The solution can be found by looking at each statement and what they imply:
Alice - "Rex is a knave"
This implies that Alice and Rex are of different types, as either Alice is a knight and rex is a knave, or Alice is a knave and Rex is not.
Rex - "It's false that Bob is a knave"
This implies that Rex and Bob are of the same type, as Rex is essentially calling bob a knight. If Rex were a knight, so would bob, and if Rex were a knave, so would bob.
Bob - "I am a knight or Alice is a knight."
This once again leads to 2 situation, either Bob is a knight, which would mean that Alice is a knave, or Bob is a knave, making Alice also a Knave.
From Alice's and Rex's remarks we can see that we have 2 situations
a) A - knight, R - knave, B - knave
b) A - knave, R - knight, B - knight.
However, according to Bob's statement:
if Bob were a knave, Alice would also be a knave. Thus, the only situation consistent with all the remarks is situation b.
Therefore, the answer is that:
Alice is a knave, and Rex and Bob are knights.