Question: If |x| > 3, which if the following must be true?

1. x > 3
2. \(x^9\) > 9
3. |x-1| > 2

1. I only
2. II only
3. I and II only
4. II and III only
5. I, II and III

“If |x| > 3, which if the following must be true?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation

Approach Solution (1):

If |x| > 3, then it must be true that either x > 3 or x < -3
(1) x > 3
This need not to be true, since it’s also possible that x < -3
For example, c could be equal -5
So, statement I need not be true
Eliminate answer choice A, C and E
Important: The remaining two answer choices (B and D) both state that statement II is true.

So, we need not analyze statement II, since we already know it must be true.
That said let’s analyze it for “fun”

(2) \(x^2\) > 9

This means that Either x > 3 or x < -3
Perfect – this matches our original conclusion that Either x > 3 or x < -3

(3) |x-1| > 2

Let’s solve two cases:
Case a) x – 1 > 2, which means x > 3 PERFECT
Or
Case b) x – 1 < -2, which means x < -1
Must it be true that x < -1?
YES.
We already learned that Either x > 3 OR x < -3
If x < -3, then we can be certain that x < -1

So, statement 3 must also be true.

Correct Answer: D

Approach Solution (2):

Original statement |x| > 3, which means either x > 3 or x < -3
Now check the options

Option 1: x > 3 – not always true as x can be smaller than -3
Thus options A, C and E are ruled out. Only B and D are left.

Option 2: \(x^2\) > 9 – Always true for x > 3 or x < -3

To check – if x = 4,5,6,7… or -4,-5,-6,-7
\(x^2\) > 9

Option 3: |x-1| > 2
Which means (x – 1) > 2
x > 3 (if x -1 > 0) – True
It also means (x -1) < -2
x < -1 (if x – 1 < 0)
x < -1 satisfies x < -3. Thus this is true

both 2 and 3 is true.

Correct Answer: D

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