Inequality | Complete Logical Comparison Tutorial with Solved Examples

28 views Ch 11: Inequality

⚖️ Chapter 11: Inequality 📊

Master Mathematical Comparisons & Logical Conclusions

📖 1. What is Inequality?

Definition

Inequality deals with comparing quantities using symbols like greater than (), less than (), greater than or equal to (≥), less than or equal to (≤), equal to (=), and not equal to (≠). These problems test your ability to draw logical conclusions from given statements.

💡 Real-World Examples

Scenario	Inequality Type	Example
Height Comparison	Direct comparison	Ram is taller than Shyam (R S)
Marks Analysis	Performance ranking	A scored more than B but less than C
Price Comparison	Financial decision	Product A costs less than Product B
Speed Measurement	Rate comparison	Train A is faster than Train B

🎯 Why is This Important?

Benefit	Application
Logical Reasoning	Understanding relationships between entities
Decision Making	Drawing valid conclusions from given facts
Competitive Exams	Very common in banking, SSC, and aptitude tests
Mathematical Thinking	Building algebraic and analytical skills
Real-Life Skills	Comparing options and making informed choices

🔑 2. Inequality Symbols & Meanings

📌 Basic Symbols

Symbol	Name	Meaning	Example	Read As
	Greater than	Left is bigger	A B	A is greater than B
	Less than	Left is smaller	A B	A is less than B
≥	Greater than or equal to	Left is bigger or same	A ≥ B	A is greater than or equal to B
≤	Less than or equal to	Left is smaller or same	A ≤ B	A is less than or equal to B
=	Equal to	Both are same	A = B	A is equal to B
≠	Not equal to	Both are different	A ≠ B	A is not equal to B

🎯 Visual Representation

Number Line Visualization:

... -3  -2  -1   0   1   2   3   4   5 ...
         ←―――――――――――――――――――――――→
       Smaller              Larger

If A  B: A is to the RIGHT of B
If A  B: A is to the LEFT of B

📊 Symbol Relationship Table

Statement	Reverse Statement	Meaning
A B	B A	Same relationship, reversed
A B	B A	Same relationship, reversed
A ≥ B	B ≤ A	Same relationship, reversed
A = B	B = A	Equality is symmetric
A ≠ B	B ≠ A	Not equal is symmetric

🧮 3. Types of Inequality Problems

📌 Type 1: Direct Conclusions

Drawing simple, direct conclusions from given statements.

✏️ Example 1: Single Chain

Problem: Statements: A B, B C
Conclusions:
I. A C
II. C A
III. B A

Which conclusions are definitely true?

Step-by-Step Solution:

Step	Given/Analysis	Conclusion
Statement 1	A B	A is greater than B
Statement 2	B C	B is greater than C
Chain	A B C	Combined relationship
Conclusion I	A C	✅ TRUE (A is definitely greater than C)
Conclusion II	C A	✅ TRUE (Same as A C, reversed)
Conclusion III	B A	✅ TRUE (Given directly: A B)

Visual Representation:

A  B  C
↑   ↑   ↑
Largest  Smallest

Answer: All conclusions (I, II, III) are TRUE ✅

✏️ Example 2: Multiple Chains

Problem:
Statements: P ≥ Q, Q R, R = S
Conclusions:
I. P S
II. P ≥ S
III. Q S

Which conclusions are definitely true?

Step-by-Step Solution:

Step	Statement	Analysis	Result
Given 1	P ≥ Q	P is greater than or equal to Q	-
Given 2	Q R	Q is greater than R	-
Given 3	R = S	R equals S	-
Combine	P ≥ Q R = S	Full chain	P ≥ Q S
Conclusion I	P S	May be true but NOT definite	❌ NOT DEFINITELY TRUE
		(P could equal Q, then P=S possible)	-
Conclusion II	P ≥ S	From chain P ≥ Q S	✅ TRUE
Conclusion III	Q S	Q R and R = S, so Q S	✅ TRUE

Detailed Analysis:

Scenario	If P = Q	If P Q
P value	P = Q	P Q
Q S	Q S (given)	Q S (given)
P vs S	P = Q S → P S	P Q S → P S
But conclusion I says	P S (definite)	Not definite if P could equal something

Wait, let me reconsider:

Since P ≥ Q and Q S, combining: P ≥ Q S means P S is definitely true!

Corrected Answer: Conclusions I, II, III all are TRUE ✅

📌 Type 2: Either-Or Cases

When complementary pairs are involved.

