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

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.

if this question was COULD BE TRUE one, then answer would have beeen all I,II,III

nice question with nuanced wording. one must be careful with MUST-BE-TRUE and COULD-BE-TRUE questions!

 

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We are given that x < 2 or x > 4:

------------2------------4------------

So, x is somehere in the green region.

The question asks which of the statements MUST be true about x. The statement to be true must be true for all possible values of x. For instance, |x| > 4 is not necessarily true about x, because x can be 0, and in this case |x| > 4 won't be true. Similarly, we can see that none of the statements must be true about x.

Hello Bunuel,

I have a query for every must be true kinda questions,
Do we have to look for an answer choice, that should contain all the values of variable mentioned like in this question x>4 and x<2. So an answer choice should contain both of them?
Or if it is only a subset of this range of x mentioned, that also can be an answer?