Inequalities

In short

An inequality question gives relations between letters — such as P ≥ Q > R = S — and asks which stated conclusions must follow. The whole topic rests on one idea: a relation travels along a chain only while the arrows point the same way, and a single reversal breaks it. It appears in two dresses: direct inequalities using >, <, ≥, ≤, = and coded inequalities where symbols like @, #, $ stand for those relations. It is a guaranteed scoring block in IBPS, SBI, RBI, RRB, SSC, LIC and insurance exams, and in campus aptitude tests (TCS NQT, Infosys, Wipro, Capgemini, Accenture, Cognizant), because every question is solvable in under 30 seconds with certainty — unlike puzzles, there is no guesswork. The same logic drives eligibility rules in software: salary bands, cut-off marks, credit-score slabs and Excel IF conditions.

Explained

The symbols and what the words mean

• > greater than; < less than (smaller than); = equal to.
• ≥ greater than or equal to — stated in exams as "not less than", "not smaller than", "at least" — the safe test: ≥ denies only "less than".
• ≤ less than or equal to — "not greater than", "not more than".
• > is also written as "neither smaller than nor equal to"; < as "neither greater than nor equal to".
• = is also written as "neither greater than nor smaller than".
• ≠ not equal to — carries no direction at all, so it is almost never useful in a chain.
• Strict signs (>, <) and non-strict / weak signs (≥, ≤, =). The difference decides every "either-or" question.

The chain rule — the only law you need

• Write all statements as one chain joined on common letters, keeping the signs pointing one way (convert A < B to B > A so the whole chain reads left-to-right as ≥ or >).
• A conclusion about two letters follows only if there is an unbroken path between them in which every link points the same direction.
• If any link on that path is strict (> or <), the overall relation is strict. If all links are weak (≥, ≤, =), the overall relation is weak. If all links are =, the result is =.
• One reversal and the relation dies. In A > B < C, both A and C are bigger than B, but which of A and C is bigger is unknown — no conclusion.
• = is a perfect conductor: it passes any relation through unchanged (A > B = C → A > C).
• ≠ is an insulator with one exception: A = B ≠ C → A ≠ C is valid, but A > B ≠ C and A ≠ B ≠ C give nothing.

Step-by-step method

1. Decode every coded symbol into >, <, ≥, ≤, = beside the question — never reason in the codes.
2. Merge the statements into a single chain on the common letters; if two chains share no letter, keep them separate (nothing can be concluded across them).
3. Normalise direction so the chain reads one way (all > / ≥).
4. For each conclusion, find the two letters and walk the path: unbroken and same-direction → it follows; otherwise it does not.
5. If the derived relation is weak (A ≥ B) and the conclusion asks for strict (A > B) or for =, mark it *not definite* and go to the either-or check.
6. Answer in the exam's option format: only I, only II, both, either, neither, all.

Either-or (complementary pairs) — definitely true vs possibly true

• Definitely (must be) true: holds in *every* arrangement allowed by the statements.
• Possibly (may be) true: holds in *at least one* allowed arrangement — "P = Q" is possible when P ≤ Q is derived.
• The either-or criterion: "either I or II follows" is the answer when (a) neither conclusion is individually definite, and (b) the two together are guaranteed — they leave no third case.
• The pairs that satisfy (b) for the same two letters are the complementary pairs: (>, ≤), (≥, <), (=, ≠), and also (>, =) when the statements establish ≥.
• Classic trigger: statements give A ≥ B; conclusions are "A > B" and "A = B". Each alone is uncertain, but A ≥ B means one of them must hold → either-or follows.
• Not a complementary pair: "A > B" and "A < B" — they miss A = B, so if the statements give nothing, the answer is *neither follows*.
• Caution: the degenerate pair (≥, ≤) and (=, ≠) is logically always true even with no derivable relation; most exam keys do mark it "either-or", but answer keys have differed, so prefer a definite conclusion if one exists before choosing either-or.

Coded inequalities

• A legend defines, say: A % B = A is not smaller than B (A ≥ B); A @ B = A is neither smaller than nor equal to B (A > B); A # B = A is neither greater than nor equal to B (A < B); A $ B = A is not greater than B (A ≤ B); A \* B = A is neither greater than nor smaller than B (A = B).
• Codes change from question set to question set — re-read the legend every time; recycling last question's meaning is the commonest avoidable error.
• The conclusions are also coded, so decode them too before testing.

Filter / reverse questions and symbol-insertion

• Filter type: "In which of the following expressions is the expression A > D definitely true?" — test each option against the chain rule; exactly one option gives an unbroken same-direction path.
• Symbol-insertion type: "P ≥ Q ? R ? S — which symbols in the blanks make P > S definitely true?" Work backwards: the path P→S must be unbroken, and at least one link must be strict.
• Two-target type: two conclusions must hold simultaneously; a correct option must satisfy both.
• Impossible-conclusion type: asks which conclusion can never be true — it is the one contradicting a derived strict relation.

