Inequalities for LIC AAO

Inequality questions are the fastest marks in the Reasoning section. A candidate who knows the rules answers each one in 20 to 30 seconds, and there is no judgement involved: the conclusion either follows or it does not. That speed matters in a prelims section of 35 questions in 20 minutes. It also means careless errors are the only way to lose these marks, so the rules below are worth learning exactly.

The one-direction rule

Two letters have a definite relation only if the chain between them runs in one direction: all "greater" signs (>, ≥) or all "smaller" signs (<, ≤), with any number of = signs mixed in.

• A > B ≥ C: one direction, so A > C.
• A > B < C: the direction changes at B, so A and C have no definite relation.
• A = B = C: equal throughout, so A = C.

Combining signs

When the chain runs one way, the result is strict (> or <) if any sign on the path is strict. It is ≥ or ≤ only if every sign on the path is ≥, ≤ or =.

Why "≥ and ≥" gives only ≥: if A ≥ B and B ≥ C, all three could be equal, so A > C is not certain.

Either–or conclusions

Choose "either I or II follows" only when all three conditions hold:

1. Neither conclusion follows on its own.
2. Both conclusions are about the same two letters.
3. Together they cover every possibility between them. The usual pairs are > with =, when the chain gives ≥, and < with =, when the chain gives ≤.

If the chain gives no relation at all, then A > C and A < C do not cover A = C. Neither follows, and it is not either–or.

Worked examples

Example 1. A ≥ B > C = D ≤ E. Conclusions: I. A > D, II. E ≥ C.

• A ≥ B > C = D runs one way and contains >, so A > D. ✓
• E ≥ D = C, so E ≥ C. ✓
• Both follow.

Example 2. P < Q ≤ R > S = T. Conclusions: I. P < R, II. Q > T.

• P < Q ≤ R gives P < R. ✓
• From Q to T: Q ≤ R, then R > S. The direction changes at R. ✗
• Only I follows.

Example 3. W = X ≥ Y ≥ Z. Conclusions: I. W > Z, II. W = Z.

• The path gives W ≥ Z. Neither > nor = is certain alone.
• Same two letters, and > with = covers ≥.
• Either I or II follows.

Example 4 (two statements). J > K ≥ L; M ≤ L < N. Conclusions: I. J > M, II. N > K.

• Join the statements at L: J > K ≥ L ≥ M, so J > M. ✓
• N > L and K ≥ L. Both are larger than L, so N and K have no relation. ✗
• Only I follows.

Example 5 (coded). Suppose @ means >, # means <, $ means =, % means ≥ and & means ≤. Statements: A @ B, B % C, C $ D. Conclusions: I. A @ D, II. D & B.

• Decode first: A > B ≥ C = D.
• A > D, which is A @ D. ✓
• B ≥ C = D, so D ≤ B, which is D & B. ✓
• Both follow.

Common mistakes

• Writing ≥ + ≥ as >. It stays ≥.
• Joining through a direction change. A > B < C tells you nothing about A and C.
• Calling "no relation" an either–or. Either–or needs the pair to cover every case, which only happens when a ≥ or ≤ relation exists.
• Reading a conclusion backwards. "E ≥ C" and "C ≤ E" are the same relation. Check which letter is on which side.

Practice set

1. M > N = O ≥ P. I. M > P, II. N ≥ P.
2. F ≤ G < H = I > J. I. F < I, II. G > J.
3. K ≥ L = M ≤ N. I. K ≥ N, II. N ≥ L.
4. R > S ≥ T; T = U < V. I. R > U, II. S ≥ V.
5. X ≤ Y ≤ Z. I. X < Z, II. X = Z.
6. A < B ≥ C. I. A < C, II. A > C.
7. Using the codes in Example 5: P % Q, Q @ R, R $ S. I. P @ S, II. S # Q.
8. Using the same codes: G & H, H $ I, I # J. I. G # J, II. G $ I.

Answers

1. Both follow. M > N = O ≥ P gives M > P; N = O ≥ P gives N ≥ P.
2. Only I follows. F ≤ G < H = I gives F < I. G < H = I > J changes direction at I.
3. Only II follows. K ≥ L = M ≤ N changes direction at M. N ≥ M = L gives N ≥ L.
4. Only I follows. R > S ≥ T = U gives R > U. S ≥ T = U < V changes direction at U.
5. Either I or II follows. The path gives X ≤ Z, and < with = covers it.
6. Neither follows. A < B ≥ C changes direction at B. A < C and A > C also leave out A = C, so this is not either–or.
7. Both follow. Decoded: P ≥ Q > R = S. So P > S, and S < Q.
8. Only I follows. Decoded: G ≤ H = I < J. So G < J. G ≤ I, so G = I is not certain.

What to do next

• Time yourself on ten inequality questions. The target is five minutes with no errors.
• Learn syllogism next; it uses the same "follows in every case" test.
• For the mains, practise coded inequalities alongside coding–decoding.