Inequalities

To test a conclusion such as P > S, find the path that joins P to S through the statements and read only the signs on that path. Everything else is noise.

1. All signs point the same way (only > ≥ =, or only < ≤ =): a relation is certain. If at least one sign on the path is strict (> or <), the result is strict; if they are only ≥ and =, the result is ≥; if all are =, the result is =.
2. The signs point both ways (a > and a < on the same path, or ≥ with ≤): P and S cannot be compared, and no definite conclusion about them follows.
3. ≥ is satisfied by > or by =. If the path gives P > S, then "P ≥ S" follows as well. But if the path gives only P ≥ S, neither "P > S" nor "P = S" follows on its own.
4. Either–or. When the path gives P ≥ S and the conclusions are I. P > S and II. P = S, neither follows alone, yet one of them must be true — the answer is either I or II. The same holds for a pair such as P ≥ S and P < S when P and S cannot be compared at all.
5. Coded inequalities. "P @ Q means P is not greater than Q" — decode every statement and both conclusions into ordinary signs first, then solve as above. "Not greater than" is ≤; "neither greater than nor equal to" is <.

The single most common mistake is jumping across a reversed sign: in A > B < C nothing at all can be said about A and C, however tempting A > C looks.

How to check: for a conclusion you think does not follow, invent numbers that obey every statement but break the conclusion. If you can, you are right.

Read only the path between the two letters · one direction → certain; both directions → no relation · any strict sign on the path → strict result · > or = makes ≥ true · path gives ≥ and the conclusions are > and = → either I or II · decode coded signs first: not greater = ≤, not smaller = ≥, neither greater nor equal = <

Statement: A ≥ B = C > D ≤ E. Conclusions: I. A > D II. E > C. → For I the path is A ≥ B = C > D: one direction with a strict sign, so A > D is true. For II the path is E ≥ D < C: the signs point both ways, so E and C cannot be compared — not definite. Answer: only I follows. Sanity check for II: A = 5, B = C = 4, D = 1, E = 2 obeys every statement, yet E is smaller than C.