D01-42

9 < x^2 < 99

Take the square root of all sides. Now when you take the square root of a number you get a positive and a negative root: √9 = ±3 and √99 ≅ ±9.95. This produces two intervals for x:

-9.95 < x < -3 and 3 < x < 9.95

Since x is an integer, its maximum value is x = 9 and its minimum value is x = -9

(Max Value of x) - (Min Value of x) = 9 - (-9) = 9 + 9 = 18

Good question.
Need to consider the negative values as well...

x can take positive, as well as negative values

One should always consider all possible cases in mind.

I think this is a high-quality question and I agree with explanation. GG. Well Played...

This is a high-quality and "tricky" question.

This is a definitely a high quality question and I agree with the explanation.

Very good, tricky and gmat kind of a question. Sad to see people are judging it as a poor quality question without properly understanding/thinking through the solution

I think this is a high-quality question and I agree with explanation.

Bunuel: Not sure if I'm misunderstanding, but I thought GMAT's mathematical rules require us to disregard negative roots?
This means that the only acceptable derivation of \(9 < x^2 < 99\) -->\( 3 < x < 9.9...\)
Which makes the maximum minus minimum = 9 - 3 = 6

Ans: B

\(\sqrt{nonnegative \ number}\geq 0\) but \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).

Hi Bunuel,

I am not sure if I am satisfied with the explanations provided (not saying that the explanations provided here are incorrect, I am sure they are correct).

I feel I still have some gaps in clarity. I found my reasoning to correct answer as the following and please correct me if I am wrong somewhere.

So we arrive at x having the following as the values after taking the square root of all the sides

√9 = ±3 and √99 ≅ ±9.95. This produces two ranges for both sides when considering both sides independently.

-3 > x > 3 and -9.95 < x < 9.95

When we plot these two ranges on a number line then we get a maximum integer value of x = 9 and a minimum value of x = -9

So since the question asked the difference between Max and Min, hence,
(Max Value of x) - (Min Value of x) = 9 - (-9) = 9 + 9 = 18


So I have a couple of questions

1. Is my reasoning and thought process correct?
2. When we are given a question where the variable has a range on both sides like we have in this question, can I consider both sides independently and derive the range and then look at both ranges together on the number line?


Thanks,
Sud

√9 = 3 only and √99 = 9.somethin only! The square root sign CANNOT given negative result.

You could take the square root from \(9 \lt x^2 \lt 99\) and you'd get \(9 \lt |x| \lt 9.something\), hence \(x\) can be: -9, -8, -7, -6, -5, -4, 4, 5, 6, 7, 8, or 9.

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

9 < x^2 < 99
9 < x^2 ≤ 81 (because x is an integer)

Since x is raised to an even exponent, x can be positive or negative.

Possible pos values of x = 4, 5, 6, 7, 8, 9
Possible neg values of x = -4, -5, -6, -7, -8, -9

Max possible value = 9
Min possible value = -9
Difference: 9-(-9) = 18

I think this is a high-quality question and I agree with explanation.