If x and y are positive integers, is (x + y + 1)^(1/2) 3(x+ 1)^(1/2)

Question

If x and y are positive integers, is  \sqrt{x + y + 1} > 3\sqrt{x+ 1} ?

(1)  y > 8x + 8

(2)  x + y > 8x + 8

anbknaga · Answer

Let's simplify the given equation,  \sqrt{x + y + 1} > 3\sqrt{x+ 1}

=>  (\sqrt{x + y + 1})^2 > (3\sqrt{x+ 1})^2 
=> x+y+1 > 9(x+1)
=> x+y+1 > 9x+9
=> y > 8x+8

We can rephrase the question as Is y > 8x+8 ?

Statement 1:  y > 8x + 8

Statement 1 and the simplified given statement is same. So, statement 1 is SUFFICIENT

Statement 2:  x + y > 8x + 8

If x = +ve, then y is not greater than 8x + 8, the answer is NO.
If x = -ve, then y is greater than 8x+8, the answer is YES.

Therefore, statement 2 is INSUFFICIENT.

The answer is  A

Archit3110 · Answer

solve the given expression  \sqrt{x + y + 1} > 3\sqrt{x+ 1} 
squaring both sides; y>8x+8
#1
sufficient
#2
y>7x+8 not sufficient
IMO A

Kinshook · Answer

IMO A

Kinshook · Answer

Given:  x and y are positive integers

Asked:  Is  \sqrt{x + y + 1} > 3\sqrt{x+ 1} ?

\sqrt{x + y + 1} > 3\sqrt{x+ 1} 
=>  (\sqrt{x + y + 1})^2 > (3\sqrt{x+ 1})^2 
=> x+y+1 > 9(x+1)
=> x+y+1 > 9x+9
=> y > 8x+8

(1)  y > 8x + 8 
SUFFICIENT

(2)  x + y > 8x + 8
y>7x+8
NOT SUFFICIENT

IMO A

anbknaga · Answer

That was a typo. thanks for notifying it to me

Kinshook · Answer

x & y are positive integers

y>7x+8 does not mean that y is NOT >8x+8
For example If y>5 than y may or may not be >8

Bunuel · Answer

Official Explanation

On the GMAT, the radical sign denotes the positive square root of something (even though, when we solve for a variable squared, there are positive and negative possibilities), so both sides of this inequality are positive. We can therefore square it and obtain:

x + y + 1 > 9(x + 1)
y > 8x + 8

This is the question we are being asked. On to the statements, separately first.

Statement (1) tells us exactly what we want to know. It says that y > 8x + 8 is a fact, so the answer to the question of whether y > 8x + 8 will be definitively "yes." Sufficient.

Statement (2) looks similar. It simplifies to y > 7x + 8. We'll analyze by cases. Say x = 1. In that case, the statement demands that y > 7 + 8. So the statement would allow, among other possibilities, y = 16 and y = 17. These two x-y possibilities yield opposite answers to the question posed, so this statement is insufficient.

The correct answer is (A).

Bunuel · Answer

Video Explanation

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If x and y are positive integers, is (x + y + 1)^(1/2) > 3(x+ 1)^(1/2)

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