M05-15

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the positive root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice though, that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).

Can we solve this question as below:-

Question stem asks if √x <2.5x−5
This means, we need to prove if 2.5x - 5>0 => if x > 2? (we don't have equality sign because question mentions that x is a positive integer. This means >0).

Statement 1. x <3
As x is a positive integer, possible values of x are 1 and 2 and none of these is larger than 2. So answer to the question asked is NO. - Sufficient

Statement 2. x is a prime number.
x can be any prime number including 2 so we don't know if x>2 or not. - Insufficient

Answer - A

hi brunel

Question stem asks if √x <2.5x−5
we know x is positive , so squaring both sides gives x< 6.25x^2 + 25 - 25x i.e 26x< 6.25x^2 + 25

so if put 1 answer is yes. for x = 2 answer is no.How can answer be A

Why are you squaring? We can only square inequality when we are certain that both sides are non-negative. We don't know that.

Also, you really do not need to square put x = 1 and x = 2 into √x <2.5x −5. What do you get?

How can you do this quickly? Without testing cases?

sarimzahid2395 - The point here is that testing cases is a time-efficient method of solving the question. I like to say that being a good reader is what helps boost your Quant scores, and this problem is a case in point. Remember, it is analytical reasoning that lies at the heart of the test, not necessarily mathematical prowess. Here, the keywords about the unknown are positive and integer. Statement (1) restricts those infinite possible positive integers to just two: 1 or 2, since 0 is not considered positive. The 1 breaks the square root down without performing any complicated analysis. The problem then reads, "is 1 < 2.5(1) - 5?" Reminder: whenever the square root symbol appears on the GMAT™, as Bunuel pointed out above in another response, only the positive root is considered. Getting back to the reformed question, it is clear that 1 is not, in fact, less than -2.5, but we have a definitive answer. The only way to overturn (A) as an answer is if 2 leads to a different scenario. In the question, "is √2 < 2.5(2) - 5," the answer will again come to no, since √2 is greater than 0. A "no" response to both 1 and 2, the only possible x values, provides a consistent picture, so the answer for now is that (A), Statement (1), is SUFFICIENT.

Looking at Statement (2), our prime positive integers, combining the information in the statement with that of the problem, will be 2, 3, 5, 7, 11, and so on, once again stretching to infinity. We have already tested 2, so we can test any other valid number in the list of primes and see whether we also get a consistent picture:

is √3 < 2.5(3) - 5?

Just by focusing on the right-hand side, we can quickly determine that 2.5(3) - 5 = 2.5, and the question thus becomes,

is √3 < 2.5?

You do not have to be a math genius to see that √3 must be less than 2.5, since √4 would give us 2. Hence, we now have a yes answer to the same question we had a no to before, with 2, from this very statement. We can say goodbye to (D), then, and choose (A) as the answer.

I am no Quant maven, even if I aspire to be one. But with a little number sense, I cracked this one in 1:16 without writing down a thing. The concepts and vocabulary that are brought to bear in the question are simple enough. It is a mistake to think that you have to figure out everything all the time when you just need to answer the question that is being asked.

Good luck in your studies.

- Andrew

I am having a hard time understand this - why can't x be -1.414? I saw the reply from bunuel saying GMAT accepts sqt(x) as only +ve, given x is +ve - but why is that?? Mathematically it is possible. It's like saying GMAT accepts only one meaning of a particular word

I find the question to be ambiguous in this case - why not just write sqrt x > 0?

Is this an Officual Question?

The question is 100% correct.

\(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.


The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;
Similarly \(\sqrt{\frac{1}{16}} = \frac{1}{4}\), NOT +1/4 or -1/4.


Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.