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Manager  Joined: 25 Jul 2010
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If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Question Stats: 52% (01:57) correct 48% (02:00) wrong based on 2023 sessions

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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. 1 and III only
D. II and III only
E. I, II and III
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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69
Orange08 wrote:
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 II

As this is a COULD be true question then even one set of numbers proving that statement holds true is enough to say that this statement should be part of correct answer choice.

Given: $$x > y^2 > z^4$$.

1. $$x>y>z$$ --> the easiest one: if $$x=100$$, $$y=2$$ and $$z=1$$ --> this set satisfies $$x > y^2 > z^4$$ as well as given statement $$x>y>z$$. So 1 COULD be true.

2. $$z>y>x$$ --> we have reverse order than in stem ($$x > y^2 > z^4$$), so let's try fractions: if $$x=\frac{1}{5}$$, $$y=\frac{1}{4}$$ and $$z=\frac{1}{3}$$ then again the stem and this statement hold true. So 2 also COULD be true.

3. $$x>z>y$$ --> let's make $$x$$ some big number, let's say 1,000. Next, let's try the fractions for $$z$$ and $$y$$ for the same reason as above (reverse order of $$y$$ and $$z$$): $$y=\frac{1}{3}$$ and $$z=\frac{1}{2}$$. The stem and this statement hold true for this set of numbers. So 3 also COULD be true.

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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Baten80 wrote:
If x > y^2 >z^4, which of the following statements could be true?

I. x > y > z

x=10000
y=10; y^2=100
z=1; z^4=1
x>y^2>z^4

II. z > y > x
z=0.5; z^4=0.0625
y=0.4; y^2=0.16
x=0.3
x>y^2>z^4

III. x > z > y
x=0.5
z=0.2; z^4=0.0016
y=0.1; y^2=0.01
x>y^2>z^4

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

We just need to remember that
1. the number decreases in value with increment in the power of the number if 0< number< 1;
if x=0.1; x>x^2>x^3>x^(100) because x is between 0 and 1.

2. the number increases in value with increment in the power of the number if number>1
if x=2;
x<x^2<x^(100) because x is more than 1.

Ans: "E"
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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2
This is a really tough problem. Here is my video explanation:
https://gmatquantum.com/gmatprep-algebr ... tatements/

Dabral

Originally posted by dabral on 28 Jun 2011, 12:19.
Last edited by dabral on 17 Sep 2018, 00:51, edited 1 time in total.
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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21
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arps wrote:
1) x > y2 > z4

which of the following is 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

I think the actual question is: Which of the following could be true?

Plugging in numbers work best for such questions. The only thing to keep in mind is that you should plug in the right numbers. How do you know the right numbers?
When I see $$x > y^2 > z^4$$, I think that $$y^2$$ and $$z^4$$ are non negative. Since $$y^2 > z^4$$, $$y^2$$ cannot be 0. Only z can be 0. x has to be positive. Also, I have to take into account two ranges: 0 to 1 and 1 to infinity. The powers behave differently in these two ranges. I will consider negative numbers only if I have to since with powers, they get confusing to deal with.

The question says: "Which of the following could be true?"
We have to find examples where each relation holds.

I. x > y > z
This is the most intuitive of course.
z = 0, y = 1 and x = 2
$$2 > 1^2 > 0^4$$

II. z > y > x
Let me consider the 0 to 1 range here. Say z = 1/2, y = 1/3 and x = 1/4
$$1/4 > 1/9 > 1/16$$

III. x > z > y
Let's stick to 0 to 1 range. z > y as in case II above but x has to be greater than both of them. Say z = 1/2, y = 1/3 and x = 1
$$1>1/9 > 1/16$$

So all three statements could be true.
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Hi Bunuel/Karishma,
Thanks for the earlier response.. I think, I am very weak in Inequalities..
Could you please post how to go about this question in algebraic way.. ...also if you could let me know how do you make sure about the "Range of the values", that will also work..
Thanks
H
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Joined: 02 Sep 2009
Posts: 59101
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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imhimanshu wrote:
Hi Bunuel/Karishma,
Thanks for the earlier response.. I think, I am very weak in Inequalities..
Could you please post how to go about this question in algebraic way.. ...also if you could let me know how do you make sure about the "Range of the values", that will also work..
Thanks
H

Plug-in method is really the best way to handle such kind of questions. No need to look for some kind of textbook or algebraic ways.

