if x > y^2 > Z^4,

[vrajesh.dave]

if x > y^2 > Z^4, which of the following statements could be true?

I. x > y > z
II. z > y > x
III. x > z > y

a. I only
b. I and II only
c. I and III only
d. II and III only
e. I, II, and III

OA: E

we can know for sure that X has to be positive for it be to greater than Y^2 and Z ^4.
and similarly |y| > |z| since Y^2 > Z^4

So I is definitely true.

But I do not understand how II and III could be true.

Please explain.

again, when they say could be true, do we have to take same values for that satisfy all 3 conditions or can we change the values of x,y and z for each statement.

[nimish.tiwari]

if x > y^2 > Z^4, which of the following statements could be true?

I. x > y > z
II. z > y > x
III. x > z > y

a. I only
b. I and II only
c. I and III only
d. II and III only
e. I, II, and III

OA: E

we can know for sure that X has to be positive for it be to greater than Y^2 and Z ^4.
and similarly |y| > |z| since Y^2 > Z^4

So I is definitely true.

But I do not understand how II and III could be true.

Please explain.

again, when they say could be true, do we have to take same values for that satisfy all 3 conditions or can we change the values of x,y and z for each statement.

In the given primary inequality: x > y^2 > z^4, all the terms would be positive (since we have even powers of y & z).

for this to hold true, x can be any +ve number (may be fraction as well) and y & z have to be necessarily fractions. Hence, I holds true.

For II. z > y > x. Consider, x=1/4, y=1/3, z=1/2. This satisfies given primary inequality. Hence, II also holds true.

For III. Consider x=1, y=1/3,z=1/2. This satisfies given primary inequality. Hence, III also holds true.

Hence, E.

[anoo.anand]

is there any criteria on which we have to choose numbers for solving these kind of problems...?

[RonPurewal]

is there any criteria on which we have to choose numbers for solving these kind of problems...?

as a takeaway from this problem, you should absorb the association between COMPARING POWERS and FRACTIONS.

basically, the idea is that fractions (i.e., numbers between 0 and 1) "act funky" when they're raised to powers.
so do negatives.
therefore, when you pick numbers, you MUST consider these sorts of numbers!

[crusade]

is there any criteria on which we have to choose numbers for solving these kind of problems...?

as a takeaway from this problem, you should absorb the association between COMPARING POWERS and FRACTIONS.

basically, the idea is that fractions (i.e., numbers between 0 and 1) "act funky" when they're raised to powers.
so do negatives.
therefore, when you pick numbers, you MUST consider these sorts of numbers!

Hi Ron -- I find myself struggling with these kind of problems; i.e. inequalities in which you have to plug in any range of numbers.

Is there something that I could practice or work on for me to be in a better position to answer such questions?

Thanks.

[RonPurewal]

is there any criteria on which we have to choose numbers for solving these kind of problems...?

as a takeaway from this problem, you should absorb the association between COMPARING POWERS and FRACTIONS.

basically, the idea is that fractions (i.e., numbers between 0 and 1) "act funky" when they're raised to powers.
so do negatives.
therefore, when you pick numbers, you MUST consider these sorts of numbers!

Hi Ron -- I find myself struggling with these kind of problems; i.e. inequalities in which you have to plug in any range of numbers.

Is there something that I could practice or work on for me to be in a better position to answer such questions?

Thanks.

well, here are a few random suggestions, in approximately increasing order of challenge (i.e., the things at the top of the list should be easier to do and/or more appropriate if you are struggling with the material).

* take a bunch of different numbers... positives, negatives, zero, fractions, huge numbers, tiny decimals, you name it, and just do stuff to them. watch whether they get bigger or smaller.
e.g.,
raise them to a whole bunch of different powers
take square roots, cube roots, etc. of them
multiply them together (by numbers of the same type, and by numbers of other types)
add them
subtract them
yada yada yada.
then just observe what happens. this is actually the single biggest secret to success here -- you just have to expose yourself to the behavior of the different kinds of numbers, so that, when you see that behavior in a problem, you'll be famiilar with it and faster on the ball.

* if any of the above behavior surprises you, make flash cards accordingly.

* try finding inequality problems in the first half of the official guide math sections, and applying the stuff you've learned.

* write a bunch of simple inequalities (e.g., x^2 < x, or x^3 > x), and then test the different kinds of numbers (and/or just recall their behavior, if that lets you resolve the issue without actually testing) to see whether they work.

* try finding inequality problems in the second half of the official guide math sections, and going for those.

