Question

If a ≠ b, prove that (a+b)/2 is strictly between a and b. I would appreciate a hint on how to take on this proof. A rough sketch if you will.

Answer 1

First convince yourself this is true. Without loss of generality, let a < b.

Put a and b on a number line. Then where is (a+b)/2 ?

Now you can see what the relation is between a and (a+b)/2 and between b and (a+b)/2.

Answer 2

If you can't see how to do that, make some experiments. For example a = 0, b = 10.
a = -5 , b = -1.