If 3 < x < 7, Which of the following must be true?

if x=3.1 then x^2<10
options A,B , E ruled out
if x=3.00000000000001
then 9< x^2< 9.0001
option C ruled out

Option D is winner having all possible values for 3<x<7

Ans D

All the answer choices involve x^2, so it would be great if the info in the stem also involved x^2. And because all the components of the inequality are known to be positive we can square all the parts of the inequality.

This leads to 9 < x^2 < 49. So we know that x^2 is somewhere between 9 and 49.

Now we just need something that is indisputably true based on that fact. And a lot of things must be true. For example, we know that x^2 must be less than a million. We also know that it is greater than 0 and great than negative three billion.

And we know that x^2 is greater than negative 100 and less than positive 100 (since everything between 9 and 49 is also between negative 100 and positive 100).

\(3 < x < 7 \implies 9 < x^2 < 49\)

Let's check each choice

(A). \(10 \leq x^2 \leq 48\).
This choice is incorrect since \(x^2\) could be \(9.5\) or \(48.5\)

(B). \(10 < x^2 < 48\)
Same as (A)

(C). \(9.0001 < x^2 < 48.9999\)
This choice is incorrect since \(x^2\) could be \(9.000000000001\) or \(48.99999999999999999\)

(D). \(-100 < x^2 < 100\)
Correct.

(E). \(10 < x^2 < 49\)
This choice is incorrect since \(x^2\) could be \(9.5\)

The answer is D.

The answer to the question must be D without solving anything here. Range in D includes all of the ranges from other options, so if say A is true then so must be D because D includes A. Since we cannot have two correct answers in PS, then the only correct answer is D.

Hope it's clear.

I felt like this question is very easy but it's tricky at the part that the range of C is included in D and therefore some people who couldn't think it through might fall for the trap of conjunction fallacy bias.

Squaring our inequality, we have:

9 < x^2 < 49

Since -100 < 9 and 49 < 100, we can also say:

-100 < x^2 < 100

Alternate Solution:

Squaring our inequality, we have:

9 < x^2 < 49

We see that x^2 is any value strictly between 9 and 49. Looking at the answer choices, the only choice that includes this entire interval is choice D, -100 < x^2 < 100.

Note that the question stem did NOT ask for the solution set for x^2; instead, it asked which answer choice is true. Since we know that x^2 is any value strictly between 9 and 49, it is also true that x^2 is between -100 and 100, since all values between 9 and 49 are contained in the interval between -100 and 100.

Answer: D

I mean..i get the -100 is less than 9, and therefore less than x, but how are we going to have x^2 be negative?
Just seems like a silly question to ask if the answer is basically a non-answer..

I mean the "solution" says less about the problem than what's actually given in the problem..

Option D has the biggest range, if apart from option D any other option is correct then D will also be correct and two options cannot be correct therefore D is the answer.
Another way to look at a problem, no calculation required.

Posted from my mobile device

Hello fireagablast,

Sometimes, I think you should not look at things in their literal sense. You need to scratch beneath the surface to understand the hidden meaning.
The range here takes care of values smaller than 9.0001 which option C doesn’t.

In fact I would argue that this is brilliant question with exceptional answer options. There are a couple of options which you know are certainly not the answers. But, it’s the sliver of a difference between options C and D that make this a challenging one. However, option C actually provides a subtle clue that there CAN be values smaller than 9.0001. Students who take this clue end up escaping from the trap answer, others don’t.
Come to think of this, the answer options actually tell a lot about the given range i.e. 3<x<7.

And many time, this is what GMAT does on difficult questions. It tests your biases and very often, students are sub-consciously biased to take easy values (that smacks of a comfort zone). The clock ticking at the back of their brains also contributes to their making such errors of judgement.
Therefore, when a range like 3<x<7 is given, it’s important to try a value which you would otherwise not try – i.e. get out of your comfort zone and try values like 3.0000000001, because that’s certainly in the given range. A range/inequality given in a question is an open invitation for you to try extreme values (pun not intended).

Hope that helps!