Data Sufficiency

Is x > y?

(1) under root x > y

(2) x3 > y

OAC

jain2016 wrote:Is x > y?

(1) under root x > y

(2) x3 > y

OAC

Hi,
we are not told anything about variables x and y, they are fractions or integers, + or -...
lets see the statements--
(1) under root x > y
underroot x >y, this means x is POSITIVE..
but we do not know if x is fration or integer..
if x=1/4 and y=1/3.. underroot x>y but x<y..
If x=16 and y = 3.. underroot x>y MEANS x>y..
Insuff

(2) x^3 > y
this tells us that x is not between 0 and 1..
but nothing if x is + or -..
if both are -ive, say x=-1/2 and y=-1/3.. x^3>y but x<y.
If x=-1 and y=-3... x^3>y and x>y..
Insuff

combined..
we know x is positive and x>1 and both square root x and x^3 are >y, so x>y
Suff
C

Is x > y ?

(1) √x > y

(2) x³ > y

Statement 1: √x > y
It's possible that x=4 and y=1, in which case x>y.
It's possible that x=1/4 and y=1/3, in which case x<y.
INSUFFICIENT.

Statement 2: x³ > y
It's possible that x=2 and y=1, in which case x>y.
It's possible that x=-1/2 and y=-1/4, in which case x<y.
INSUFFICIENT.

Statements 1 and 2 combined:
Since we can't take the square root of a negative, statement 1 implies that x≥0.
Thus, when we combine the two statements, if y<0, we know that y<x.
Our concern is what happens when y≥0.
One approach is to memorize the shapes of some basic graphs:



Only in the yellow region is y<√x and y<x³.
The entire yellow region is below the graph of y=x, implying that y<x throughout the entire region.
Thus, combining the two statements, we know that y<x.
SUFFICIENT.

The correct answer is C.

An alternate approach would be to use algebra to test the 3 cases: y=x, y>x, and y<x.

Case 1: y=x.
Statement 1: If y=x and y<√x, then x < √x.
Statement 2: If y=x and y<x³, then x < x³.
No value for x will work here: a number cannot be less than both its root and its cube.
Thus, y≠x.

Case 2: y>x.
Statement 1: If x<y and y<√x, then x < √x.
Statement 2: If x<y and y<x³, then x < x³.
No value for x will work here: a number cannot be less than both its root and its cube.
Thus, it is not possible that y>x.

Since it is not possible that y=x or that y>x, we know that y<x.
SUFFICIENT.