Is x^4 + y^4 > z^4? (1) x^2 + y^2 > z^2 (2) x + y > z

Is \(x^4+y^4>z^4\)?

The best way to deal with this problem is plugging numbers. Remember on DS questions when plugging numbers, goal is to prove that the statement is not sufficient. So we should try to get a YES answer with one chosen number(s) and a NO with another.

(1) \(x^2+y^2>z^2\)
It's clear that we get YES answer very easily with big x and y (say 10 and 10), and small z (say 0).

For NO answer let's try numbers from Pythagorean triples:
\(x^2=3\), \(y^2=4\) and \(z^2=5\) (\(x^2+y^2=7>5=z^2\)) --> \(x^4+y^4=9+16=25=z^4\), so we have answer NO (\(x^4+y^4\) is NOT more than \(z^4\), it's equal to it).

Not sufficient.

(2) \(x+y>z\). This one is even easier: again we can get YES answer with big x and y, and small z.

As for NO try to make z some big enough negative number: so if \(x=y=1\) and \(z=-5\), then \(x^4+y^4=1+1=2<25=z^4\).

Not sufficient.

(1)+(2) As we concluded YES answer is easily achievable. For NO try the case of \(x^2=3\), \(y^2=4\) and \(z^2=5\) again: \(x+y=\sqrt{3}+\sqrt{4}>\sqrt{5}\) (\(\sqrt{3}+2\) is more than 3 and \(\sqrt{5}\) is less than 3), so statement (2) is satisfied, we know that statement (1) is also satisfied (\(x^2+y^2=7>5=z^2\)) and \(x^4+y^4=9+16=25=z^4\). Not sufficient.

Answer: E.

Hope it's clear.
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Great question. Yes you can use algebra, but as shown by shrouded1 above you'd better not. Below is great note from ManhattanGMAT tutor Ron Purewal about this problem:

"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."
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Yes ! and it is fairly simple as well ...

(1) \(x^2+y^2>z^2\)
Both sides are positive, so we can square them up
\(x^4+y^4+2x^2y^2>z^4\)
For statement to be true or false, we need to know the magnitude of \(2x^2y^2\). So insufficient

(2) \(x+y>z\)
Let me take the special case, x,y,z>0. If we cannot answer for that case, we can certainly not answer in general.
Now simply take both sides to the fourth power :
\(x^4+y^4+4x^3y+6x^2y^2+4xy^3>z^4\)
Again unless we know the extra terms, we cannot say for sure. So insufficient

(1+2) Again consider the special case for positive numbers only. Even then we don't have enough information to say anything. Do the same manipulation as above and it is easy to see that inequality two has the same terms on the LHS as inequality 1 and some extra terms on top. So clearly its enough to look at 1, which we know is not sufficient to give us an answer, so insufficient

Answer is (e)
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This is the approach I used. It isn't algebraic and it isn't completely based on plugging numbers either (though I am explaining using plugs). It seemed intuitive to me, but your opinion could be different.

Stmnt 1: \(x^2 + y^2 > z^2\)
I do not like \(x^4 + y^4 > z^4\), powers of 4 since I do not know how to work with them in multiple ways. I prefer squares.
So I look at the question in this way:
Stmnt 1: X + Y > Z (X, Y and Z are all positive)
Is \(X^2 + Y^2 > Z^2\)?
Now it is intuitive to say "Yes, it is." because if X = 8, Y = 9 and Z = 1, it is.
The problem is, is there a case in which I would answer "No".
\(X^2 + Y^2 > Z^2\) reminds me of pythagorean triplets where \(X^2 + Y^2 = Z^2\). I check the easiest one 3, 4, 5. I get "No". Hence this is not sufficient.
The toughest part of the question is over.

Stmnt 2: x + y > z
Again, intuitive to say "Yes" because if X = 8, Y = 9 and Z = 1, the answer is yes.
And very easy to say "No" because if x = root(3), y = root(4) and z = root(5), x^4 + y^4 is not greater than z^4

We have used the same examples in both the cases to get a "Yes" and a "No" hence using both together, we will not get the answer.
Answer (E).
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Nice question. Here is my reasoning that allowed me to solve the problem pretty fast.

The inequity true for X=Y=Z=1 --> 1+1>1

Now, let's increase Z (X=Y=1). Z^2 will grow faster than Z but slower than Z^4. So, at some point Z^4 will be greater than 2 but Z^2 and Z will be lesser than 2 and still both conditions will be satisfied. So E.

Damn i felt stupid when i saw this question. Should you have the knowledge to solve this kind of questions to hit a 700+ score, before starting your prep to gmat?

This is definitely a hard question, but some practice should help to deal with such kind of questions in the future.

Try the following inequality questions to practice:

PS: search.php?search_id=tag&tag_id=184
PS: search.php?search_id=tag&tag_id=189

Hard inequality and absolute value questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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Having a problem in statement (1)

(1) \(x^2+y^2>z^2\)
Both sides are positive, so we can square them up
\(x^4+y^4+2x^2y^2>z^4\)
\(2x^2y^2>=0\). so why isnt this sufficient?

