Inequality

Here is a detailed explanation of the Inequality Reasoning content from your PDF, including core concepts, solved examples, and additional practice questions with explanations¹²³.

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Detailed Explanation of Inequality Reasoning

Inequality Reasoning involves comparing two or more elements using symbols such as $>$, $<$, $=$, $\geq$, $\leq$, and $\neq$. The goal is to deduce definite relationships between elements based on the given statements and to evaluate the validity of the conclusions drawn from them.

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Key Concepts and Symbols

Symbol	Meaning
$A > B$	A is greater than B
$A < B$	A is smaller than B
$A = B$	A is equal to B
$A \geq B$	A is greater than or equal to B
$A \leq B$	A is less than or equal to B
$A \neq B$	A is not equal to B
$A \not \leq B$	A is greater than B
$A \not < B$	A is greater than or equal to B

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How to Solve Inequality Questions

1. Combine Statements: Write all inequalities in a single chain if possible.
2. Analyze Relationships: Check if a direct relationship can be established between the elements in the conclusions.
3. Evaluate Conclusions: For each conclusion, see if it logically follows from the combined statements.
4. Special Cases:
   • Either-Or Case: If two conclusions form a complementary pair (e.g., $A > B$ and $A \leq B$), and both are individually false, then “either-or” is true.
   • No Direct Relationship: If you cannot establish a definite relationship, the conclusion is invalid.

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Solved Examples

Example 1: Direct Inequality

Statement: $H < A < T = G > U \geq V \geq B$

Conclusions: I. $T > B$ II. $G > H$

Solution:

• Combine: $H < A < T = G > U \geq V \geq B$
• Conclusion I: $T > B$ is true because $T = G > U \geq V \geq B$
• Conclusion II: $G > H$ is true because $H < A < T = G$
• Answer: Both conclusions I and II follow.

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Example 2: No Direct Relationship

Statement: $F > Y \geq X < Z$, $C \leq X < W$

Conclusions: I. $Z > C$ II. $F > W$

Solution:

• Combine: $F > Y \geq X \geq C$ and $F > Y \geq X < W$
• Conclusion I: $Z > C$ is true because $X \geq C$ and $X < Z$, so $Z > X \geq C$
• Conclusion II: $F > W$ is false because $F > Y \geq X < W$ does not establish a relationship between $F$ and $W$
• Answer: Only conclusion I follows.

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Example 3: Either-Or Case

Statement: $B = K \geq H = T > U \leq I$

Conclusions: I. $H > I$ II. $H \leq I$

Solution:

• Combine: $B = K \geq H = T > U \leq I$
• Conclusion I: $H > I$ is false because $H = T > U \leq I$ does not mean $H > I$
• Conclusion II: $H \leq I$ is false because $H = T > U \leq I$ does not mean $H \leq I$
• But: Both conclusions are complementary and both are false individually, so “either-or” is true.
• Answer: Either conclusion I or II follows.

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Example 4: Coded Inequality

Symbols:

• $A @ B$: $A \leq B$
• $A # B$: $A \geq B$
• $$ A \$ B$: $A = B $$
• $A % B$: $A < B$
• $A * B$: $A > B$

Statements:

$$ R \$ J; J \% Y; C @ Y $$

Conclusions: I. $C % J$ II. $R * Y$ III. $R * C$

Solution:

• Decode: $R = J; J < Y; C \leq Y$
• Combine: $R = J < Y \geq C$
• Conclusion I: $C < J$ is false (no direct relation)
• Conclusion II: $R > Y$ is false ($R = J < Y$
• Conclusion III: $R > C$ is false (no direct relation)
• Answer: None follows.

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Practice Questions

Question 1

Statements: $W < X; X \geq Y; Y = Z; Z \geq A$

Conclusions: I. $X \geq A$ II. $A < Z$ III. $A > X$

Solution:

• Combine: $W < X \geq Y = Z \geq A$
• I: $X \geq A$ is true.
• II: $A < Z$ is possible but not definite.
• III: $A > X$ is false.
• Answer: Only I is true.

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Question 2

Statements: $P \geq U; Q \geq S; S \leq P; R \geq U$

Conclusions: I. $P \geq R$ II. $Q \leq U$ III. $S \geq R$

Solution:

• Combine: $Q \geq S \leq P \geq U \leq R$
• I: $P \geq R$ is false (no direct relation)
• II: $Q \leq U$ is false (no direct relation)
• III: $S \geq R$ is false (no direct relation)
• Answer: None is true.

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Question 3

Statements: $A \leq B = D \leq E; F < L = K; E \geq F$

Conclusions: I. $F \geq B$ II. $E \geq A$

Solution:

• Combine: $A \leq B = D \leq E \geq F < L = K$
• I: $F \geq B$ is false (no direct relation)
• II: $E \geq A$ is true
• Answer: Only II is true.

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Question 4

Statements: $X = Y > Z \leq A; A = Q; Q > R \geq S < W$

Conclusions: I. $W > R$ II. $A \geq Y$ III. $Q \geq Z$ IV. $Q \geq X$

Solution:

• Combine: $X = Y > Z \leq A = Q > R \geq S < W$
• I: $W > R$ is false (no direct relation)
• II: $A \geq Y$ is false (no direct relation)
• III: $Q \geq Z$ is true
• IV: $Q \geq X$ is false (no direct relation)
• Answer: Only III is true.

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Tips for Solving Inequality Questions

• Always combine all statements if possible.
• Look for direct relationships between elements in the conclusions.
• Be cautious about “either-or” cases where both conclusions are individually false but together cover all possibilities.
• Practice with both direct and coded inequalities.

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Summary Table

Question Type	Key Point	Example Answer
Direct Inequality	Combine and check direct relationships	Both I and II follow
No Direct Relation	“Can’t say” if no direct relation	Only I follows
Either-Or Case	Both conclusions are complementary	Either I or II follows
Coded Inequality	Decode symbols, then proceed as above	None follows

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This comprehensive guide, with explanations, examples, and practice questions, will help you master inequality reasoning for any competitive exam¹²³.