Notice that there are are certain GMAT questions which are pretty much only solvable with plug-in or trial and error methods (well at leas in 2-3 minutes). Many difficult inequality problems will often require some sort of plug-ins, as part of your technique or else you'll spend too much time solving them with algebra. Which means that you MUST make plug-in methods part of your arsenal if you want to get a decent score.

Inequality questions to practice.
DS: search.php?search_id=tag&tag_id=184
PS: search.php?search_id=tag&tag_id=189

Hope it helps.

P.S. I'm not sure understood the following part of your post: "how do you make sure about the "Range of the values", that will also work.. "
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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1
This is what I mean when I said range - Red Part in Karishma's response"
VeritasPrepKarishma wrote:
arps wrote:
1) x > y2 > z4

which of the following is 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

I think the actual question is: Which of the following could be true?

Plugging in numbers work best for such questions. The only thing to keep in mind is that you should plug in the right numbers. How do you know the right numbers?
When I see $$x > y^2 > z^4$$, I think that $$y^2$$ and $$z^4$$ are non negative. Since $$y^2 > z^4$$, $$y^2$$ cannot be 0. Only z can be 0. x has to be positive. Also, I have to take into account two ranges: 0 to 1 and 1 to infinity. The powers behave differently in these two ranges. I will consider negative numbers only if I have to since with powers, they get confusing to deal with.

The question says: "Which of the following could be true?"
We have to find examples where each relation holds.

I. x > y > z
This is the most intuitive of course.
z = 0, y = 1 and x = 2
$$2 > 1^2 > 0^4$$

II. z > y > x
Let me consider the 0 to 1 range here. Say z = 1/2, y = 1/3 and x = 1/4
$$1/4 > 1/9 > 1/16$$

III. x > z > y
Let's stick to 0 to 1 range. z > y as in case II above but x has to be greater than both of them. Say z = 1/2, y = 1/3 and x = 1
$$1>1/9 > 1/16$$

So all three statements could be true.
"
Math Expert V
Joined: 02 Sep 2009
Posts: 59101
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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imhimanshu wrote:
This is what I mean when I said range - Red Part in Karishma's response"
VeritasPrepKarishma wrote:

First notice that since x>z^4 (x is greater than some nonnegative value) then x>0.

Now, as Karishma correctly noted, numbers in powers behave differently in the range {0. 1} and {1. +infinity}. For example:

If 0<a<1 then a, a^2 and a^4 will be ordered as follows: 0--(a^4)--(a^2)--(a)--1

If a>1 then a, a^2 and a^4 will be ordered as follows: 1--(a)--(a^2)--(a^4)--

So, we should take the above difference in ordering into account when picking numbers for x, y, and z, since we need to find the values which satisfy 3 different statements.

Hope it's clear.
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Restrictions are not provided on the variables so I planned to check different values I and used x=y=z=1/2.
As if I take 1/2 for each variables, its given condition would be satisfied and it will become 1/2>1/4>1/8.

So according to me none of the conditions are satisfied.
Am I doing anything wrong here?
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Posts: 59101
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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chetan86 wrote:
Restrictions are not provided on the variables so I planned to check different values I and used x=y=z=1/2.
As if I take 1/2 for each variables, its given condition would be satisfied and it will become 1/2>1/4>1/8.

So according to me none of the conditions are satisfied.
Am I doing anything wrong here?

Notice that the question asks "which of the following statements could be true" NOT "which of the following statements must be true"
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Bunuel wrote:
Orange08 wrote:
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 II

As this is a COULD be true question then even one set of numbers proving that statement holds true is enough to say that this statement should be part of correct answer choice.

