⚖️ 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

Symbol | Meaning              | Example
-------|---------------------|----------
>      | 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! ⚖️