If |x| > 3, which of the following must be true?
1) x > 3
2) x^2 > 9
3) |x - 1| > 2
I only
II only
I and II only
II and III only
I, II, and III
Plug in a value that satisfies the condition that |x| > 3.
Let x=-4.
Eliminate any statement that is not valid for x=-4.
Statement I: x > 3
-4 > 3
Not true.
Eliminate any answer choice that includes I.
Eliminate A, C and E.
Compare the remaining answer choices.
B and D each include II.
Thus, II must be true, since it is included in both of the remaining answer choices.
Thus, we need to evaluate only statement III.
Statement III: |x-1| > 2
|x-1| > 2 implies that the distance between x and 1 is more than 2 units.
|x| > 3 implies that x is more that 3 units from 0.
Since x is more than 3 units from 0 -- in other words, x<-3 or x>3 -- the distance between x and 1 must be more than 2 units.
Thus, III must be true.
Eliminate B, which doesn't include III.
The correct answer is
D.
Please note the following:
If |x| > 3, then statement III must be true: |x-1| > 2.
This does NOT imply the reverse.
If |x-1| > 2, it does NOT have to be true that |x| > 3.
For example, x=-2 satisfies |x-1| > 2 but not |x| > 3.
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