✏️ Example 3: Either-Or Logic

Problem:
Statements: A ≥ B, B = C
Conclusions:
I. A C
II. A = C

Which is true?

Step-by-Step Solution:

Step	Analysis	Result
Given	A ≥ B, B = C	-
Combine	A ≥ B = C → A ≥ C	-
Meaning	A is greater than OR equal to C	Two possibilities
Conclusion I	A C	❌ Not definitely true (might be equal)
Conclusion II	A = C	❌ Not definitely true (might be greater)
Either-Or	Either I OR II is true	✅ TRUE

Either-Or Rule:

Condition	When Applicable	Example
Either A or B	When A ≥ B or A ≤ B	Either A B OR A = B
Both together	Cannot be both	Cannot have both A B AND A = B
At least one	One must be true	Complementary conclusions

Answer: Either Conclusion I OR Conclusion II is TRUE ✅

📌 Type 3: Coded Inequalities

Symbols are replaced with special characters or codes.

🔤 Common Coding Patterns

Symbol	Possible Codes	Examples
	@, #, %, $	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

✏️ Example 4: Coded Inequality

Problem:
Code Definitions:

A @ B means A B
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
II. P @ R

Decode and solve:

Step-by-Step Solution:

Step	Coded Statement	Decoded Statement	Meaning
Given 1	P @ Q	P Q	P is greater than Q
Given 2	Q $ R	Q ≥ R	Q is greater than or equal to R
Given 3	R % S	R = S	R equals S
Combine	P Q ≥ R = S	Full chain	P Q ≥ S

Analyzing Conclusions:

Conclusion	Statement	Analysis	Result
I	P @ S (P S)	From P Q ≥ S	✅ TRUE
II	P @ R (P R)	From P Q ≥ R	✅ TRUE

Answer: Both Conclusions I and II are TRUE ✅

📌 Type 4: Complex Chains

Multiple interconnected relationships.

✏️ Example 5: Complex Chain

Problem:
Statements:

M N
N ≥ O
O = P
P Q

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

Step-by-Step Solution:

Step	Statement	Chain Building
Statement 1	M N	M N
Statement 2	N ≥ O	M N ≥ O
Statement 3	O = P	M N ≥ O = P
Statement 4	P Q	M N ≥ O = P Q

Complete Chain: M N ≥ O = P Q

Visual Representation:

    M
    ↓ ()
    N
    ↓ (≥)
    O = P
        ↓ ()
        Q

Analyzing Conclusions:

Conclusion	Required	From Chain	Result
I. M P	M compared to P	M N ≥ O = P → M P	✅ TRUE
II. Q N	Q compared to N	Q P = O and O ≤ N	❌ CANNOT SAY
		(N could be O or N = O)	(If N = O, then Q N, but not definite)
III. M Q	M compared to Q	M and Q not directly comparable	❌ FALSE
		(Q P and M P, can't determine M vs Q)	-

Detailed Analysis for Conclusion II:

Scenario	If N = O	If N O
Chain	Q P = O = N	Q P = O N
Q vs N	Q N ✓	Q vs N unclear
Definite?	Not in all cases	❌

Answer: Only Conclusion I is definitely TRUE ✅

🎓 4. Problem-Solving Strategy

8-Step Master Approach 📝

Step	Action	Details	Example
Step 1	Decode Symbols	If coded, convert to standard symbols	@ = , # =
Step 2	List Statements	Write all given inequalities	A B, B ≥ C
Step 3	Build Chain	Connect related statements	A B ≥ C
Step 4	Draw Diagram	Visual representation helps	Use arrows/number line
Step 5	Check Each Conclusion	Test against the chain	Is A C true?
Step 6	Watch for ≥ and ≤	These create "either-or" situations	Could be equal OR greater
Step 7	Look for Gaps	Can't connect unrelated chains	If A B and C D, can't compare A to C
Step 8	Mark Answers	Definitely true, false, or can't say	Use logic, not assumptions

📝 5. Practice Problems

Set A: Basic Level 🎯

Q#	Problem
Q1	Statements: A B, B C, C DbrConclusions: I. A D II. D B III. C AbrWhich conclusions are TRUE?

