SOLUTION:

\(|x - 3| > 1\) means that:
\(x - 3 > 1\) --> \(x > 4\)
\(-(x - 3) > 1\) --> \(x < 2\)

So, we are given that \(x < 2\) or \(x > 4\).

I. \(|x| > 4\). Not necessarily true. Consider x = 0.
II. \(x^2 > 16\). Not necessarily true. Consider x = 0.
III. \(x > 4\). Not necessarily true. Consider x = 0.

Answer: E.

To understand the underline concept better practice other Trickiest Inequality Questions Type: Confusing Ranges.
_________________

Refer attached Image.

Ans. E

Easy. What |x−3|>1 depicts is that x is at a distance of more than 1 from 3 on the number line, and thus, x>4 OR x<2

For x=1, |x−3|>1, |1-3| = |-2| = 2 and 2>1 and hence, x=1 satisfies the inequality

Now quickly looking at all the options, x=1 does not fit in any of the three, and hence <1 min., you can arrive at your answer, i.e. (E) None

If |x−3|>1
then which of the following must be true?

I. |x|>4
II. x^2>16
III. x>4

A. I only
B. II only
C. III only
D. I, II and III
E. None

|x-3|>1
x>4 or x<2

I. |x|>4
Take for example x =1 => |x-3| = |1-3| = 2>1 satisfies the equation. But |1| is not >4. NOT NECESSARILY TRUE.
II. x^2>16
Take for example x =1 => |x-3| = |1-3| = 2>1 satisfies the equation. But 1^2 is not >16. NOT NECESSARILY TRUE.
III. x>4
Take for example x =1 => |x-3| = |1-3| = 2>1 satisfies the equation. But 1 is not > 4. NOT NECESSARILY TRUE.

IMO E
_________________

Two ways to solve :

Method 1) Find a counter example (easiest), X=-1 will satisfy the stem

I. |x|>4 - since x = -1 will satisfy this is not true
II. x^2>16 - since x = -1 will satisfy this is not true
III. x>4 - since x = -1 will satisfy this is not true

IMO E None

Method 2: solve using modulus properties to arrive at X cannot lie from 2 to 4. it can take any other value.

If |x−3|>1 then which of the following must be true?

I. |x|>4
II. x^2>16
III. x>4


|x−3|>1

I.e. x > 4 or x < 2

So none of them is essentially true as x may be 1 which negates all

Posted from my mobile device
_________________

|x – 3| > 1 will have two cases:
Case 1: x – 3 > 1 => x > 4
Case 2: -x + 3 > 1 => -x > -2 => x < 2

So we have x such that x<2 and x>4. Thus, x can take values such as -5, -1, -0.5, 0, 0.5, 1, 5, 10

I. |x|>4
We can have x = 0.5. Therefore, this need not be true.

II. x^2>16
Again we can have x = 0.5, where x^2 = 0.25. Therefore, this need not be true.

III. x>4
We can have x = -1. Therefore, this need not be true.

Answer E.

If |x−3|>1 then which of the following must be true?
I. |x|>4
II. x^2>16
III. x>4

Given that |x−3|>1, it follows that x<2 or x>4. Possible values of x include 0,1,5,...

I. |x|>4
Given that |x|>4, it follows that x<-4 or x>4.
For every value that |x−3|>1 can take (e.g. x=0,1,5), must it be true that x<-4 or x>4 ? Nope.
Although x=5 fits well within |x|>4, both x=0 and x=1 never do.
Statement I is not necessarily true

II. x^2>16
Given that x^2>16, it follows that (x-4)(x+4)>0 and then x<-4 or x>4.
For every value that |x−3|>1 can take (e.g. x=0,1,5), must it be true that x<-4 or x>4 ? Nope.
Although x=5 fits well within x^2>16, both x=0 and x=1 never do.
Statement II is not necessarily true

III. x>4
For every value that |x−3|>1 can take (e.g. x=0,1,5), must it be true that x>4 ? Obviously not. Both x=0 and x=1 never satisfy x>4.
Statement III is not necessarily true

Answer is (E) None.

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.
_________________