Transforming Equal Output to Include Inequality in Mathematica

Questions
1

set = {{2, a, 2 + a}, {1, 1, a}}
Equal @@ Divide @@ %

Execute the code to generate this equation.

2 == a == (2 + a)/a

However, I need the output in this specific form as follows. How can I adjust the code to achieve it?

2 != a == (2 + a)/a
2

Most likely there are more elegant versions, but this should work:

#1 @@ #2 & @@@ Transpose[{{Unequal, Equal}, Partition[Divide @@ set, 2, 1]}]

EDIT:
Special requirements need special solutions, I guess …

#1 != #2 == #3 & @@ (Divide @@ set)
3

Thank you! This approach can resolve this issue.