一條數(香港青少年數學精英選拔賽)

a + b < 0………….(1)
ab < 0………..(2)
a < b…………(3)

from (1)
a + b < 0
a < -b……………….(4)
also by (1) x -1
-a > b………………..(5)

From (2)
ab < 0
that means one is +ve and the other is negative
if a> 0 and b < 0
then b < 0 < a, which will contradict with (3)

therefore, b > 0 and a < 0
b > 0
then –b < 0
-b < b…………..(6)

By (3), (4), (5), (6)
a < -b < b < -a

a<-b<-a

2007-01-31 13:22:19 補充：
a

2007-01-31 13:22:52 補充：
a

2007-01-31 13:24:04 補充：
a＜-b＜b＜-a

(1) If a+b＜0, than a<-b and b<-a.

(2) If ab＜0, either (a<0 and b>0) or (a>0 or b<0). That is,
(i) a<0

2007-01-30 18:14:02 補充：
(1) If a b ＜ 0, than a

2007-01-30 18:15:38 補充：
(1) If a b ＜ 0, than a ＜ -b and b ＜ -a.(2) If ab ＜ 0, either (a ＜ 0 and 0 ＜ b) or (0 ＜ a or b ＜ 0). That is,(i) a ＜ 0 ＜ b, or(ii) b ＜ 0 ＜ a.

2007-01-30 18:15:54 補充：
(3) If a ＜ b, then from (2), only (i) is satisfied.From (1), (2) and (3), we have a ＜ -b, b ＜ -a and a ＜ 0 ＜ b.Therefore, a ＜ -b ＜ 0 ＜ b ＜ -a, or a ＜ -b ＜ b ＜ -a.