D01-15

Replying to a pm:

You cannot decide on a particular method for all 'such and such questions'. Your methodology will change according to the question. I anyway do not endorse the positive/negative approach for mod questions. It's very time consuming and there are easier methods available in most cases.
What is the first thing you think about when you see a mod? I think that this term is either 0 or positive.
K^3 is too cumbersome to be dealt algebraically. So I try to deal with it arithmetically.

Is K a positive number? (Or rephrase - Is it that K cannot be 0 or negative?)

(1) |K^3| + 1 > K
0/Positive + Positive > K
K can be positive but of course K can be 0 and negative too here. This relation will still hold. Not sufficient.

(2) K + 1 > |K^3|
K + positive > 0/positive
Here, my first thought is that if K = 0, this relation still holds (and of course it holds for some positive values of K too)
It becomes 0+1 > 0
(Some negative values also satisfy this inequality but I don't need to go there. I need just one value and I got it.)
Not sufficient.

Using both statements, there are positive values of K that satisfy both equations and 0 satisfies them both too. So both together are not sufficient.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

I think this is a high-quality question and I agree with explanation.