Question 348246: Let x,y belongs to Z. prove that if x>0 and x+y < xy,then y>0.
Answer by nyc_function(2741)   (Show Source):
You can put this solution on YOUR website! Starting with the given,
x+y < xy
Subtract y from each side,
x < xy - y
x < y ( x - 1 )
Since x > 0,
0 < x < y(x-1)
0 < y(x-1)
By the trichotomy principle, y is either 0, negative, or positive.
Consider each case:
(a) y=0
0 < 0(x-1)=0 which is a contradiction.
(b) y<0
0 < y(x-1), divide by y to get
0 > (x-1) (recall that the inequality sign must change since we assumed y < 0)
1 > x, which is impossible since x is in Z and x > 0.
(c) The only possible choice, therefore, is y > 0.
That proves it.
|
|
|
