How to prove that (a^2 + b^2 -2)/(a + b - 2) < 1 + b if a<b and a,b>1?

adarascarlet80 2022-10-01 Answered
How to prove that a 2 + b 2 2 a + b 2 < 1 + b if a < b and a , b > 1?
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Answers (2)

Giancarlo Phelps
Answered 2022-10-02 Author has 10 answers
Let a = 1 + x and b = 1 + y
Thus,
1 + b a 2 + b 2 2 a + b 2 = 2 + y x 2 + y 2 + 2 x + 2 y x + y = x ( y x ) x + y > 0.
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smetuwh
Answered 2022-10-03 Author has 2 answers
your inequality is equivalent to
a 2 + b 2 2 < ( 1 + b ) ( a + b 2 )
and this is equivalent to
( 1 a ) ( a b ) > 0
this is true!
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