If |x - 1| 2, which of the following must be true?

Question

If |x - 1| > 2, which of the following must be true?

(I)  |x|>3  
(II)  x^{2} >9  
(III)  x>3

A. I only 
B. II only
C. I and II only
D. II and III only 
E. None of the above.

Bunuel · Accepted Answer

|x - 1| > 2; x - 1 > 2 or x - 1 < -2; x > 3 or x < -1. If x = -2, none of the options must be true. Answer: E.

Bunuel · Answer

Solution is given here: https://gmatclub.com/forum/if-x-1-2-whi ... l#p2033362 Theory is below 10. Absolute Value Theory Math: Absolute value (Modulus) Absolute Value: Tips and hints Properties of Absolute Values on the GMAT Absolute modulus : A better understanding GMAT Question Patterns: Abolute Value The Reason Behind Absolute Value Questions on the GMAT Questions Inequality and absolute value questions from my collection The E-GMAT Question Series on ABSOLUTE VALUE DS Questions PS Questions For more check Ultimate GMAT Quantitative Megathread If still not clear please ask specific questions.

rahul16singh28 · Answer

Take  x = -1.1 ..... All the conditions will fail... Hence E.

renjana · Answer

can someone please explain this question ! i cannot understand this one

GMATWhizTeam · Answer

Solution: We are given |x - 1| > 2 . So there can be 2 case: (1) x-1>2 and (2)-(x-1)>2 Case 1: x-1>2 ⇒ x>3 Case 2: -(x-1)>2 ⇒ -x+1>2 ⇒ -x>2-1 ⇒ -x>1 ⇒ x<-1 So we get the range of x for |x - 1| > 2 as ⇒ x>3 and ⇒ x<-1 . Now lets look into the statements: 1. |x|>3 Not always true. Lets say we have x=-2 (following the range) then |x| is not >3 2. x^{2} >9 Not always true. Lets say we have x=-2 (following the range) then x^2 is not >9 3. x>3 Not always true. Because we got the range as ⇒ x>3 AND ⇒ x<-1 . We see that none of 3 statement must always be true. Hence the right answer is Option E .

nazii · Answer

Dear Friends I write this for those in future maybe have the same as my problem in understanding this question
Before this question I was solving another question, which I share the link here
https://gmatclub.com/forum/if-6-4-x-5-w ... l#p3113405
I had the problem with these two.
Here is the simple logic
for such questions this is the solution I learned from my mistakes
First: find the proper region of x from the question stem
Second and the most important things to remember: do not solve the equations (in(l),(ll), (lll)) for finding the region of X, instead, try to replace different value of X based on the appropriate region you found from the question stem. Then check that do the values you assigned for each part, make that statement always true or not. If not, Omit that answer choice.
My own problem before was that I tried to solve other equations in the answer choices and then find the intersection between each and the proper region of X from question stem. This is very confusing for me.
The important note is that the proper value of X is determined by question stem (which you have to solve to find it). In answer choices you should look that , do the value from X , obtained by the question stem, satisfy the equation always or not
I hope it was clear

BrushMyQuant · Answer

|x - 1| > 2, which of the following must be true?

As we have \|x - 1\| in the equation so we will have two cases
-Case 1: x - 1 ≥ 0 => x ≥ 1 => \|x - 1\| = x - 1 => x - 1 > 2 => x > 3 But condition was x ≥ 1 and x > 3 is in the same range => x > 3	-Case 2: x - 1 ≤ 0 => x ≤ 1 => \|x - 1\| = -(x - 1) = 1 - x => 1 - x > 2 => x < -1 But condition was x ≤ 1 and x < -1 is in the same range => x < -1

=> x > 3 and x < -1

(I) \(|x|>3\)
=> x > 3 and x < -3
But x can be between -3 and -1 also based on the solution
=> This should NOT be TRUE ALWAYS

(II) \(x^{2} >9\)
Taking square roots on both the sides we will get |x| > 3
This is same as Statement 1
=> This should NOT be TRUE ALWAYS

(III) \(x>3\)
This need not be true for the cases when < -1
=> This should NOT be TRUE ALWAYS

So, Answer will be E
Hope it helps!

Watch the following video to MASTER Absolute Values

Abdullah9097 · Answer

Here the question says ' Must be true'.
So to be 'Must be true', the options should be satisfied by the both two answers of the stem?

Bunuel · Answer

You’re thinking in terms of “two answers,” but the inequality doesn’t give two numbers, it gives two ranges: x > 3 or x < -1.

“Must be true” means a statement has to hold for every x in the entire solution set (the union of both ranges). If it fails for even one permissible x from either side, it’s not must-be-true. That’s why showing a single allowed counterexample is enough to rule an option out, and why none of I, II, III qualifies here.

Hope it's clear.

If |x - 1| > 2, which of the following must be true?

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If |x - 1| > 2, which of the following must be true?

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