Inequalities Notes for Reasoning

Inequalities is one of the quickest and most scoring topics in the reasoning section of bank PO and clerk exams (IBPS, SBI, RRB), insurance exams (LIC, NIACL) and several state-level exams. A typical set has 3 to 5 questions, and a well-prepared student can solve each in about 30 seconds. In UPSC CSAT, pure inequality questions are rare, but the same logic of comparing quantities through chains of relations appears in ranking and comparison puzzles. The topic is mechanical: once you know the rules for combining symbols, every question follows the same steps. That is exactly why it rewards practice so well.

Quick Facts at a Glance

• Six symbols: >, <, ≥, ≤, = and their combinations.
• Same-direction chain: Conclusions can be drawn (A > B ≥ C gives A > C).
• Opposite-direction signs: No definite relation (A > B < C means A and C cannot be compared).
• Priority: If the chain has any strict sign (> or <), the result is strict.
• Either-or: Applies when two conclusions with the same pair of elements together cover all possibilities, and neither alone is definitely true.

1. The Symbols and Their Meaning

2. Rules for Combining Relations

Rule 1: Same direction signs can be combined

If all signs in the path between two elements point the same way (all "greater" type or all "less" type), a conclusion follows.

• A > B > C → A > C
• A ≥ B ≥ C → A ≥ C
• A > B ≥ C → A > C (strict sign dominates)
• A ≥ B = C → A ≥ C
• A = B = C → A = C

Rule 2: Opposite signs break the chain

If the path contains signs in opposite directions, such as > and <, or ≥ and ≤, no definite relation exists between the end elements.

• A > B < C → no relation between A and C.
• A ≥ B ≤ C → no relation between A and C.

Rule 3: The equal sign is neutral

The "=" sign does not change direction. It simply passes the relation forward. A > B = C ≥ D gives A > D.

Rule 4: Priority of signs

When combining same-direction signs, the result takes the strongest sign in the path: if any ">" appears, the result is ">"; if only "≥" and "=" appear, the result is "≥"; if only "=" appears, the result is "=".

3. Direct Inequalities

In direct inequalities, statements are given with the actual symbols.

Example 1: Statements: P ≥ Q > R = S ≤ T
Conclusions: I. P > S   II. T ≥ R
Solution: P ≥ Q > R = S gives P > S, so I is true. For T and R: R = S ≤ T gives T ≥ R, so II is true. Both follow.

Example 2: Statements: A > B ≥ C < D ≤ E
Conclusions: I. A > D   II. E > C
Solution: Between A and D, the path has >, ≥ and then <, so the direction changes; no relation. I is false. For E and C: C < D ≤ E gives E > C. II is true. Only II follows.

Combining Two or More Statements

Sometimes statements are given separately, such as "M > N ≥ O; O = P < Q". Join them through the common element (here O): M > N ≥ O = P < Q. Then apply the rules as usual.

4. Either-Or Cases

This is the most important trap in inequalities. "Either I or II follows" is the answer when:

1. Both conclusions involve the same two elements.
2. Neither conclusion is definitely true on its own.
3. Together, they cover all possible relations between the two elements.

Common either-or pairs:

Example: Statements: X ≥ Y = Z
Conclusions: I. X > Z   II. X = Z
Solution: X ≥ Z is the actual relation. Neither I nor II is definite, but together they cover the "≥" relation. Either I or II follows.

Important: If the statements show no relation at all between two elements (because signs are opposite), the pair "A > B" and "A ≤ B" still forms an either-or case, since all possibilities are covered.

5. Coded Inequalities

In coded inequalities, symbols such as @, #, $, % and * are given meanings. For example:

• 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

Method: Decode every statement and conclusion into normal symbols first, then solve like direct inequalities. Some coded sets describe the meaning in words, such as "P © Q means P is neither smaller than nor equal to Q", which simply means P > Q. Translate carefully: "neither smaller than" excludes <; "nor equal to" excludes =; so only > remains.

