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

This question was provided by Crack Verbal for the Game of Timers Competition

 

_________________

Answer should be none.
As x=1 satisfies the exuation |x-3|>1
But does not satisfy any of the options.

|x-3|>1

Then we should consider two cases:
x-3>1 >> x>4
x-3<-1 >> x<2

So the solution is : (-∞;2) to (4;+∞)

I. |x|>4
then x>4; x<-4
Must be true

II. x^2>16
x>4; x<-4
the same as the 1st case
Must be true

III. x>4
Must be true

Answer: D. I, II and III

As explained in the attached picture, x<2 & x>4

All the given choices CAN BE TRUE, but not MUST BE TRUE.

Hence Ans should be (E)

|x-3|>1
i.e.
x-3>1 or -(x-3)<1
x>4 or x <-4
i.e. |x|>4
imo A.

Posted from my mobile device

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
_________________

|x−3|>1

if x>3
x-3>1
x>4

If x<3
x-3<-1 (multiply by -1)
x<2

So we have x in the range (x<2) and (x>4)

Now lets check the options

1)|x| > 4
x>4 for x>0
x<-4 for x<0
AND for x > 0
x>4

So this falls in the range.

2)x^2 > 16
sqrt of both side gives
|x| > 4
This is same as 1) above so again this is also fine.

3)x>4 again falls in the range.

So all the above falls in the range.

I would chose D as OA.

|x-3|>1

Either (x-3)>1 or (x-3)<-1

Either x>4 or x<2


IMO, Answer is (E)
All three statements only take x>4 into consideration but we also have x<2

So none of the statements MUST BE true

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.
_________________

Given |x - 3| > 1
Solving for x;
x - 3 > 1
x > 1 + 3
x > 4
Also, negating the right side of the original equation
x - 3 > -1
x > -1 + 3
x > 2
From the options available, only III fulfilled the condition, hence answer choice "C"

Posted from my mobile device
_________________