u have to be a bit creative & make a calculated guess.
Thanks!
no, don't get me wrong, it's a great solution.
it's just very abstract, and is thus beyond the purview of many students (for whom number picking, by contrast, is still quite accessible).
- kevin
kylo Wrote:i think there is a much simpler approach for this problem -
stat1 - x^2 + y^2 = z^2
squaring & solving we get -
x^4 + y^4 > z^4 - 2*x^2*y^2
now there are 2 possibities -
1) x^4 + y^4 > z^4 & this condition satisfies the above equation.
2) z^4 - 2*x^2*y^2 < x^4 + y^4 < z^4 & this condition satisfies the above equation.
similarly we can derive the same logic from stat2.
hence IMO E.
Thanks!
Great solution! Note that (1) tells us more than (2), so if (1) is insufficient, the answer is E
- RonPurewal
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kevin Wrote:Note that (1) tells us more than (2)
not really. there are triples of numbers satisfying (1) but not (2), such as (x, y, z) = (-3, -3, 4).
such triples all include negative numbers, but, still, they do exist.
if you assume that x, y, and z are all positive, then, yes, the numbers satisfying (1) are a subset of the numbers satisfying (2), and so (1) insufficient would imply (e).
luckily, you can get away with that sort of assumption here, because the answer to the problem is (e) even if you limit yourself to the universe of positive numbers. however, in general, making unwarranted assumptions that numbers are positive, without justification, can destroy you.
be careful.