⚖️ Inequality (Coded Inequalities) - Complete Theory

Master logical comparisons - the fastest-scoring reasoning topic!

🎯 What is Coded Inequality?

Coded Inequality questions test your ability to:

• Decode symbols representing relationships (>, <, =, ≥, ≤)
• Combine multiple inequality statements
• Draw logical conclusions about relationships

Example:

Statements: A > B, B > C Conclusions: I. A > C II. C < A

Both conclusions are TRUE (A is greater than B, B is greater than C, so A must be greater than C)

📐 Basic Symbols

Standard Inequality Symbols

 | Greater than        | A > B (A is greater than B)

< | Less than | A < B (A is less than B) = | Equal to | A = B (A is equal to B) ≥ | Greater than or | A ≥ B (A is greater than or equal to B) | equal to | ≤ | Less than or | A ≤ B (A is less than or equal to B) | equal to | ≠ | Not equal to | A ≠ B (A is not equal to B)

🔤 Coded Symbols (IBPS Pattern)

In IBPS exams, symbols are coded. You need to decode them first!

Common Coding Pattern:

@ means “greater than” (>)

means “less than” (<)

$ means “equal to” (=) % means “greater than or equal to” (≥) & means “less than or equal to” (≤)

• means “not equal to” (≠)

Example:

Given: A @ B means A > B A # B means A < B A $ B means A = B

Statement: P @ Q $ R Decoded: P > Q = R Means: P > Q and Q = R, therefore P > R

🔗 Combining Inequalities

Rule 1: Transitive Property (Same Direction)

If A > B and B > C, then A > C

If A < B and B < C, then A < C

If A = B and B = C, then A = C

Example:

P > Q, Q > R Conclusion: P > R ✓ (Definitely true)

Rule 2: Mixed Symbols (No Direct Conclusion)

If A > B and B < C, we CANNOT conclude relationship between A and C

Possible cases:

• A > C (if A is much greater)
• A < C (if C is much greater)
• A = C (by coincidence)

Example:

P > Q, Q < R Conclusion: P > R? CANNOT SAY ✗ Conclusion: P < R? CANNOT SAY ✗

Rule 3: Greater/Equal Combined (≥)

If A ≥ B, then:

• Either A > B OR A = B
• Both are possible

Cannot conclude definitely which one!

Example:

P ≥ Q, Q ≥ R Possible conclusions:

• P ≥ R ✓ (Definitely true)
• P > R ✗ (May or may not be true)
• P = R ✗ (May or may not be true)

Rule 4: Complementary Pairs

“Either-Or” applies when:

1. Both conclusions are individually false
2. Both form a complementary pair

Complementary pairs:

• A > B and A = B (either one must be true if A ≥ B)
• A < B and A = B (either one must be true if A ≤ B)

💡 Solved Examples

Example 1: Basic Chain

Q: Statements: M > N, N > O Conclusions: I. M > O II. O < M

Solution:

Step 1: Draw chain

M > N > O (M is greatest, O is smallest)

Step 2: Test conclusions

Conclusion I: M > O From chain: M > N > O, so M > O ✓ TRUE

Conclusion II: O < M Same as M > O ✓ TRUE

Answer: Both I and II are true

Example 2: Coded Symbols

Q:

Given codes: A @ B means A > B A # B means A < B A $ B means A = B

Statements: P @ Q, Q $ R, R @ S

Conclusions: I. P @ S (P > S) II. S # P (S < P)

Solution:

Step 1: Decode statements

P @ Q → P > Q Q $ R → Q = R R @ S → R > S

Combined: P > Q = R > S

Step 2: Simplify

Since Q = R: P > Q and R > S P > R > S (substituting Q = R)

Chain: P > R > S Therefore: P > S

Step 3: Test conclusions

Conclusion I: P @ S (P > S) ✓ TRUE Conclusion II: S # P (S < P) ✓ TRUE (same as P > S)

Answer: Both conclusions true

Example 3: No Direct Relation

Q: Statements: A > B, C < B Conclusions: I. A > C II. C < A

Solution:

Step 1: Analyze statements

A > B C < B (means B > C)

Can we relate A and C? A > B > C? Not sure!

Could be: Case 1: A = 10, B = 8, C = 5 (A > B > C) → A > C ✓ Case 2: A = 10, B = 8, C = 7 (A > B, B > C) → A > C ✓

Actually, we CAN conclude! Let me reconsider:

A > B and B > C Therefore A > C ✓

Step 2: Test conclusions

Conclusion I: A > C ✓ TRUE Conclusion II: C < A ✓ TRUE (same as A > C)

Answer: Both true

Note: If one is > and other is < with SAME middle element, we CAN conclude!

Example 4: Equal to Symbol

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

Solution:

Step 1: Analyze

P ≥ Q means P > Q OR P = Q Q = R (definite)

Combining: If P > Q and Q = R → P > R ✓ If P = Q and Q = R → P = R ✓

Both are POSSIBLE, but neither is DEFINITE!

