First, I think you are misunderstanding what "Linear and Abstract Algebra" means. This probably belongs in the "Pre-University Algebra" section.
I presume you know that an "iff" (if and only if) statement has to be proved both ways. So you must prove both "if |x|< |y| then " and "if then |x|< |y|".
Now to prove the first, look at |x|< |y|. As is almost always the case with absolute value, break this into parts. If , |x|= x while if |x|= -x. Since each of x and y can be "non-negative" or "negative", there will be four cases.
1) Suppose x and y are both non-negative. Then |x|< |y| means . Now what can you say?
2) Suppose x and y are both negative. Then |x|< |y| means . Now what can you say about their squares?
3) Suppose x is non-negative and y is negative. Then |x|< |y| means . Now what can you say about their squares?
4) Suppose x is negative and y is non-negative. Then |x|< |y| means . Now what can you say about their squares?
I rather suspect that you can do the other way by looking at the same four cases.