The scale and the different ball

1. As 3<6, there is not enough information capacity to distinguish all cases.
2. Model the possible states: and also rare = heavier or lighter (2 options).
3. Model the possible states: odd ball = 1, 2 or 3 (3 options).

Back to the problem
Answer: No, it is impossible in a single weighing.
Model the possible states:

• rare ball = 1, 2 or 3 (3 options),
• and also rare = heavier or lighter (2 options).

Total:

$$ 3\times2=6\ \text{estados}. $$

A single weighing of a dish scale only has 3 results:

1. left low,
2. balance,
3. low right.

Like $3<6$, there is not enough information capacity to distinguish all cases.
Conclusion: in a weighing you cannot simultaneously determine identity and direction (plus/minus).