If x and y are positive integers, is x y? (1) (x)(x)

Question

If x and y are positive integers, is x > y?

(1) x^2 < y
(2) The square root of x is less than y

unraveled · Answer

If x and y are positive integers, is x > y?

(1) \(x^2\) < y
(2) The square root of x is less than y

Since x > 0 and y > 0.
Statement 1) \(x^2 < y\)
x*x < y so x would definitely be less than y always.

ALTERNATIVELY:
Verifying using numbers and fractions.
Case 1: x = \(\frac{1}{2}\) and y > 1(2 OR 3 OR any value).
Case 2: x = \(\frac{1}{2}\) and 0 < y < 1 such that \(x^2\) < y always (y = \(\frac{2}{3}\) OR \(\frac{4}{9}\))
Case 3: x > 1 and y > 1 such that \(x^2\) < y always. Take x = 2 so that \(x^2\) = 4 and y ≥ 5

Only Case 3 is required here as per question
Sufficient.

Statement 2) \(\sqrt{x}\) < y
As both x and y are positive we can square the inequality
x < \(y^2\)

Take x = 1 and y = 2, So x > y ? NO
OR
x = 2 and y = 2, So x > y ? NO
OR
x = 3 and y = 2, So x > y ? YES

Insufficient.

Answer (A)

CareerGeek · Answer

x and y are positive integers Possible values of x = {1, 2, 3, 4, . . . .. } Possible values of y = {1, 2, 3, 4, . . . .. } (1) x^2 < y If x = 1, y > 1 --> x < y ( Answer - Definite NO ) If x = 2, y > 4 --> x < y ( Answer - Definite NO ) If x = 3, y > 9 --> x < y ( Answer - Definite NO ) --> Sufficient (2) The square root of x is less than y √x < y If x = 1, y > 1 --> x < y ( Answer - Definite NO ) If x = 4, y > 2 --> Possible values (x, y) = (4, 3), (4, 5) --> Can't say --> Insufficient IMO Option A Pls Hit Kudos if you like the solution

HoudaSR · Answer

Testing cases on inequalities questions can be a nightmare. I often try to solve using algebra or by visualizing values on the number line as below:

If x and y are positive integers, is x > y? (1) (x)(x) < y (2) The squ

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