असमानता

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Inequality (असमानता)

Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Read and interpret all inequality symbols instantly (>, <, ≥, ≤, =, ≠).
2. Convert coded statements into standard inequality form.
3. Combine two or more statements to find a definite relationship.
4. Decide between "definitely true", "definitely false", and "either-or" conclusions.
5. Solve an inequality question in under 30 seconds.

The Hook: One Symbol, One Mark, Ten Seconds

Inequality is a regular feature of the reasoning sections of banking exams (IBPS PO/Clerk, SBI PO) and also appears in SSC and other government exams. Each question takes a trained student barely 15–30 seconds — while an untrained student burns 2 minutes and still picks the wrong option. The difference is not intelligence. It is one small skill: knowing when a conclusion is *definitely* true and when it is only *maybe* true. Master that one idea, and these become some of the fastest marks in your reasoning section.

The Common Struggle (You Are Not Alone)

Most students stumble on Inequality for three reasons:

• They try to "feel" the answer instead of chaining the symbols logically.
• They forget that ≥ means "greater OR equal" — so "A > B" is NOT guaranteed by "A ≥ B".
• They panic when the symbols are coded (like `A ★ B` meaning `A > B`).

Every one of these is fixable in the next 20 minutes. Let's go chunk by chunk.

Chunk 1: The Six Symbols — Your Alphabet

Golden rule of reading: The symbol always opens toward the bigger quantity. `A > B` means A is bigger.

🧠 Active Recall: Without looking up — what does `P ≤ Q` mean in words? Say it aloud: "P is less than or equal to Q." Now write the symbol for "X is at least Y". (Answer: X ≥ Y.)

Chunk 2: Combining Statements — The Chain Rule

Examiners give you statements like:

> A > B, B ≥ C, C = D

and ask whether a conclusion like `A > D` follows.

The method: build one chain.

A > B ≥ C = D

Now compare A and D by travelling along the chain. The rule:

• If the path from A to D contains only >, ≥, = (all "forward-friendly"), then A > D holds only if at least one strict `>` exists on the path. Here A > B is strict, so A > D is definitely true.
• If the path contains both > and < facing each other (e.g., A > B < C), the chain breaks — no definite relation between A and C.

The Gate Rule (mnemonic): Think of `>` and `<` as gates.

• Gates opening the same way (>, ≥, =) → you can walk through; strict gate (>) anywhere means strict result.
• Gates clashing (A > B < C) → the door slams shut. Relation: cannot be determined.

🧠 Active Recall: Statements: M ≥ N, N > O. Does M > O follow? (Yes — same-direction gates, and N > O is strict.) Does M = O follow? (No — M is strictly greater.)

Chunk 3: The Three Verdicts

Every conclusion gets exactly one of three verdicts:

1. Definitely True — follows in every possible case.
2. Definitely False — contradicted in every possible case.
3. Cannot Be Determined (CBD) — true in some cases, false in others.

The trap is verdict 3 disguised as verdict 1. Example:

> Statement: A ≥ B. Conclusion: A > B.

Many students mark "true". Wrong! A could equal B. Since A > B is not true in *every* case, the verdict is CBD (as a standalone conclusion, it does not follow).

Mantra: *"≥ gives you 'at least', never 'strictly more'."*

🧠 Active Recall: Statement: X ≤ Y. Conclusion: X < Y. Follows? (No — X could equal Y.)

Chunk 4: Either-Or (Complementary Pair)

Sometimes neither conclusion follows alone, but together they cover all possibilities. Then the answer is "either I or II follows".

Conditions for either-or:

1. Both conclusions have the same two elements.
2. Both are individually not definite.
3. Together they are exhaustive (e.g., A > B and A ≤ B cover everything).

Example:

> Statements: P ≥ Q, Q = R. Conclusions: I. P > R II. P = R

P ≥ R is definite, but P > R alone isn't, and P = R alone isn't. Together they exhaust all cases → Either I or II follows.

Chunk 5: Coded Inequalities

Banking exams love codes:

> 'A @ B' means A > B; 'A # B' means A < B; 'A $ B' means A = B; 'A % B' means A ≥ B; 'A & B' means A ≤ B.

Strategy: Before touching the conclusions, decode every statement into plain symbols on your rough sheet. Never solve in coded form — that's where silly errors breed.

🧠 Active Recall: If 'A % B' means A ≥ B and 'A @ B' means A > B, decode: "P % Q, Q @ R". (P ≥ Q, Q > R → so P > R follows.)

Chunk 6: The 30-Second Solving Ritual

1. Decode (if coded) — write plain symbols.
2. Chain — merge statements into one line.
3. Check each conclusion — same-direction gates? Strict gate present?
4. Check either-or — same elements, both indefinite, exhaustive?
5. Mark and move — never re-read.

⚠️ Caution: Examiner's Trap

Trap 1 — Multiple possible relations shown in options.
The examiner writes a conclusion like "A ≥ B" when the chain only proves "A > B". Students think "greater means greater-or-equal is also satisfied" and mark it true. Technically A > B does imply A ≥ B is true — but the reverse trap is deadlier: chain proves A ≥ B, option says A > B. That does NOT follow. Always check the *direction* of the implication.

Trap 2 — Confusing coded language.
Codes like '@', '#', '$' are deliberately chosen to look alike, and the meanings are scrambled across questions in the same paper. Decoding mentally (instead of on paper) causes sign-flips. Write the decoded chain every single time — it costs 5 seconds and saves the mark.

Trap 3 — Double-relation (either-or) questions.
The examiner gives two conclusions that *look* like they both fail, pushing you toward "neither follows". But if the pair is complementary (A > B / A ≤ B, or A ≥ B / A < B), the correct answer is "either-or". Before marking "neither", always run the 3-condition either-or check.

Summary Table

📌 Review this chapter after 2 days. Redo the Active Recall prompts from memory — if you can answer all of them cold, Inequality is permanently yours.