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

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.

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

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

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

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.