Please remember to post the source (author and book / number / location of problem). Instructors cannot respond until we have this. Luckily, I recognize this one as one of ours.
I'm not following this:
1. Tells us that x / |x| < x
or 1/|x| < 1
The first bit is what the problem says; you rephrase it to 1/|x| < 1. How did you do this? It seems like you divided each side by x, but I'm not sure. If that is what you did, the statement is incomplete because we don't know if x is positive or negative. When we divide an inequality by a negative, we have to flip the inequality sign. So I would get two options:
if x is positive, then 1/|x| < 1
if x is negative, then 1/|x| > 1
And BOTH options now have to be tested.
You can see that this is the case more intuitively if you try some real numbers. From the question stem, x is not zero but could be anything else.
First, let's try x = -0.5. If I plug this into x / |x| < x I get -.5/.5 < -.5 or -1 < -.5 which is true, so -0.5 is a valid value for x. In this case, the absolute value of x is less than 1.
Next, let's try x = 2. If I plug this into x / |x| < x I get 2/2 < 2 or 1 < 2 which is true, so 2 is a valid value for x. In this case, the absolute value of x is greater than 1.
Contradictory results = insufficient.
Re: your point about the second statement, assume NOTHING. You can't assume that x is an integer. It is only limited to integers if the question tells you so; otherwise, decimals are possible.