Step 2: Test conclusions

Conclusion I: P > R → POSSIBLE but not definite ✗ Conclusion II: P = R → POSSIBLE but not definite ✗

Step 3: Check Either-Or

Since P ≥ Q and Q = R: P ≥ R (definite)

Either P > R OR P = R (one must be true)

Answer: Either I or II is true

Example 5: Complex Chain

Q:

Statements: A > B ≥ C = D < E

Conclusions: I. A > D II. A > E III. C < E

Solution:

Step 1: Break down chain

A > B B ≥ C C = D D < E

Combining: A > B ≥ C = D < E

Step 2: Test each conclusion

Conclusion I: A > D

A > B ≥ C = D Since A > B and B ≥ C and C = D: A > D ✓ TRUE (definitely)

Conclusion II: A > E

A > B ≥ C = D < E A vs E: Cannot determine!

A could be > E, = E, or < E ✗ FALSE (cannot conclude)

Conclusion III: C < E

C = D < E Since D < E and C = D: C < E ✓ TRUE (definitely)

Answer: Conclusions I and III are true, II is false

Example 6: Multiple Statements

Q:

Statements: I. M > N II. N > O III. O = P IV. P < Q

Conclusions: I. M > Q II. N > P

Solution:

Step 1: Combine all statements

M > N > O = P < Q

Chain: M > N > O = P < Q

Step 2: Test conclusions

Conclusion I: M > Q

M > … > P < Q Cannot determine relationship between M and Q ✗

Conclusion II: N > P

N > O = P Therefore N > P ✓ TRUE

Answer: Only conclusion II is true

⚡ Quick Rules & Shortcuts

Rule 1: Same Direction Chain

All symbols pointing same direction (all > or all <): Can conclude first > last

Example: A > B > C > D Conclusion: A > D ✓

Rule 2: Mixed Direction (Breaking Point)

If chain reverses direction (… > … < …): Cannot connect across the reverse point

Example: A > B > C < D Cannot conclude about A and D relationship ✗

Rule 3: Equal Sign Substitution

If A = B, then A and B are interchangeable

Example: A > B = C > D Same as: A > C > D ✓

Rule 4: Greater/Equal (≥) Handling

If A ≥ B appears: Can conclude A > B OR A = B (either-or) Cannot conclude definitively which one!

For chains with ≥: Only use ≥ in final answer, not > or =

Rule 5: Either-Or Conditions

When conclusion has ≥ or ≤ in statements: Two complementary conclusions may form Either-Or

Example: If statements say A ≥ B:

• Conclusion I: A > B
• Conclusion II: A = B Answer: Either I or II

⚠️ Common Mistakes

❌ Mistake 1: Ignoring Direction Change

Wrong: A > B < C, so A > C ✗ Right: Cannot conclude A vs C (direction changed) ✓

❌ Mistake 2: Treating ≥ as >

Wrong: A ≥ B definitely means A > B ✗ Right: A ≥ B means A > B OR A = B (both possible) ✓

❌ Mistake 3: Reversing Symbols

Wrong: A > B is same as B > A ✗ Right: A > B means B < A (reversed symbol!) ✓

❌ Mistake 4: Not Decoding First

Wrong: Directly using coded symbols in conclusions ✗ Right: First decode all symbols, then solve ✓

❌ Mistake 5: Assuming Transitivity Everywhere

Wrong: A > B and C < D, so A > D ✗ Right: Can only use transitivity with connected chain ✓

📝 Practice Problems

Level 1: Basic

1. Statements: A > B, B > C Conclusions: I. A > C, II. C < A

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

3. Statements: X < Y, Y < Z Conclusions: I. Z > X, II. X < Z

Level 2: Medium

4. Statements: M ≥ N, N > O Conclusions: I. M > O, II. M = O

5. Statements: A > B = C > D Conclusions: I. A > D, II. B > D

6.

Codes: @ means >, # means <, $ means = Statements: P @ Q $ R # S Conclusions: I. P @ S, II. S # P

Level 3: Hard

7. Statements: A ≥ B > C = D ≤ E Conclusions: I. A > D, II. E > B, III. A > E

8. Statements: P > Q ≥ R, S = R, T < S Conclusions: I. P > T, II. Q > T

9. Statements: M > N = O, P < O, Q ≥ P Conclusions: I. M > P, II. N > Q

🎯 Exam Strategy

Time Management:

• Per question (5 conclusions): 60 seconds
• For 5 inequality questions: 5 minutes

Quick Approach:

1. Decode symbols (10 sec) - if coded
2. Draw combined chain (15 sec)
3. Test each conclusion (5-7 sec each)
4. Mark answer (5 sec)

Priority:

• ✅ Simple chains (all >) - 40 sec
• ✅ Equal sign chains (A = B > C) - 50 sec
• ✅ Coded inequalities - 60 sec
• ⏭️ Complex ≥/≤ with either-or - 75+ sec

🔗 Related Topics

Uses Concepts From:

• Syllogism - Logical deduction
• Basic mathematical inequalities

Related Reasoning Topics:

• Data Sufficiency - Uses inequality logic
• Coded relationships

Practice:

• Inequality Questions

Master Inequalities - Draw the chain, watch the direction! ⚖️