Algebraic inequalities (the companion quant block)

• Adding or subtracting the same number keeps the sign: a > b → a + k > b + k.
• Multiplying or dividing by a positive number keeps the sign; by a negative number reverses it: a > b and k < 0 → ak < bk.
• Taking reciprocals of two numbers of the same sign reverses the sign: 2 < 5 but 1/2 > 1/5.
• Squaring is safe only for non-negative quantities: −5 < 2 but 25 > 4.
• Compound inequality: 5 < x ≤ 9 with x an integer → x ∈ {6, 7, 8, 9}.
• Quadratic comparison (x vs y type): solve both equations by factorisation; compare each root of one with each root of the other. If every x-root relates the same way to every y-root, a relation exists; if the ranges overlap, the answer is cannot be determined / no relation.
• Modulus: |x| < a → −a < x < a; |x| > a → x < −a or x > a.

Key points

• Chain rule - a relation passes between two letters only along an unbroken, same-direction path.
• One strict link - any > or < in a valid path makes the final relation strict.
• All-weak path - a path of only ≥, ≤ and = yields ≥ or ≤, never >.
• Equality conducts - A > B = C gives A > C; = never weakens a relation.
• Direction reversal kills - A > B < C yields no relation between A and C.
• ≠ is directionless - usable only as A = B ≠ C → A ≠ C.
• Not greater than = ≤ and not smaller than = ≥ - the two most misread phrases.
• Neither greater nor equal to = <; neither greater nor smaller than = =.
• Definitely true - holds in every case; possibly true - holds in at least one case.
• Either-or - chosen only when neither conclusion alone is definite *and* together they exhaust all cases.
• Complementary pairs - (>, ≤), (≥, <), (=, ≠), and (>, =) when ≥ is derived.
• (>, <) is not complementary - the case of equality is missing.
• Unconnected letters - two chains with no common letter give no cross conclusions.
• Decode first - rewrite coded symbols in standard form; legends change per set.
• Multiply by a negative - reverse the inequality sign (algebraic questions).
• Overlapping root ranges - in x-versus-y quadratics, the answer is "relation cannot be established".
• Speed - each question should take 20-30 seconds; accuracy here should be 100%.

Formulas and examples

Example 1 — direct inequality with an either-or Statements: P ≥ Q > R = S ≤ T Conclusions: I. P > S II. T > R III. Either T > R or T = R IV. Q < T

• Path P→S: P ≥ Q > R = S, one strict link → P > S. I follows.
• Path T→R: T ≥ S = R (reading S ≤ T backwards) → all weak → T ≥ R only, so "T > R" is not definite. II does not follow.
• But T ≥ R means T > R or T = R, and neither alone is definite → III follows (either-or).
• Path Q→T: Q > R = S ≤ T — direction reverses at S ≤ T → no relation. IV does not follow.

Answer: I and III follow.

Example 2 — coded inequality Legend: X % Y = X ≥ Y; X @ Y = X > Y; X # Y = X < Y; X $ Y = X ≤ Y; X \* Y = X = Y. Statements: M @ N, N % O, O # P, P $ Q Decoded chain: M > N ≥ O < P ≤ Q

• I. Q @ O (Q > O): O < P ≤ Q → O < Q → follows.
• II. M @ O (M > O): M > N ≥ O → follows.
• III. M % P (M ≥ P): path M > N ≥ O < P reverses at O < P → does not follow.
• IV. N # Q (N < Q): N ≥ O < P ≤ Q reverses → does not follow.

Answer: only I and II.

Example 3 — filter type (choose the expression) Which expression makes both "M > P" and "N ≥ Q" definitely true? (a) M > N ≥ P = Q (b) M ≥ N > P ≤ Q (c) M > N ≤ P ≥ Q (d) M = N ≥ P < Q

• (a) M > N ≥ P → M > P ✓; N ≥ P = Q → N ≥ Q ✓ → both true.
• (b) M ≥ N > P → M > P ✓; but N > P ≤ Q reverses → N vs Q unknown ✗.
• (c) M > N ≤ P reverses → M vs P unknown ✗.
• (d) M = N ≥ P → M ≥ P, not M > P ✗.

Answer: (a).

Example 4 — symbol insertion Fill the blanks so that A > E is definitely true: A ≥ B _ C _ D _ E

• The path must stay in one direction and contain at least one strict sign, because A ≥ B alone is weak.
• Putting ≥, ≥, ≥ gives A ≥ E — not enough. Putting >, =, ≥ gives A ≥ B > C = D ≥ E → A > E ✓.
• Any option containing <, ≤ or ≠ in the path breaks it.

Answer: a set such as (>, =, ≥) — i.e. at least one strict sign, and no reversal.

Example 5 — possibility question and x-versus-y (i) Statements: A ≤ B < C = D ≥ E. Which is possibly true?

• "C > A" → A ≤ B < C gives A < C → definitely true (not merely possible).
• "D > B" → B < C = D → definitely true.
• "A > E" → path A ≤ B < C = D ≥ E reverses at D ≥ E → no relation → possibly true (e.g. A = 3, B = 4, C = D = 9, E = 1 gives A > E; A = 3, B = 4, C = D = 9, E = 8 gives A < E).