Given: $$x > y^2 > z^4$$.

1. $$x>y>z$$ --> the easiest one: if $$x=100$$, $$y=2$$ and $$z=1$$ --> this set satisfies $$x > y^2 > z^4$$ as well as given statement $$x>y>z$$. So 1 COULD be true.

2. $$z>y>x$$ --> we have reverse order than in stem ($$x > y^2 > z^4$$), so let's try fractions: if $$x=\frac{1}{5}$$, $$y=\frac{1}{4}$$ and $$z=\frac{1}{3}$$ then again the stem and this statement hold true. So 2 also COULD be true.

3. $$x>z>y$$ --> let's make $$x$$ some big number, let's say 1,000. Next, let's try the fractions for $$z$$ and $$y$$ for the same reason as above (reverse order of $$y$$ and $$z$$): $$y=\frac{1}{3}$$ and $$z=\frac{1}{2}$$. The stem and this statement hold true for this set of numbers. So 3 also COULD be true.

Isn't it stated in the exam that assume all numbers are integers? We can't try fractions unless they say they are not integers.
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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SaramiR wrote:
Bunuel wrote:
Orange08 wrote:
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 II

As this is a COULD be true question then even one set of numbers proving that statement holds true is enough to say that this statement should be part of correct answer choice.

Given: $$x > y^2 > z^4$$.

1. $$x>y>z$$ --> the easiest one: if $$x=100$$, $$y=2$$ and $$z=1$$ --> this set satisfies $$x > y^2 > z^4$$ as well as given statement $$x>y>z$$. So 1 COULD be true.

2. $$z>y>x$$ --> we have reverse order than in stem ($$x > y^2 > z^4$$), so let's try fractions: if $$x=\frac{1}{5}$$, $$y=\frac{1}{4}$$ and $$z=\frac{1}{3}$$ then again the stem and this statement hold true. So 2 also COULD be true.

3. $$x>z>y$$ --> let's make $$x$$ some big number, let's say 1,000. Next, let's try the fractions for $$z$$ and $$y$$ for the same reason as above (reverse order of $$y$$ and $$z$$): $$y=\frac{1}{3}$$ and $$z=\frac{1}{2}$$. The stem and this statement hold true for this set of numbers. So 3 also COULD be true.

Isn't it stated in the exam that assume all numbers are integers? We can't try fractions unless they say they are not integers.

No that's not true at all. All numbers on the test represent real numbers: Integers, Fractions and Irrational Numbers. You cannot assume a variable is integer if you are not explicitly told so.
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GMAT 1: 720 Q50 V37 Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Algebraic solution:

In the question we are given: x>y2>z4, hence from concepts of inequalities we break it into 2 parts: x>y2 and y2>z4.
1. x>y2 means -x(1/2)<y<x(1/2)
2. y2>z4 means -y<z2<y, but a square cannot be negative so 0<z2<y, this implies -y(1/2)<z<y(1/2).

Now we plot all these points on number line with the intersection of there ranges. But before that we need to understand that we will only be taking x,y,z as positive since if we take y as negative for example(easiest one) the value z(2) becomes negative, whereas a square can never be negative.

hence we plot all of them on the positive x-axis. From above 1 & 2 point we get a general range as such 0<z<y(1/2)<y<x(1/2)<x.
Now, we need to see that we haven't in reality considered various values of x,y,z but have come up with a general idea of how they look on the number line.
Now we define the ranges, since we know about a^x graph varies for values 0<a<1 and a>1, we also take such cases for all three of them.
1. 0<x<1 and x>1
2. 0<y<1 and y>1
3. 0<z<1 and z>1

Hence looking at the combinations we find we have 8 possibilities (2*2*2). taking the 2 general ones:
1. x>1 y>1 z>1. In the general formula we simply put x,y,z and get x>y>z. (Would have figured initially).
2. 0<x<1, 0<y<1 and 0<z<1. In this possibility put x as 1/x, y as 1/y and z as 1/z in general formula we get z>y>x.
3. x>1 0<y<1 and 0<z<1. In this put y as 1/y and z as 1/z. Keep x as x in general formula, we see x>1/y>1/z. since only 1/y>1/z are in reciprocal hence z>y by inequalities. thus x>z>y.

