If |x|/|3| > 1, which of the following must be true?

If |x|/|3| > 1, which of the following must be true?

A. x > 3
B. x < 3
C. x = 3
D. x ≠ 3
E. x < -3

Notice that if x = 3, then \(\frac{|x|}{|3|} = \frac{|3|}{|3|} = 1\), so \(\frac{|x|}{|3|}\) is NOT more than 1, it's equal to 1. Thus if x = 3, the given inequality does NOT hold true.

As for the other options.

First, simplify the given inequity: \(\frac{|x|}{|3|} > 1\) --> \(\frac{|x|}{3}> 1\) --> \(|x| > 3\) --> \(x < -3\) or \(x > 3\). This is given as a fact.

Now, if x < -3 or x > 3, then which of the options MUST be true?

A. x > 3 --> this option is not necessarily true since x could be less than -3, for example -4, which will make this options not true.

B. x < 3 --> this option is not necessarily true since x could be more than 3, for example 4, which will make this options not true.

C. x = 3 --> this option is NEVER true since we know that x < -3 or x > 3.

D. x ≠ 3 --> we know that x < -3 or x > 3. Thus x cannot be 3. Thus this option is true.

E. x < -3 --> this option is not necessarily true since x could be more than 3, for example 4, which will make this options not true.

Answer: D.

Similar questions to practice:
https://gmatclub.com/forum/if-x-x-x-whic ... 68886.html
https://gmatclub.com/forum/if-4x-12-x-9- ... 01732.html

All must or could be true questions: https://gmatclub.com/forum/search.php?se ... tag_id=193

Hope it helps.
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(D) for me

lxl / l3l > 1
<=> |x| > 3
<=> x > 3 or x < -3

So, the only answer choice that we are sure of is x ≠ 3

I dont know why I am not convinced with the answer. I eliminated B,C and D. D eliminated for the same reasons as mentioned above by few GMAT friends. Can someone please tell me why is A or B wrong ?

A says that any value of X is above 3, which actually makes the LHS greater than 1.
Example, I take the value of X as 10 (greater than 3),--------> |10|/|3| > 1

Same is the case with B. B states that any value of X is below -3, which again makes the LHS greater than 1
Example, I take the value of X as -10 (smaller than 3),--------> |10|/|3| > 1

Can someone please explain me, why these two answer choices does not come under MUST category ? Thanks

The question asks "which of the following must be true". Thus, the answer choice must be valid and satisfy the given condition ALL the times.

We can't choose A as because a value of x<-3 also satisfies the given condition. Take x = -5, we still have the condition satisfied. Thus, it is not absolutely necessary that x has to be greater than 3. If the question would have asked which of the following choices COULD BE TRUE, then you could have selected A.

The same goes for E as well. Any value of x, greater than 3, say x = 6 also satisfies the given condition.

But for x =3, we can never get the given inequality, as then it equals one. Thus, x can never be equal to 3.
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Now I get it. I have to eliminate an option (even if its true) in a MUST BE TRUE questions, if there is any other choice that satisfies the conditions equally (Like and A and B). The only option that has no other alternative and satisfies the condition of the question is considered correct (Like D). Thanks Vinaymimani and Bunuel.

Hi,

It seems I have got a Problem here. what If X equals -3 ( Condition in D is satisfied since X is not 3 it is -3 ; I am fine if the variable in the condition is Mod X instead of X).

Can Someone Pls clarify?

\(\frac{|x|}{|3|}>1\) you can see it as \(|x|>3\) or x>3, x<-3

If \(x=-3\)
\(\frac{|-3|}{|3|}>1\) and this is 1>1 which is false.

x cannot be -3, it's inside the range of values (\(-3\leq{}x\leq{}3\)) that x cannot assume

Hope it's clear now. Let me know

Hi,

Thanks for the quick response.

I got that can not be -3, I just proposed that as a contradiction to the OA which is D here. Option D says It must be true that X does not equal 3. while It is correct, IMO It is not the only restriction we have here. X can not equal -3 as well ( Just as you explained ). so IMO Correct answer should reflect |x| > 3. Here None of the options reflect that condition. Am I missing something very basic here?

Pavan.

It's simple: from the quesion we know that x cannot assume values \(-3\leq{}x\leq{3}\)

Of course D is not the only possible answer.
You said "Here None of the options reflect that condition", but here we are not asked to find the range of possible values!
We have to check if the values A,B,... fit with the condition above.

Also x≠2 or ≠1 must be true for example, but we have to answer by looking at the possible answer.

So which must be true?
A. x > 3
B. x < 3
C. x = 3
D. x ≠ 3 <---This one is correct
E. x < -3

Hope it's clear now

Modulus of a number always tells us the absolute value or put simply the positive value of a number.
Modulus of a positive number will always be the same number
Modulus of a negative number will be just the positive part of the number

In this question, 3 is a positive number, hence |3| = 3
Hence we have \(|x|/3 > 1\)

or |x| > 3.
This means x > 3 or x <-3

Of the given options, the only option that does not hold good is Option D

Solving modulus inequality
If |x| > a, then the inequality will pan out as x > a and x< -a
If |x| < a, then the inequality will pan out as -a < x <a

Check similar questions here: Trickiest Inequality Questions Type: Confusing Ranges.
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Manipulating the given inequality, we have:

|x| > |3|

|x| > 3

When the value of x is positive, we have:

x > 3

When the value of x is negative we have:

-x > 3

x < -3

Thus, either x > 3 or x < -3

Therefore x cannot equal 3.

Answer: D
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Given that \(\mid \frac{x}{3} \mid > 1\) and we need to find the range of values of x

\(\mid \frac{x}{3} \mid > 1\) can be written as \(\frac{\mid x \mid }{\mid 3 \mid } > 1\)
And we know that | 3 | = 3 as 3 is a positive number
=> \(\frac{\mid x \mid }{ 3} > 1\)
=> | x | > 3

Now, this is of the form | x | > a and we know that we can open it as
Either x > a or x < -a

=> x > 3 or x < -3
Now, lets look at the answer choices

A. \(x > 3\) -> this is partially true it covers only a part of the answer which is x > 3 and does not cover x < -3. So this cannot be "Must be true"

B. \(x < 3\) -> this is NOT TRUE as it contains a lot of numbers which are not in the range. Such as 0

C. \(x = 3\) -> this is NOT TRUE as x is not equal to 3

D. \(x ≠ 3\) -> this is true as x cannot be equal to 3

E. \(x < -3\) -> Same as option A. This is partially true it covers only a part of the answer which is x < -3 and does not cover x > 3. So this cannot be "Must be true"

So, Answer will be D
Hope it helps!

Watch the following video to learn the Basics of Absolute Values


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