6. Finding the Relation in Reverse (Fill-in-the-Blank Type)

Newer exam patterns ask: "Which symbol should replace the question mark so that A > D is definitely true in A ? B ≥ C = D?" Try each option. Only signs in the same direction work: ">" gives A > D (true); "≥" gives A ≥ D (not strictly >); "=" gives A ≥ D. So ">" is the answer. Another variant asks you to arrange given letters with symbols so that a conclusion becomes true; the same rules apply.

7. Speed Tricks

• Write the combined chain once, neatly, and solve all conclusions from it.
• Read the path between two elements from left to right and only look for direction changes.
• For either-or, check the three conditions quickly: same elements, neither true, together complete.
• In coded sets, write the code table in a corner of your rough sheet before reading questions.

Common Mistakes to Avoid

1. Combining signs across a direction change.
2. Writing ≥ when the chain contains a strict >.
3. Marking "either-or" when one conclusion is already definitely true.
4. Mis-decoding verbal descriptions in coded inequalities.
5. Reading the relation in the wrong direction (A < B is the same as B > A; flip both carefully).

Practice Questions with Answers

1. Statements: J > K = L ≥ M. Conclusions: I. J > M; II. L > M. Answer: Only I follows (L ≥ M, so II is not definite).
2. Statements: R ≤ S < T = U. Conclusions: I. U > R; II. S ≤ U. Answer: I follows (R ≤ S < T = U gives U > R). II: S < U, so S ≤ U is also true. Both follow.
3. Statements: D ≥ E < F. Conclusions: I. D > F; II. D ≤ F. Answer: Either I or II.
4. Statements: W = X ≥ Y. Conclusions: I. W > Y; II. W = Y. Answer: Either I or II.

Preparation Strategy

• Learn the four rules and the either-or conditions on day one.
• Solve 50 direct inequality questions over three days with a timer.
• Move to coded and fill-in-the-blank types after you can do direct questions in under 30 seconds each.
• Revise with one mixed set of 10 questions twice a week until your exam.

Why Inequalities Matter for Your Score

In a bank prelims paper of 35 reasoning questions to be solved in about 20 minutes, every second counts. Inequality questions are among the few where accuracy close to 100% is realistic, because the rules never change. Students who finish an inequality set in two minutes gain time for difficult puzzles. That is why toppers practise this topic until it becomes automatic, and why it is one of the first chapters you should master when starting reasoning preparation.

Related Notes

• Syllogism Notes for Reasoning
• Ranking, Order and Mathematical Operations Notes
• Coding-Decoding Notes for Reasoning
• All Reasoning Notes

Frequently Asked Questions

What is inequality in reasoning?

It is a question type where relations between elements are given using symbols like >, <, ≥, ≤ and =, and you decide which conclusions follow.

When can two inequality signs be combined?

Only when they point in the same direction. The equal sign can be combined with either direction.

What happens when signs are in opposite directions?

No definite relation can be drawn between the elements on either side of the direction change.

What is the result of combining > and ≥?

The result is the strict sign >.

What is an either-or case?

It applies when two conclusions involve the same elements, neither is definitely true, and together they cover all possible relations.

What are coded inequalities?

Questions where symbols like @, # or $ stand for inequality signs. You decode them first and then solve normally.

What does "neither smaller than nor equal to" mean?

It means greater than (>).

What does "not greater than" mean?

It means less than or equal to (≤).

How many inequality questions come in bank exams?

Bank prelims often include a set of about 3 to 5 inequality questions.

Is inequality asked in UPSC CSAT?

Direct inequality questions are rare in CSAT, but the logic is useful for comparison and ranking puzzles.

How can I solve inequality questions faster?

Combine the statements into one chain, scan for direction changes and apply the sign priority rule.

What is the most common mistake in inequalities?

Drawing a conclusion across signs that point in opposite directions.