Inequalities

Overview

Inequalities is one of the most reliable scoring topics in IBPS PO Prelims Reasoning section. Unlike puzzles that can consume 10+ minutes, inequality questions—when mastered—take 15–30 seconds each. You'll typically see 3–5 questions in the exam, appearing either as direct inequalities (symbols given explicitly) or coded inequalities (symbols replaced by codes like @, #, $, %).

The core skill is simple: chain multiple two-variable relationships and determine whether a definite relationship exists between two variables that aren't directly compared. Once you internalize the chaining rules, this becomes a near-guaranteed scoring area. Most errors come from rushing or misreading coded symbols—not from conceptual difficulty.

Key Concepts

• **The Six Symbols**: The relationships are > (greater than), < (less than), = (equal to), ≥ (greater than or equal), ≤ (less than or equal), and ≠ (not equal). Every question tests your ability to chain these.

• **Chaining Rule**: You can only chain relationships when the connecting variable sits at opposite ends. A > B and B > C chains to A > B > C. A > B and C > B does NOT chain directly—you must reverse one.

• **Definite vs. Indefinite**: A conclusion is definite only if ALL elements in the chain point the same direction (all > or all <, with = allowed in between). If directions conflict or "either-or" is needed, the relationship is indefinite.

• **Equal Sign Behavior**: The = sign acts as a "pass-through." A ≥ B = C means A ≥ C. The equal sign doesn't block the relationship—it transfers it.

• **Complementary Pairs (Either-Or)**: When neither X > Y nor X < Y is individually true, but together they cover all possibilities, "either (i) or (ii)" becomes the answer. This happens when the chain shows X and Y are definitely not equal, but direction is unclear.

• **Coded Inequalities**: The symbols are replaced with arbitrary codes. Your first job is to decode and write down the symbol meanings. Then solve exactly as direct inequalities.

• **Negation Trap**: "A is not greater than B" means A ≤ B. "A is neither greater than nor equal to B" means A < B. Read negations carefully.

Formulas / Key Facts

| Combination | Can Chain? | Result | |-------------|------------|--------| | > with > | Yes | > | | < with < | Yes | < | | > with ≥ | Yes | > | | ≥ with ≥ | Yes | ≥ | | < with ≤ | Yes | < | | ≤ with ≤ | Yes | ≤ | | > with = | Yes | > | | < with = | Yes | < | | > with < | No | Indefinite | | ≥ with < | No | Indefinite |

**Priority Rule**: When chaining ≥ and =, result is ≥. When chaining > and =, result is >. The "strict" symbol dominates over "equal."

**Either-Or Condition**: Conclusions X > Y and X < Y form a valid either-or pair only when X ≠ Y is definite but direction is not.

Worked Examples

### Example 1: Direct Inequality **Statement**: P > Q ≥ R = S < T ≤ U

**Conclusions**: (i) P > S (ii) T > R

**Solution**:

• For P > S: Trace P > Q ≥ R = S. Direction is consistent (all leftward pointing). P > Q ≥ R = S gives P > S. **True**.
• For T > R: From the chain, R = S < T. So T > S = R, meaning T > R. **True**.

**Answer**: Both (i) and (ii) are true.

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### Example 2: Coded Inequality **Codes**: @ means >, # means <, $ means =, % means ≥, & means ≤

**Statement**: A @ B $ C # D % E

**Decode**: A > B = C < D ≥ E

**Conclusions**: (i) A > C (ii) E < C

**Solution**:

• For A > C: A > B = C. Chain works. A > C. **True**.
• For E < C: From C < D ≥ E, we get C < D and D ≥ E. For E and C: C < D and E ≤ D. This means C < D and E ≤ D, but we can't determine if E is greater than, less than, or equal to C. **False** (not definite).

**Answer**: Only (i) is true.

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### Example 3: Either-Or Situation **Statement**: M ≥ N > O; O < P ≤ Q

**Conclusions**: (i) M > P (ii) M < P

**Solution**:

• Chain for M and P: M ≥ N > O and O < P. So M ≥ N > O < P.
• Direction breaks at O (> then <). No definite relationship between M and P.
• But can we say M ≠ P? Not necessarily—M could equal P if values work out.
• Neither conclusion is individually true, and they don't form a valid either-or (M = P is possible).

**Answer**: Neither (i) nor (ii) is true.

Common Mistakes

• **Reversing without flipping the symbol**: When you need to write B > A instead of A < B to chain, students forget that the inequality direction must flip. A < B is the same as B > A—always rewrite to make chains flow one direction.

• **Treating ≥ as strictly >**: A ≥ B does NOT mean A > B. If the conclusion asks "A > B" and the statement only gives A ≥ B, the conclusion is **not definite** (A could equal B).

• **Misreading coded symbols**: Under exam pressure, students decode @ as > once, then accidentally read it as < later. Write the decoded statement fully before solving.

• **Ignoring the "either-or" option**: When both individual conclusions are false but they're complementary (one must be true), the answer is "either (i) or (ii)." Students often mark "neither" without checking this.

• **Assuming relationships that don't exist**: If A > B and C > B, you CANNOT conclude anything about A and C. Both are greater than B, but their relationship to each other is unknown.

Quick Reference

• Same-direction symbols chain; opposite directions break the chain.
• Strict symbol (> or <) dominates when chained with ≥, ≤, or =.
• "Either-or" is valid only when both conclusions are individually false but one MUST be true.
• For coded inequalities: decode first, solve second—never decode on the fly.
• "Not greater" = ≤; "Not less" = ≥; "Neither greater nor equal" = <.
• When in doubt, pick two test values that satisfy the statement and check the conclusion.