Therefore we can get 8 possibilities and the fact is all of them are correct.
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Hi vinnisatija,

Yes, if you see a "could be" question, then just one example that satisfies the given condition is sufficient. In case of "must be" questions, the required condition must hold true under all circumstances along with whatever additional constraint is given in the problem.

Dabral
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Quote:
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 II

We are given that x > y^2 > z^4 and need to determine which statements must be true. Let’s test each Roman Numeral.

I. x > y > z

Notice that the order of arrangement of x, y, and z in the inequality x > y > z is the same as the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive integers in this case.

x = 10

y = 3

z = 1

Notice that 10 > 3 > 1 for x > y > z AND 10 > 9 > 1 for x > y^2 > z^4.

We see that I could be true.

II. z > y > x

Notice that the order of arrangement of x, y, and z in the inequality z > y > x differs from the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive proper fractions in this case. This is because we need to decrease the value of y and z to make them work within the given inequality.

x = 1/5

y = 1/3

z = 1/2

Notice that 1/2 > 1/3 > 1/5 for z > y > x AND 1/5 > 1/9 > 1/16 for x > y^2 > z^4.

We see that II could be true.

III. x > z > y

Notice that the order of arrangement of y and z in the inequality x > z > y differs from the order of arrangement of y^2 and z^4 in the inequality x > y^2 > z^4, so we once again want to test positive proper fractions. This is because we need to decrease the value of z to make it work within the given inequality (that is, we want to swap the order of z4 and y2 even if z > y).

x = 1/2

y = 1/4

z = 1/3

Notice that 1/2 > 1/3 > 1/4 for x > z > y AND 1/2 > 1/16 > 1/81 for x > y^2 > z^4.

We see that III could be true.

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GMAT 1: 730 Q51 V36 Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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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 II

Solution:
To answer questions like this, use ZONEF.
Z = zero
O = one
N = negative integers
E = Extreme Integers (Read as Positive Integers > 1)
F = Fractions (think both positive and negative)

For x > y^2 > z^4, the following numbers work: x = 10, y = 3, z = 1. So , I is true.
Eliminate (D).

The easiest way to prove II true is to multiply the above numbers by -1, but as you are dealing with even exponents, negative numbers will not work.
That leaves us with F(Fractions).

Let z = 0.9, y = 0.7 and x = 0.1
These numbers keep both x > y^2 > z^4 and z > y > x true.
Eliminate (A) and (C).

Now, proving the third is easy. Just take x = any big positive number, say 10
Now, you have x = 10, y = 0.7 and z = 0.9
These values keep both x > y^2 > z^4 and x > z > y true.

So, all three are true.
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Orange08 wrote:
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. 1 and III only
D. II and III only
E. I, II and II

Question is about "COULD BE" true ..so we just need one set of numbers which ensures that the above conditions are true..

Take x = 100, y = 5 and z = 2
So statement 1 and the statement in the question stem are true..

Take x = 0.4, y = 0.5 and z = 0.6
So statement 2 and the statement in the question stem are true

Take x = 100, y = 0.5 and z = 0.6
So statement 3 and the statement in the question stem are true

So option E
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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Top Contributor
1
Orange08 wrote:
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. 1 and III only
D. II and III only
E. I, II and II

If we CAN find a set of values that satisfies a statement AND yields values such that x > y² > z⁴, then we'll keep that statement.

Statement I. x > y > z
If x = 2, y = 1, and z = 0, then x > y² > z⁴
KEEP statement I

Statement II. z > y > x
If x = 1/4, y = 1/3, and z = 1/2, then x > y² > z⁴
KEEP statement II

Statement III. x > z > y
If x = 2, y = -1, and z = 0, then x > y² > z⁴
KEEP statement III

Cheers,
Brent
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Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

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