* lastly, try writing your own inequality problems, especially in the DS format. See whether you can write different DS problems with all the possible correct answers (i.e., one problem with answer (a), one problem with answer (b), etc.)

that should keep your hands full for a while.

[shishir_jolly]

if x > y^2 > Z^4, which of the following statements could be true?

I. x > y > z
II. z > y > x
III. x > z > y

a. I only
b. I and II only
c. I and III only
d. II and III only
e. I, II, and III

OA: E


If if we chane the question a bit from statements "could be true" to "must be true" will the answer change?????????
plz help.....

[crusade]

if x > y^2 > Z^4, which of the following statements could be true?

I. x > y > z
II. z > y > x
III. x > z > y

a. I only
b. I and II only
c. I and III only
d. II and III only
e. I, II, and III

OA: E


If if we chane the question a bit from statements "could be true" to "must be true" will the answer change?????????
plz help.....

The fact that all three statements "could be true" implies that none of them individually "must be true".

[shishir_jolly]

The fact that all three statements "could be true" implies that none of them individually "must be true".

so for what values these inequalities will not work??

[RonPurewal]

The fact that all three statements "could be true" implies that none of them individually "must be true".

so for what values these inequalities will not work??

well, if you just pick random values for the variables, then there a pretty good chance they won't work.

as you'll note in the discussion above, it's actually a decent amount of effort to find numbers that do work in these inequalities. so, finding numbers that won't work in them should be a lot easier.

[ghong14]

This is a question that is based on the special properties of fractions and integers between -1<x<1. As Ron said it is pretty funky. I noticedno one have used an example of the pick a number approach so here goes one.

I. X>Y>Z: 2>1>1/2 therefore X>Y^2>Z^4: 2>1^2>1/2^4

II. Z>Y>X: 1/2>1/3>1/4 therefore X>Y^2>Z^4: (1/2)^2>(1/3)^2>(1/4)^2

III. X>Z>Y: 2>(-1/4)>(-1/2) therefore X>Y^2>Z^4: 2>(-1/2)^2>(-1/4)^4

This is definitely a pen to paper problem.....

[ghong14]

This is a question that is based on the special properties of fractions and integers between -1<x<1. As Ron said it is pretty funky. I noticed no one have used an example of the pick a number approach so here goes one.

I. X>Y>Z: 2>1>1/2 therefore X>Y^2>Z^4: 2>1^2>1/2^4

II. Z>Y>X: 1/2>1/3>1/4 therefore X>Y^2>Z^4: (1/2)^2>(1/3)^2>(1/4)^2

III. X>Z>Y: 2>(-1/4)>(-1/2) therefore X>Y^2>Z^4: 2>(-1/2)^2>(-1/4)^4

This is definitely a pen to paper problem.....

[RonPurewal]

This is definitely a pen to paper problem.....

EVERY problem should be a "pen to paper problem".

you should not be trying to juggle stuff in your head on this test -- it's slower than writing things down, and you'll make way, way more mistakes. (for every 1 mistake people make when they write stuff down, they make 20 to 30 mistakes doing problems mentally.)

[rajbharat.87]

Ron,
Is there any way to figure out the boundary conditions,for values to be plugged in ,either based on the question stem or the answer choices as such so solving becomes much easier?

x > y^2 > z^4

As in,
i)positive / negative
ii)Fraction between 0 and 1 or 0 and -1
iii)Fractions >1 or fractions <-1
iv) a bunch of whole numbers>1 and <-1?
v) 0 , 1 and -1

[RonPurewal]

Ron,
Is there any way to figure out the boundary conditions,for values to be plugged in ,either based on the question stem or the answer choices as such so solving becomes much easier?

i don't know what "boundary conditions" are, so i'm not sure whether i understand the nature of the question.

if you're just asking "is there a simple rule for which numbers to plug in?" -- no, not really. that's the whole point of number-property questions: the kinds of numbers that matter (or don't matter) will vary from problem to problem.

still, you can generally conquer these problems by doing two things:

1/
make associations between "signals" and certain types of numbers.
for instance, if you see absolute value bars, or even powers, or negative signs, or product < 0 or > 0, or quotient < 0 or > 0, then any of these things would indicate that signs are probably an issue.
if you see comparisons involving different powers of the same number, then that indicates that small vs. big numbers (i.e., "fractions" vs. "normal" numbers) are probably an issue.
etc.

2/
most importantly...
if you're not sure, TRY.
this is really the key here -- if you have even the slightest hint that a certain type of number might be involved, then just go ahead and try it.
how long does it take to plug in a specific value? five, six, maybe ten seconds?
there's really no reason to hesitate -- if you don't know how certain numbers will behave, just plug 'em in and watch what happens.