Lesson: Inequalities and Absolute Value

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Is |x−1|<1|x−1|<1 ?
(1) (x−1)2≤1(x−1)2≤1
(2) x2−1>0

Could someone help me this?
gmat-admin's picture

Are you sure you've correctly transcribed the question?
Also, what's the source of this question?

Cheers,
Brent

Hi Brent, have a query in the below question

https://gmatclub.com/forum/if-4-7-x-3-which-of-the-following-must-be-true-168681.html

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

While I got the equations right, i.e in st2: x<-5 or x>-1 and st 3 is always true, the question asks which of the statement must be true and since st2 yields 2 possible solutions for x, the answer for me is B. Really confused how people have taken st2 to be always true.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-4-7-x-3-which-of-the-following-must-be-tru...

Great question, Jalaj!

From the given information, we know for certain that x < -5

Given that x < -5, must statement II be true?
From statement II, you determined that EITHER x < -5 OR x > -1
So, if x < -5, must it be true that EITHER x < -5 OR x > -1?
In order to answer YES to the above question we only need one of the parts to be true: x < -5 OR x > -1
Since we can be certain that x < -5.

Notice that the question does NOT read "If x < -5, must it be true that BOTH x < -5 AND x > -1?"

Here's an analogous question:
Let's say you have a rock in your pocket, and someone asks you "Is it true that you have EITHER a rock in your pocket OR an elephant in your pocket?
The answer to that question is YES, it is true that I have EITHER a rock in my pocket OR an elephant in my pocket.

Does that help?

Cheers,
Brent

Hi Brent, in the question below statement III has 2 possible solutions: x < -1 and x > 3.
Therefore, x can have a value of -2, which negates the inequality question |x| > 3.
Therefore my answer is B. Can you help me identify where am I going wrong in this?

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

https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-138652-40.html
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-13...

Be careful! We cannot say "statement III has 2 possible solutions x < -1 AND x > 3"
The value of x cannot be less than -1 AND greater than 3 at the same time.
Instead, we can say that EITHER x < -1 OR x > 3

This is very similar to your question above.

I should also point out the error in your comment that "Therefore, x can have a value of -2, which negates the inequality question |x| > 3."
You are approaching this in the wrong direction.
We are told that it is 100% true that |x| > 3, and our job is to identify other statements (from I, II and III) that are 100% true.

Instead, you are assuming that statement III is true and trying to determine whether the given information (|x| > 3) is true.

Does this help?

Cheers,
Brent

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