IN USUAL NOTATION

Question

P.T. /A-B/>/A/-/B/

Answer 1

(1) Let A=-a² and B=b², then A-B=-a²-b²=-(a²+b²), and |A-B|=a²+b²; |A|-|B|=a²-b²; a²+b²>a²-b²⇒|A-B|>|A|-|B|.

(2) Let A=a² and B=-b², then A-B=|A-B|=a²+b²; |A|-|B|=a²-b²; a²+b²>a²-b²⇒|A-B|>|A|-|B|.

(3) Let A=a² and B=b², then |A|-|B|=a²-b², which can be positive or negative; but |A-B|=|a²-b²|>0 always. (3.1) When A<B, |A|-|B|<0, so |A-B|> |A|-|B|.

(3.2) When A>B, |A-B|=A-B; |A|-|B|=A-B, so |A-B|=|A|-|B|.

(4) Let A=-a² and B=-b², then |A-B|=|-a²+b²|; |A|-|B|=a²-b², which can be positive or negative.

(4.1) When A>B, -a²>-b², b²>a². |A-B|=b²-a²>0; |A|-|B|=a²-b²<0, |A-B|>|A|-|B|.

(4.2) When A<B, -a²<-b², b²<a². |A-B|=a²-b²; |A|-|B|=a²-b², |A-B|=|A|-|B|.

From these results the following conditions satisfy the inequality: (1), (2), (3.1), (4.1).

(i) A and B have opposite signs satisfies the inequality.

(ii) A and B are both positive then A<B satisfies the inequality.

(iii) A and B are both negative then A>B satisfies the inequality.