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If |x| > 3, which of the following must be true? 07 Aug 2017, 21:32
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ssislam
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corvinis
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

Responding to a pm:

|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.

II. X^2 > 9
This is the same as |x| > 3 so it must be true

III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.

hi

very beautiful explanation ....I must say ..

anyway ...I quote you " Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take."

For option # 3
III. |x-1|>2

we got "-1>x>3"

but the question stem says "-3>x>3"

So, according to the question stem, if we suppose x = -4, and according to the option #3, if you suppose x = -2, then these two values may not match, but the range provided by the option # 3, that is "-1>x>3" will cover the range provided by the question stem that is "-3>x>3", because -4 is certainly less than -1 and so is x...

please correct me if I am missing something you meant...

thanks in advance .....
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VeritasPrepKarishma
hey

according to the question stem, x can never equal to -2, as it must be less than -3. I got it. I meant according to the option 3 (answer choice 3), x can equal to -2...
It doesn't matter, however, as -2 is always less than -1 and so is -3, and hence the option # 3 holds true always...thanks I got it ...
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If |x| > 3, which of the following must be true? 08 Aug 2017, 01:14
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ssislam

VeritasPrepKarishma
hey

according to the question stem, x can never equal to -2, as it must be less than -3. I got it. I meant according to the option 3 (answer choice 3), x can equal to -2...
It doesn't matter, however, as -2 is always less than -1 and so is -3, and hence the option # 3 holds true always...thanks I got it ...
Show more


The point is - what is given and what is asked?

You are GIVEN that x is less than -3. (question stem gives you that)
You are ASKED whether x will always be less than -1 too. (this is point III. You need to find this. You are not given this.)
So x cannot be -2.
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If |x| > 3, which of the following must be true? 07 Mar 2018, 16:47
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corvinis
If |x| > 3, which of the following must be true?

I. x > 3

II. x^2 > 9

III. |x - 1| > 2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

Show Spoiler
I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks
Show more
Main Idea: We need to use the normal expressions corresponding to the modulus expression given and solve for x and compare with the choices given for truth of the choices.

Details: By topic specific problem solving aid , which is :

- For converting the modulus to the normal form, one would be just remove the modulus and the other would be to remove the modulus and add a negative sign before the expression. For example |x-3| can correspond to (x-3) or –(x-3).
 , we have

X >3 or x<-3

Checking the choices we see:

(i) Need not be true
(ii) must be true
(iii) implies x-1>2 => x>3 or –(x-1) >2 => x<-1. This is consistent with what is given. Hence true
Hence D.
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If |x| > 3, which of the following must be true? 15 Aug 2022, 00:39
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KarishmaB
corvinis
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

Responding to a pm:

|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.

II. X^2 > 9
This is the same as |x| > 3 so it must be true

III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.
Show more

Hi KarishmaB, I have a few doubts regarding this question and other it's variations.

I understood that for |x| > 3 to be true x <-3 or x> 3 i.e. x = -3.1 , -4 , 4 , 4.1. and so on, but not 3,-3, -2, -1 etc.

Now, in (I) x > 3, why is this not a must be true statement?. How do we differentiate between must be / could be true where we have 2 applicable ranges for the question stem.
Per my understanding, x>3 will always be true for the given question stem because whatever value x>3 gives, it will always satisfy |x| > 3

In (III) we get the range for |x-1| > 2, as x <-1 or x>3,
x>3 satisfies the question stem, but I don't see how x <-1 satisfies it.
What if x takes the value -2 ? in that case |-2| >/ 3.
How is this a must be true scenario. All the values from -1 to -3 inclusive will fail the question stem.

Could you please point out where I'm going wrong? I'm almost always incorrect on questions of this form.
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If |x| > 3, which of the following must be true? 15 Aug 2022, 23:53
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KartikSingh09
KarishmaB
corvinis
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

Responding to a pm:

|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.

II. X^2 > 9
This is the same as |x| > 3 so it must be true

III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.

Hi KarishmaB, I have a few doubts regarding this question and other it's variations.

I understood that for |x| > 3 to be true x <-3 or x> 3 i.e. x = -3.1 , -4 , 4 , 4.1. and so on, but not 3,-3, -2, -1 etc.

Now, in (I) x > 3, why is this not a must be true statement?. How do we differentiate between must be / could be true where we have 2 applicable ranges for the question stem.
Per my understanding, x>3 will always be true for the given question stem because whatever value x>3 gives, it will always satisfy |x| > 3

In (III) we get the range for |x-1| > 2, as x <-1 or x>3,
x>3 satisfies the question stem, but I don't see how x <-1 satisfies it.
What if x takes the value -2 ? in that case |-2| >/ 3.
How is this a must be true scenario. All the values from -1 to -3 inclusive will fail the question stem.

Could you please point out where I'm going wrong? I'm almost always incorrect on questions of this form.
Show more


The relation that is given to us is the one that will provide the valid values of x.
So |x| > 3 is given to us. We know it holds. So we know that x can be -4 or 5 or 100 etc.
The set of VALID values of x is {... -5,.. -4,... -3.1, ,3.05,... 4.90,... 100,... 178...}
.
When we ask "which of the following must be true?" we need to find the option that is true for all VALID values of x.

