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If |x| > 3, which of the following must be true?

I. x > 3

II. x^2 > 9

III. |x-1|>2


Option A:

I only

Option B:

II only

Option C:

I and II only

Option D:

II and III only

Option E:

I, II, and III

Difficulty Level



Option D is the correct answer.

Option Analysis

Given, |x| > 3, which means either x>3 or x<-3

Now check the statements

I. x > 3 – not always true as x can be smaller than -3.Thus option A,C & E is ruled out. Only B & D are left.

II. X^2 > 9 – Always true for x>3 or x<-3 .To check – if x = 4,5,6,7…. or -4,-5,-6,-7, x^2>9

III. |x-1|>2, which means (x-1)>2 —> x>3 (if x-1>0) – True it also means (x-1)<-2—- >x<-1

(if x-1<0) X<-1 satisfies x<-3.

Thus true Both II and III, is true