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Rephase the question: The question is really asking if -8<k<-2?

(1) k^2 - 7k -18 >0 is the same as (k-9)*(k+2) > 0
From this, we know that the critical points are k=9 and k=-2. The range of k where (k-9)*(k+2) > 0 are k<-2 and k>9. INSUFFICIENT.

(2) 1/k > 1/2
From this, we know that 0<k<2. This is SUFFICIENT since k will never be between -8 and -2.

Here, we have so: (k+2)*(k-9) > 0 <k> 9 or k < -2.

As |k| < 8, we are sure that -8 < k < -2.

No, this is incorrect. |k|<8 is the same as -8<k<8. We don't know for sure that k is between -8 and 8. The answer only say k<-2. If k=-13, then k will NOT be between -8 and 8.

Here, we have so: (k+2)*(k-9) > 0 <k> 9 or k < -2.

As |k| < 8, we are sure that -8 < k < -2.

No, this is incorrect. |k|<8 is the same as -8<k<8. We don't know for sure that k is between -8 and 8. The answer only say k<-2. If k=-13, then k will NOT be between -8 and 8.

We analyse the inequation without constraint and then we add |k| < 8 to conclude.... So, no, we do not say k = -13

[u]Statement 1[/u]
k^2-7k-18>0
(k-9)(k+2)>0
it implies that either k-9 and k+2 both are positive or negative. If they are positive then, k>9 (which is not possible according to the stem), and if they are negative, then k<2.
Since, k<2 does not clearly tells us whether k<2>1/2
This statement tell us two things. 1) that K is positive and 2) that k<2. Since K is positive and smaller than 2, so this statement definitely tells us that K cannot be <-2. So it is SUFFICIENT.
Therefore, the answer is B