Now think, "is x always greater than 3 for all valid values of x?"
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If |x| > 3, which of the following must be true? 25 Aug 2023, 02:48
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Asked: If |x| > 3, which of the following must be true?

Either x > 3 or x < -3

I. x > 3
Not true when x < -3

II. x^2 > 9
|x| > 3
(|x|)ˆ2 = xˆ2 > 9
Must be true

III. |x - 1| > 2
Either x-1>2; x> 3
or x-1 < -2; x < -1; If x<-3 then x<-1
Must be true

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

IMO D
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If |x| > 3, which of the following must be true? 26 Jun 2025, 18:55
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Bunuel JeffTargetTestPrep
Just wanted to confirm when must questions are given with absolute value ranges, we have to ensure the entire range is covered, right?
In this case, I thought option (i) x > 3 was also right because it was meeting the second part of the range.
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If |x| > 3, which of the following must be true? 26 Jun 2025, 23:41
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sahitiyalavarthi
Bunuel JeffTargetTestPrep
Just wanted to confirm when must questions are given with absolute value ranges, we have to ensure the entire range is covered, right?
In this case, I thought option (i) x > 3 was also right because it was meeting the second part of the range.
Show more

It has nothing to do with absolute values.

|x| > 3 means that \(x < -3\) or \(x > 3\):

---------(-3)---------0---------(3)---------
So, x is somewhere in green segments. Notice that, x can be ONLY in green segments, for example, x cannot equal -2, or -1.5, or 0, or, 2.7, ...

The question asks which of the following options is ALWAYS true about x.

I. \(x > 3\):


We know that x could be less than -3 (for example, if say x = -10), so this option is not always true.

Please review the thread and follow the links to similar questions for more.
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If |x| > 3, which of the following must be true? 13 Oct 2025, 09:49
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Let's break down what \(|x| > 3\) actually means

When you see \(|x| > 3\), you need to think about what absolute value represents—it's the distance from zero on the number line. So \(|x| > 3\) means x is more than 3 units away from zero, which happens in two situations:

  • \(x > 3\) (to the right of zero)
  • \(x < -3\) (to the left of zero)

This is the critical insight: you must consider both positive AND negative values. Let's use some concrete examples: \(x = 4\), \(x = 10\), \(x = -4\), and \(x = -10\) all satisfy our condition.

Now let's test each statement systematically

Statement I: \(x > 3\)

Let's check:
  • When \(x = 4\): Is \(4 > 3\)? Yes ✓
  • When \(x = -4\): Is \(-4 > 3\)? No ✗

Notice how Statement I fails for negative values! Since we found a counterexample (\(x = -4\) satisfies \(|x| > 3\) but doesn't satisfy \(x > 3\)), Statement I does NOT must be true.

Statement II: \(x^2 > 9\)

Let's check:
  • When \(x = 4\): \(4^2 = 16 > 9\) ✓
  • When \(x = -4\): \((-4)^2 = 16 > 9\) ✓

Here's the key: when you square x, the negative sign disappears! If \(|x| > 3\), then \(x^2 = |x|^2 > 3^2 = 9\). This works for both positive and negative values, so Statement II must be true.

Statement III: \(|x - 1| > 2\)

Let's test both cases:
  • When \(x = 4\): \(|4 - 1| = |3| = 3 > 2\) ✓
  • When \(x = -4\): \(|-4 - 1| = |-5| = 5 > 2\) ✓

Think about it: if \(x > 3\), then \(x - 1 > 2\), so \(|x - 1| > 2\). If \(x < -3\), then \(x - 1 < -4\), so \(|x - 1| > 4 > 2\). Both cases work! Statement III must be true.

The answer is D: II and III only

We eliminated Statement I because it fails for negative values, but Statements II and III work for all values satisfying \(|x| > 3\).

Want to master these types of problems?

The approach I showed you works for this question, but there's a systematic framework for tackling all "must be true" problems with absolute values. You can check out the complete solution on Neuron by e-GMAT to understand the underlying patterns and common error traps that trip up most students. The full solution also shows you how to quickly identify which statements to test first and reveals the typical variations of this problem type you'll see on test day. You can also explore comprehensive explanations for other official GMAT questions on Neuron with detailed analytics into your specific weaknesses.

Hope this helps you think through absolute value inequalities more systematically!
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If |x| > 3, which of the following must be true? 19 Oct 2025, 11:59
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KarishmaB

For option C, when we say X<-2, doesn;t it mean it can take the value of x = -3 which is not allowed and hence fails. Thus I removed option C
KarishmaB


Note that -1>x>3 is not correct.
x cannot be less than -1 and greater than 3 at the same time.
It should be x < -1 OR x > 3
The question stem tells us that x > 3 OR x < -3. So every valid value of x is either greater than 3 OR less than -3.

Now, every valid value that is greater than 3 is of course greater than 3.
Every valid value that is less than -3 (e.g. -4, -5.67 etc) is obviously less than -1 too. x cannot be - 2 since it must be less than -3.
That is why option III is always true.
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If |x| > 3, which of the following must be true? 21 Oct 2025, 05:37
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Yes, this is the trouble with must be true questions. I have discussed it in detail in the blog post and video here: https://anaprep.com/algebra-game-must-b ... questions/

Rickooreoisb
KarishmaB

For option C, when we say X<-2, doesn;t it mean it can take the value of x = -3 which is not allowed and hence fails. Thus I removed option C

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