Solution Table:

Step	Analysis	Result
Chain	A B C D	Complete chain
Conclusion I	A D	✅ TRUE (Direct from chain)
Conclusion II	D B	✅ TRUE (Same as B D)
Conclusion III	C A	✅ TRUE (Same as A C)

Answer: A) All conclusions I, II, III are TRUE ✅

Q#	Problem
Q2	Statements: P ≥ Q, Q = R, R SbrConclusions: I. P S II. P = R III. Q SbrWhich conclusions are TRUE?

Solution Table:

Step	Analysis	Chain
Combine	P ≥ Q = R S	P ≥ R S
Conclusion I	P S	✅ TRUE (P ≥ R S means P S)
Conclusion II	P = R	❌ NOT DEFINITELY TRUE (P could be R)
Conclusion III	Q S	✅ TRUE (Q = R S means Q S)

Answer: B) Conclusions I and III are TRUE ✅

Set B: Coded Inequalities 🔤

Q#	Problem
Q3	Code: A @ B = A B, A # B = A B, A $ B = A ≥ B, A % B = A ≤ BbrStatements: M @ N, N $ O, O # PbrConclusions: I. M @ P II. M @ O III. P @ NbrWhich are TRUE?

Decoding Table:

Coded	Decoded	Meaning
M @ N	M N	M is greater than N
N $ O	N ≥ O	N is greater than or equal to O
O # P	O P	O is less than P

Chain: M N ≥ O P

Analysis:

Conclusion	Decoded	From Chain	Result
I. M @ P	M P	M N ≥ O and O P	❌ CANNOT SAY
		(M and P not directly comparable)	-
II. M @ O	M O	M N ≥ O	✅ TRUE
III. P @ N	P N	P O and O ≤ N	❌ CANNOT SAY

Answer: C) Only Conclusion II is TRUE ✅

Set C: Advanced Level 🔥

Q#	Problem
Q4	Statements: A ≥ B C ≤ D EbrConclusions: I. A C II. E C III. B D IV. A EbrWhich are definitely TRUE?

Chain Analysis:

A ≥ B  C ≤ D  E
↓       ↓       ↓
Chain 1: A ≥ B  C
Chain 2: C ≤ D  E

Detailed Solution:

Conclusion	Required Comparison	Analysis	Result
I. A C	A vs C	A ≥ B C → A C	✅ TRUE
II. E C	E vs C	E D ≥ C → E C	✅ TRUE
III. B D	B vs D	B C and C ≤ D	❌ CANNOT SAY
		(If C = D, then B D, but not definite)	-
IV. A E	A vs E	No direct connection	❌ CANNOT SAY

Answer: D) Conclusions I and II are TRUE ✅

Q#	Problem
Q5	Statements: P Q ≥ R = S TbrConclusions: I. P S II. T Q III. R T IV. P TbrWhich are TRUE?

Chain Analysis Table:

Part	Chain	Connections
Main Chain	P Q ≥ R = S T	All connected
Left Side	P Q ≥ R = S	P down to S
Right Side	R = S T	S connects to T

Conclusion Analysis:

Conclusion	Comparison	From Chain	Result
I. P S	P vs S	P Q ≥ R = S → P S	✅ TRUE
II. T Q	T vs Q	T S = R and R ≤ Q	❌ CANNOT SAY
		(Q could equal R or Q R)	-
III. R T	R vs T	R = S T → R T	✅ TRUE
IV. P T	P vs T	No definite relation	❌ CANNOT SAY

Answer: E) Conclusions I and III are TRUE ✅

🎯 6. Special Cases & Rules

📌 Either-Or Rule

When to Apply	Condition	Example
Complementary Conclusions	If A ≥ B, then either A B OR A = B	Exactly one must be true
Symbol ≥ or ≤	Creates two possibilities	Not both, not neither

✏️ Example 6: Either-Or Application

Statements: A ≥ B ≥ C
Conclusions:
I. A C
II. A = C

Analysis:

Possibility	Scenario	Result
Case 1	A B C	Conclusion I is TRUE
Case 2	A = B = C	Conclusion II is TRUE
Case 3	A B = C	Conclusion I is TRUE
Case 4	A = B C	Conclusion I is TRUE

Either-Or Check:

Test	Result
Can both be true together?	❌ NO (A cannot be both C and = C)
Can both be false together?	❌ NO (One must be true from A ≥ C)
Is exactly one true?	✅ YES (Either-Or case)

