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## Comment on

Introduction to Square Roots## Can you please help me with

Is x > y ?

(1) ✓x > y

(2) x² > y

## Tricky question! Just curious

Tricky question! Just curious, what's the source?

TARGET QUESTION: Is x > y?

STATEMENT 1: ✓x > y

There are several values x and y that satisfy statement 1. Here are two:

CASE A: x = 9 and y = 2 (notice that ✓9 = 3, and 3 is greater than 2. In this case, x > y

CASE B: x = 1/4 and y = 0.3 (notice that ✓(1/4) = 1/2, and 1/2 is greater than 0.3. In this case, x < y

Since we cannot answer the TARGET QUESTION with certainty, statement 2 is NOT sufficient.

STATEMENT 2: x² > y

There are several values x and y that satisfy statement 2. Here are two:

CASE A: x = 3 and y = 2 (notice that 3² = 9, and 9 is greater than 2. In this case, x > y

CASE B: x = 3 and y = 4 (notice that 3² = 9, and 9 is greater than 4. In this case, x < y

Since we cannot answer the TARGET QUESTION with certainty, statement 2 is NOT sufficient.

STATEMENTS 1 and 2 COMBINED

It's easy to show values of x and y that satisfy BOTH statements where x > y. For example, the following values of x and y satisfy BOTH statements:

x = 100 and y = 1

x = 16 and y = 1

x = 25 and y = -3

x = 1 and y = -30

etc

In all of the above cases, x > y

So, the correct answer MAY be C.

That said, the answer will E if there are values of x and y that satisfy BOTH statements AND where x < y

Is it possible to find such values for x and y?

No.

The tricky part is finding finding a situation where ✓x > y YET x < y

Important concept: in most cases, the square root of a number is less than the original number. For example:

✓9 < 9

✓4 < 4

✓36 < 36

The only time ✓x is GREATER THAN x is when 0 < x < 1. For example:

✓(1/4) > 1/4

✓(1/25) > 1/25

etc

So, if 0 < x < 1 then ✓x > x

The problem is that, if 0 < x < 1 then x² < x

So, any value on this range will satisfy statement 1, but will NOT satisfy statement 2.

As such, we CANNOT find values for x and y that satisfy BOTH statements such that x < y

So, it MUST be the case that, when both statements are combined, x > y

Answer: C

## Thanks Brent :)

## Hi Brent,

I couldn't find your answer to this question:

https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-following-must-be-82080.html

Please let me know if it is already somewhere, if not, I would appreciate if you help and answer it, :

My answer is D. I and II only

There is too much debates there, it was confusing for me.

Thanks

Aladdin

## Hi Aladdin,

Hi Aladdin,

Here's my step-by-step solution: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...

Cheers,

Brent

## Thank you so much.

## Hi Brent, could you please

If x and y are positive, which of the following must be greater than 1x+y−−−−√1x+y?

I. x+y−−−−√2x+y2

II. x√+y√2x+y2

III. x√−y√x+yx−yx+y

A. I only

B. II only

C. III only

D. I and II only

E. None

## Hi Jalaj,

Hi Jalaj,

I'd love to help you, but some of your entries are confusing and/or ambiguous.

For example, in statement I (x+y−−−−√2x+y2), is the square root over BOTH 2x and y²?

Likewise, in statement II (x√+y√2x+y2), what is inside the first square root symbol? It looks empty.

If possible, it would be great if you could find the question online and send me the link.

Alternatively, please add some brackets and spaces to prevent ambiguity.

By the way, what's the source of this question?

Cheers,

Brent

## Hi Brent, below is the source

https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-following-must-be-82080.html

## That helps a lot.

That helps a lot.

Here's my full solution: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...

Cheers,

Brent

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