The following is the problem on 63rd page of Zakon's "Basic Mathematics"
(iii) If x > y ≥ 0, then
x2 > y2 and x3 > y3 ≥ 0 (where x3 = x2x);
x4 > y4 ≥ 0 (where x4 = x3x).
Which (if any) of these propositions remain valid also if x or y is negative? Give proof.
I am able to solve the first part of the problem. Got stuck in the second. If somebody can give me a hint of how to determine whether the proposition is valid if x or y is negative.
Please you all have a look at it and let me know if I understood what Tonio tried to explain me. Please let me know if my understanding of Tonio's solution is Okay or Not.
x2 > y2 => (x-y)(x+y) > 0
=> (x-y) > 0 & (x+y) > 0 OR (x-y) < 0 & (x+y) < 0
(Both the terms +ve) OR (Both the terms –ve)
=> x > y & x > -y OR => x < y & x < -y
=> x > |y| OR => x < |y|
=> x > 0 OR => x < 0
Thus x2 > y2 holds true only when:
1. x > |y| if x > 0
2. x < |y| if x < 0