(ii) I. x² − 11x + 30 = 0 → (x − 5)(x − 6) = 0 → x = 5, 6. II. y² − 15y + 56 = 0 → (y − 7)(y − 8) = 0 → y = 7, 8. Every y-root exceeds every x-root → x < y. (iii) I. 2x² − 7x + 3 = 0 → (2x − 1)(x − 3) = 0 → x = 0.5, 3. II. y² − 5y + 6 = 0 → y = 2, 3. x = 3 > y = 2 but x = 0.5 < y = 2 → relation cannot be established.

Where it is used

• Banking and insurance recruitment: IBPS PO/Clerk/RRB, SBI PO/Clerk, RBI Assistant and Grade B, LIC AAO, NIACL — 3 to 5 marks almost every paper, usually the fastest set in the reasoning section.
• Campus placement aptitude rounds: TCS NQT, Infosys, Wipro Elite, Capgemini, Accenture, Cognizant, Zoho and most service-company tests, where sectional cut-offs make these sure marks decisive.
• Government and PSU exams: SSC CGL/CHSL, RRB NTPC, state PSC prelims, and the general-aptitude part of GATE-adjacent screening tests.
• Software and data roles: eligibility and validation logic — SQL WHERE marks >= 60 AND backlogs <= 1, BETWEEN ranges, nested IF/IFS in Excel, business-rule engines; an off-by-one between > and ≥ is a real production bug, for example a GST slab or an EMI cut-off applied at the boundary.
• Credit, HR and policy slabs: CIBIL score bands (for instance "score ≥ 750 → pre-approved"), income-tax rules where a threshold reads "up to ₹ X" (≤) rather than "below ₹ X", scholarship rules such as family income not exceeding ₹ 8,00,000, and campus shortlists like "CGPA ≥ 7.0, no active backlog".
• Analyst and QA work: writing boundary-value test cases exactly on the ≥ / > edge, and reading tender or SLA clauses such as "not less than 95% uptime".

Interview questions

• Q: In P > Q < R, what can you say about P and R? A: Nothing — both are greater than Q, but their mutual relation is undetermined because the direction reverses at Q.
• Q: From A ≥ B > C, does A > C follow? A: Yes. The path is same-direction and contains a strict link, so the result is strict: A > C.
• Q: From A ≥ B ≥ C, can you conclude A > C? A: No, only A ≥ C; A = B = C is allowed, so the strict conclusion is not definite.
• Q: What does "P is not smaller than Q" mean? A: P ≥ Q — it rules out only P < Q, so equality is permitted.
• Q: When is the answer "either I or II follows"? A: When neither conclusion is individually definite but together they cover every possibility — typically a ≥ relation split into > and =.
• Q: Are "A > B" and "A < B" a complementary pair? A: No. They omit A = B, so if nothing is derivable the correct answer is that neither follows.
• Q: Difference between "definitely true" and "possibly true"? A: Definitely true holds in every arrangement consistent with the statements; possibly true holds in at least one.
• Q: Can anything be concluded from A > B ≠ C? A: No. The ≠ sign carries no direction, so C may be above, below or unrelated in magnitude to A.
• Q: Then is ≠ ever usable? A: Yes, with equality: A = B ≠ C gives A ≠ C.
• Q: Two statements share no common letter — say A > B and C > D. What follows? A: Nothing across the chains; conclusions are possible only within each chain.
• Q: x² = 49 and y = 7 — what is the relation between x and y? A: x = ±7, so x ≤ y; neither x < y nor x = y is definite individually, making it an either-or case.
• Q: If a > b and you multiply both sides by −3, what happens? A: The inequality reverses: −3a < −3b.
• Q: How does this topic matter on the job? A: Business rules and queries are inequalities; confusing ≥ with > shifts an entire eligibility boundary — for example excluding a candidate who exactly meets a 7.0 CGPA cut-off.

Common mistakes

• Reading "not greater than" as ≥ (it is ≤) and "not less than" as ≤ (it is ≥).
• Concluding a strict relation (A > C) from an all-weak path (A ≥ B ≥ C) — the commonest single error.
• Treating the derived ≥ case as "neither follows" instead of spotting the either-or with > and =.
• Choosing either-or for the non-complementary pair (>, <), or choosing it when one conclusion is already definitely true.
• Carrying a relation across a direction reversal, as in A ≥ B ≤ C → A ≥ C.
• Reusing the previous question's symbol legend in coded sets instead of re-reading the new one.
• In x-versus-y quadratics, comparing only the larger roots and declaring a relation when the root ranges overlap.
• Forgetting to reverse the sign after multiplying or dividing by a negative quantity, and squaring an inequality containing negative values.

Related topics

Syllogisms · Statements and conclusions · Data sufficiency · Coding-decoding · Quadratic equations and quantity comparison · Seating arrangement and puzzles