|n| < 4 ??

[rahulg83]

If n is not equal to 0, is |n| < 4 ?

(1) n2 > 16

(2) 1/|n| > n

[DanaJ]

1. n^2 > 16 tells us that n is either smaller than -4 or greater than 4, since 16 = 4^2 = (-4)^2. Knowing this, we can safely say that |n| > 4, so the answer to the question in the stem will be a clear NO. 1 is therefore sufficient.

2. 1/|n| > n immediately makes me think of negative numbers. Since 4 is your "flag" here, pick two numbers on each side of -4:
a. n = -6, then 1/|n| = 1/6. In this case, |n| = 6 > 4.
b. n = -2, then 1/|n| = 1/2. However, this time you get |n| = 2 < 4.
Since there are two possible outcomes, we'll note that 2 is insufficient.

The answer will therefore be A.

[rahulg83]

1. n^2 > 16 tells us that n is either smaller than -4 or greater than 4, since 16 = 4^2 = (-4)^2. Knowing this, we can safely say that |n| > 4, so the answer to the question in the stem will be a clear NO. 1 is therefore sufficient.

2. 1/|n| > n immediately makes me think of negative numbers. Since 4 is your "flag" here, pick two numbers on each side of -4:
a. n = -6, then 1/|n| = 1/6. In this case, |n| = 6 > 4.
b. n = -2, then 1/|n| = 1/2. However, this time you get |n| = 2 < 4.
Since there are two possible outcomes, we'll note that 2 is insufficient.

The answer will therefore be A.

Hey Dana, thanks..this approach works..
I don't know what i was up to, but it seemed to me that two statements are contradicting each other
Statement 1: n^2>16 that means n>4 or n<-4(excluding), right?
Statement 2: 1/|n|>n
i.e. n*|n|<1, now if n is negative, this inequality becomes n*(-n)<1 or n^2>-1, which is true for any negative value of n..

and if n is positive, n^2<1, that means n is b/w 1 and -1 (excluding)

now if we see statement 1 and 2, n has different set of values altogether. I am sure i am missing something here, but what??

Moreover, i usually solve modulus questions by this approach only. But lately, i am getting more incorrect than correct . So is it a better idea to plug-in values rather than scratching head by using terms (as in n or x) giving in the question?
Help!!

[DanaJ]

The two statements are NOT CONTRADICTORY as long as there is some sort of overlap in the two intervals. And you can clearly see that there is:
- statement 1: n < -4
- statement 2: n is negative

Basically, the overlap of these two statements is an infinite interval, i.e. every number that's smaller than -4.

Plugging in numbers is not a favorite strategy of mine. I usually use it for counterexamples only (i.e. when I'm trying to prove that something is false), which was the case here. It's not a full-proof strategy otherwise and picking the wrong numbers might get you in trouble. My advice: use it wisely!