Does the Property Hold?

Algebra Level 2

We all know this must be true:

If a = b a=b and a + x = b + y , a+x=b+y, then x = y . x=y.

Does this mean that the following must be true?

If a , b , x , y a, b, x, y are positive integers, a ≥ b , a\geq{b}, and a + x ≥ b + y , a+x\geq{b+y}, then x ≥ y . x\geq{y}.

Yes No

Aidan Poor by Aidan Poor , 18, USA

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1 solution

Joseph Newton
Oct 13, 2018

Consider a = 1000 , b = 0 , x = 1 , y = 2 a=1000,b=0,x=1,y=2 . Then, despite y y being greater than x x , the inequality a + x ≥ b + y a+x\geq b+y holds true because 1000 + 1 ≥ 0 + 2 1000+1\geq0+2 . Therefore the answer is no , not necessarily.

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