असमानता

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Inequality (असमानता)

Outline & Learning Goals

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

1. Read and understand the five inequality symbols instantly.
2. Decode coded inequalities (where symbols are replaced by words or signs).
3. Find the definite relationship between two elements using a chain.
4. Handle "either-or" (complementary pair) conclusions confidently.
5. Solve a standard 5-question inequality set in under 3 minutes.

The Hook: One Symbol Can Cost You a Government Job

In SSC CGL, IBPS PO, and RRB NTPC, inequality questions look like the easiest marks in the reasoning section — and they are, *if* you respect them. Many aspirants lose marks here not because they didn't know the rules, but because they read `>` as `<` in a hurry, or forgot the "either-or" case. This chapter will make sure that never happens to you. Inequality is pure logic — no formulas to memorise, no calculations. Just symbols, a chain, and discipline.

The Common Struggle (You're Not Alone)

Most students struggle with inequality for three reasons:

• Symbol confusion under pressure — `>` and `<` look similar when the clock is ticking.
• Broken chains — when the chain goes `A > B < C`, students force a relationship that doesn't exist.
• The either-or panic — when neither conclusion follows individually, students don't know what to mark.

We will fix all three, one chunk at a time.

Chunk 1: The Five Symbols — Your Alphabet

The Crocodile Rule 🐊: A hungry crocodile always opens its mouth toward the bigger meal. So in `A > B`, the mouth faces A — A is bigger. In `A < B`, the mouth faces B — B is bigger.

🧠 Active Recall: Without looking above — in `P < Q`, which element is bigger? Which way does the crocodile's mouth open?

Chunk 2: The Priority Order (The Golden Rule)

When two symbols fight, the stronger one wins. Remember this priority:

> `>` and `<` (strict) beat `≥` and `≤` (flexible), which beat `=`

Think of it like a family hierarchy:

• Strict symbols (`>`, `<`) = the strict father — his decision is final.
• Flexible symbols (`≥`, `≤`) = the adjustable mother — she allows equality too.
• Equal (`=`) = the child — adjusts to everyone.

Example: If `A ≥ B` and `B > C`, then `A > C` (strict `>` wins over flexible `≥`).

🧠 Active Recall: If `X > Y` and `Y ≥ Z`, what is the definite relation between X and Z — strict or flexible?

Chunk 3: Building the Chain

Inequality questions give you statements and ask you to check conclusions. The method:

Step 1: Write all statements in one line, joining common elements.

Step 2: Trace a path from the first element of the conclusion to the second.

Step 3: Apply the priority rule along the path.

Example:
Statements: `A > B ≥ C = D`
Conclusion I: `A > D` → Path: A > B ≥ C = D. The strict `>` dominates → True.
Conclusion II: `B > D` → Path: B ≥ C = D. Only flexible symbols → `B ≥ D`, not `B > D` → False.

The Gate Rule 🚪: If the path between two elements has opposite-facing signs (like `A > B < C`), the gate is closed — no definite relationship exists. Never force one.

🧠 Active Recall: In `M > N < O ≥ P`, can you find any definite relation between M and O? Why or why not?

Chunk 4: The Either-Or (Complementary Pair) Case

This is where examiners trap the most students. Sometimes neither conclusion follows individually, but together they cover all possibilities. Then the answer is "either I or II follows."

Three conditions must ALL be true:

1. Both conclusions have the same two elements.
2. Neither conclusion is definitely true on its own.
3. The two conclusions together form a complementary pair — one of these combinations:

• `>` and `≤`
• `<` and `≥`
• `=` and `≠` (equal vs. not equal)

Note: `>` and `<` alone are not a complementary pair, because they leave out the possibility of equality. A valid pair must cover *every* possible relationship between the two elements.

Example:
Statement: `A = B`
Conclusion I: `A > B` — False.
Conclusion II: `A ≤ B` — True (since A = B satisfies ≤).
Here II follows directly, so no either-or needed.

Truer either-or example:
Statement: `A ≥ B`
Conclusion I: `A > B` — not definite.
Conclusion II: `A = B` — not definite.
Both same elements, neither true, and together (`>` or `=`) they cover everything under `≥` → Either I or II follows.

🧠 Active Recall: Statement: `P ≤ Q`. Conclusions: I. `P < Q` II. `P = Q`. What is the answer?

Chunk 5: Coded Inequalities

Here symbols are disguised in words:

> 'A @ B' means 'A is greater than B'
> 'A # B' means 'A is less than or equal to B'

Strategy: Before solving, make a small translation table on your rough sheet:

Then rewrite every statement and conclusion in real symbols, and solve normally. Never try to solve in coded form in your head — that's exactly where silly mistakes happen.

🧠 Active Recall: If 'A $ B' means 'A ≥ B' and 'A % B' means 'A < B', translate: `P $ Q % R`.

Mnemonic Trick: "Crocodile Eats Big, Strict Dad Decides, Opposite Gates Close"

• 🐊 Crocodile eats big → mouth opens toward the bigger element.
• 👨 Strict dad decides → `>`/`<` override `≥`/`≤`.
• 🚪 Opposite gates close → `A > B < C` means no relation between A and C.

Say this line to yourself before every practice set.

Summary Table

📅 Review this chapter after 2 days. Redo one either-or example and one coded inequality from memory — if you can do both without notes, this chapter is permanently yours.

⚠️ Caution: Examiner's Trap

Trap 1: Multiple possible relations shown in options.
The examiner designs conclusions where a *possible* relation looks like a *definite* one. Example: from `A ≥ B`, the conclusion `A > B` is *possible* but not *definite*. Unless the question says "possibility," mark only what is 100% certain. When in doubt, ask: "Can I find even one case where this fails?" If yes, it's false.

Trap 2: Confusing language to rattle you.
Phrases like "A is not smaller than B" simply mean `A ≥ B`, and "A is not greater than B" means `A ≤ B`. Examiners use double negatives to slow you down. Translate every phrase into a symbol immediately — never reason in words.

Trap 3: Double-relation (either-or) questions.
Students see that neither conclusion is true and mark "neither follows" — forgetting to check the either-or conditions. Always run the 3-point checklist: same elements? neither true? complementary pair? If all three are yes, the answer is "either I or II follows." Missing this check is the single most common reason for losing marks in this chapter.