Ta có: \(a(b +2001) = ab + 2001a\)
\(b(a +2001)=ab + 2001b\)
Vì \(b >0\) nên \(b + 2001 > 0\).
a) Nếu \(a > b\) thì \(2001a > 2001b\)
\(\Rightarrow ab + 2001a > ab + 2001b\)
\(\Rightarrow a\left( {b + 2001} \right) > b\left( {a + 2001} \right) \)
\(\Rightarrow \displaystyle {a \over b} > {{a + 2001} \over {b + 2001}}\)
b) Nếu \(a < b\) thì \(2001a < 2001b\)
\(\Rightarrow ab + 2001a < ab + 2001b \)
\(\Rightarrow a\left( {b + 2001} \right) < b\left( {a + 2001} \right)\)
\(\displaystyle \Rightarrow {a \over b} < {{a + 2001} \over {b + 2001}}\)
c) Nếu \(a = b\) thì \(ab + 2001a = ab + 2001b\)
\(\Rightarrow a\left( {b + 2001} \right) = b\left( {a + 2001} \right)\)
\(\displaystyle \Rightarrow{a \over b} = {{a + 2001} \over {b + 2001}}\).