Answer: Either Conclusion I OR Conclusion II is TRUE ✅

📌 Combination Rules

Rule	Formula	Example
Greater than chain	A B, B C → A C	Transitive property
Mixed symbols	A ≥ B C → A C	Greater still holds
Equal in chain	A B, B = C → A C	Substitute equal value
Cannot combine	A B and C D	A vs C unknown

🏆 7. Key Takeaways & Pro Tips

#	Tip	Why It Matters
1	⚖️ Build complete chains	Connect all related statements before analyzing
2	🔍 Watch for ≥ and ≤	These create either-or situations
3	❌ Separate chains can't be compared	If A B and C D, can't say A vs C
4	✅ Decode carefully	In coded problems, convert all symbols first
5	📊 Draw diagrams	Visual representation prevents errors
6	🎯 Test each conclusion separately	Don't assume; verify against chain
7	⏰ Check for "definitely true"	"Possibly true" ≠ "Definitely true"
8	🔄 Reverse is same	A B is same as B A

📚 8. Common Mistakes to Avoid

Mistake	Why It's Wrong	Correct Approach
Assuming transitivity across gaps	Can't connect unrelated chains	Only combine connected statements
Treating ≥ as	These are different!	≥ means "could be equal too"
Ignoring either-or cases	Missing possible answers	Check for complementary conclusions
Reversing symbols wrongly	becomes , not ≥	Be careful with direction
Not building complete chain	Missing relationships	Always write full chain first
Assuming both conclusions true	In either-or, only one is true	Cannot have both A B AND A = B

💡 9. Advanced Strategies

Quick Decision Table

Chain Pattern	Definite Conclusion	Uncertain Conclusion
A B C	A C ✓	A = C ✗
A ≥ B ≥ C	A ≥ C ✓	A C ?
A B, C D	(separate)	A vs C ?
A B = C	A C ✓	B vs C ? No! B = C
A ≥ B C	A C ✓	-

Symbol Conversion Chart

If Given	Can Conclude	Cannot Conclude
A B	B A ✓	A = B ✗, A ≥ B ✓
A ≥ B	B ≤ A ✓	A B ?, A = B ?
A = B	B = A ✓, A ≥ B ✓, A ≤ B ✓	A B ✗, A B ✗
A ≠ B	B ≠ A ✓	A B ?, A B ?

📊 10. Quick Reference Guide

Inequality Combination Table

Statement 1	Statement 2	Conclusion	Valid?
A B	B C	A C	✅ YES
A ≥ B	B C	A C	✅ YES
A B	B ≥ C	A C	✅ YES
A ≥ B	B ≥ C	A ≥ C	✅ YES
A ≥ B	B = C	A ≥ C	✅ YES
A B	C D	A vs C?	❌ NO

Either-Or Identification

Situation	Conclusions Given	Answer Type
A ≥ B given	I. A B, II. A = B	Either I or II
A ≤ B given	I. A B, II. A = B	Either I or II
Both definite	Both can be proven	Both true
Neither definite	Cannot prove either	Neither true

Verbal Intelligence

Inequality | Complete Logical Comparison Tutorial with Solved Examples

⚖️ Chapter 11: Inequality 📊

Master Mathematical Comparisons & Logical Conclusions

📖 1. What is Inequality?

Definition

💡 Real-World Examples

🎯 Why is This Important?

🔑 2. Inequality Symbols & Meanings

📌 Basic Symbols

🎯 Visual Representation

📊 Symbol Relationship Table

🧮 3. Types of Inequality Problems

📌 Type 1: Direct Conclusions

✏️ Example 1: Single Chain

✏️ Example 2: Multiple Chains

📌 Type 2: Either-Or Cases

✏️ Example 3: Either-Or Logic

📌 Type 3: Coded Inequalities

🔤 Common Coding Patterns

✏️ Example 4: Coded Inequality

📌 Type 4: Complex Chains

✏️ Example 5: Complex Chain

🎓 4. Problem-Solving Strategy

8-Step Master Approach 📝

📝 5. Practice Problems

Set A: Basic Level 🎯

Set B: Coded Inequalities 🔤

Set C: Advanced Level 🔥

🎯 6. Special Cases & Rules

📌 Either-Or Rule

✏️ Example 6: Either-Or Application

📌 Combination Rules

🏆 7. Key Takeaways & Pro Tips

📚 8. Common Mistakes to Avoid

💡 9. Advanced Strategies

Quick Decision Table

Symbol Conversion Chart

📊 10. Quick Reference Guide

Inequality Combination Table

Either-Or Identification

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