1. The problem statement, all variables and given/known data
Given:
[itex]|x-y| < K[/itex]
[itex]x+y > K - 2[/itex]
[itex]0 < K < 1[/itex]
Prove:
[itex]\frac{|1-K+x|}{|1+y|} < 1[/itex]
3. The attempt at a solution
I have tried using the fact that [itex]|x-y| < K \Rightarrow -K < x-y < K \Rightarrow y-K < x < y+K[/itex] to write [itex]\frac{1-K+x}{1+y} < \frac{1+y}{1+y} = 1[/itex]
But I can't figure out how to show that the absolute value is less than one.
I have also been trying various applications of the triangle inequality with little success.
Any help would be greatly appreciated.
Given:
[itex]|x-y| < K[/itex]
[itex]x+y > K - 2[/itex]
[itex]0 < K < 1[/itex]
Prove:
[itex]\frac{|1-K+x|}{|1+y|} < 1[/itex]
3. The attempt at a solution
I have tried using the fact that [itex]|x-y| < K \Rightarrow -K < x-y < K \Rightarrow y-K < x < y+K[/itex] to write [itex]\frac{1-K+x}{1+y} < \frac{1+y}{1+y} = 1[/itex]
But I can't figure out how to show that the absolute value is less than one.
I have also been trying various applications of the triangle inequality with little success.
Any help would be greatly appreciated.
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