I also got E with the following approach:
From the question |a| > |b| or, a^2 > b^2
or, (a-b)(a+b) > 0 and this means, either a > b and a > -b or, a < b and a < -b
Now, a > b and a > -b is possible only if a and b are both positive
and a < b and a < -b is possible only if a < 0.
Now, if a > 0 then a.|b| > 0 and a-b > 0 but, a.|b| < a-b may not be true.
Similarly, if a < 0 then both a.|b| and a-b will be < 0 but again, inequality may not be true.
Now stmt1 does not give any extra information. Insufficient.
Stmt2 also does not give any extra information. Insufficient.
When you say that a > b and a > -b is possible only if a and b are both positive, does that mean in all cases or only in some? If a =2 and b =-1 it holds true as well.