If \(2x^2y^2>=0\) then either X or Y equals 0 (but not both!). which implies that if x=0, \(y^2>z^2\) ===> \(y^4>z^4\) or y=0 \(x^2>z^2\) ===> \(x^4>z^4\) , in both cases sufficient. What I am doing wrong? What am I not considering?

Ok, say you know that a+b > c and that b is positive. Can you say that a is definitely greater than c?

Given that \(x^4+y^4+2x^2y^2>z^4\) and you know that \(2x^2y^2\) is positive or 0, can you say that \(x^4+y^4>z^4\) ?
Isn't it possible that \(2x^2y^2\) part of the left hand side is making the left hand side greater than the right hand side?

Assume, \(x = \sqrt{3}\), \(y = 2\) and\(z = \sqrt{5}\)
\(x^2+y^2>z^2\) but \(x^4+y^4=z^4\)
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Brunuel: Is this reasoning right? Bcz we are given with the fact that statement 1 (condition) is true. Then why should we check its validity bringing in Pythagoras theorm.

We are not checking whether the first statement is true, it is. We are trying to get an YES and a NO answer to the question (is \(x^4+y^4>z^4\)) to prove its insufficiency.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is x^4 + y^4 > z^4?

(1) x^2 + y^2 > z^2
(2) x + y > z

From the original condition, we get 3 variables (x,y,z), and in order to match the numbers, we need 3 equations. We are only given 2, so there is high chance (E) will be our answer. Looking at the conditions together, the answer to the question is ‘yes’ for x=y=z=1, but ‘no’ for x=y=0.6, z=0.8. So the conditions are insufficient, and the answer becomes (E).

For cases where we need 3 more equation, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

We want to determine whether x^4 + y^4 > z^4.

Statement One Alone:

x^2 + y^2 > z^2

If we square both sides of the inequality, we will have:

x^4 + 2(x^2)(y^2) + y^4 > z^4

x^4 + y^4 > z^4 – 2(x^2)(y^2)

We still cannot determine whether x^4 + y^4 > z^4, since 2(x^2)(y^2) is a nonnegative quantity.

For example, if x = 0, y = 2, and z = 1, then x^4 + y^4 > z^4, since 0^4 + 2^4 > 1^4.

However, if x = 3, y = 3, and z = 4, then x^4 + y^4 is not greater than z^4, since 3^4 + 3^4 is not greater than 4^4.

Statement one is not sufficient to answer the question.

Statement Two Alone:

x + y > z

We cannot determine whether x^4 + y^4 is greater than z^4.

For example, if x = 0, y = 2, and z = 1, then x^4 + y^4 > z^4, since 0^4 + 2^4 > 1^4.

However, if x = 3, y = 3, and z = 4, then x^4 + y^4 is not greater than z^4, since 3^4 + 3^4 is not greater than 4^4.

Statement two is not sufficient to answer the question.

Statements One and Two Together:

Using our two statements, we still cannot determine whether x^4 + y^4 is greater than z^4. Using the same numerical examples used earlier, we have:

If x = 0, y = 2, and z = 1, then x^4 + y^4 > z^4.

However, if x = 3, y = 3, and z = 4, then x^4 + y^4 is not greater than z^4.

Answer: E
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Hi All,

This question can be solved by TESTing VALUES. When a DS prompt involves lots of exponents, it's likely that you'll have to consider values OTHER than positive integers though (including 0, negatives, fractions, etc.).

We're asked if X^4 + Y^4 > Z^4. This is a YES/NO question.

Since Fact 2 is easier to deal with than Fact 1, I'm going to start there (we might also be able to use our work on Fact 2 to more easily deal with Fact 1)....

2) X + Y > Z

IF....
X = 1, Y = 0 and Z = 0, then the answer to the question is YES.
X = 1/2, Y = 1/2 and Z = .7, then the answer to the question is NO.
Fact 2 is INSUFFICIENT

1) X^2 + Y^2 > Z^2

The TESTs that we used in Fact 2 also "fit" the information in Fact 1...

IF....
X = 1, Y = 0 and Z = 0, then the answer to the question is YES.
X = 1/2, Y = 1/2 and Z = .7, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

Combined, we have two sets of numbers that fit both Facts and produce different answers to the given question (one YES and one NO).
Combined, INSUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

for this problem, remember the simple rule

if a number is smaller than 1, the more square, we get smaller number

Hi thangvietnam,

You have to be careful with your "generalizations." I think what you mean to say is "squaring any positive fraction (meaning 0 < X < 1) will result in a value for X^2 that is smaller than X." Squaring 0 will NOT lead to a smaller result; squaring a negative value will lead to a LARGER result.

GMAT assassins aren't born, they're made,
Rich

One trick to maximize efficiency on difficult DS questions is to subconsciously test values that satisfy BOTH statements so that time need not be wasted.

(x,y,z) = (1,3,2) & (3,3,4) --> Answer is E

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