Inequalities

1. Theory: About Inequalities

Inequalities in reasoning tests your ability to understand and deduce relationships between different elements (variables). These relationships are represented by inequality symbols. The goal is to determine if a given conclusion is definitively true based on one or more statements.

Key Concepts and "Diagrams"

The best "diagrams" for this topic are tables that define the rules and logical structures for solving the problems.

Diagram 1: The Basic Symbols

Symbol	Meaning
`A > B`	A is Greater than B
`A < B`	A is Less than B
`A = B`	A is Equal to B
`A ≥ B`	A is Greater than or Equal to B
`A ≤ B`	A is Less than or Equal to B
`A ≠ B`	A is Not Equal to B

Diagram 2: Priority of Signs (The "Magic Box")

When moving from one element to another in a combined statement, the conclusion is based on the lowest priority sign in the path. This table shows the resulting relationship when combining two signs.

Sign 1	Sign 2	Resulting Relation
`>`	`>`	`>`
`>`	`≥`	`>`
`>`	`=`	`>`
`≥`	`≥`	`≥`
`≥`	`=`	`≥`
`<`	`<` or `≤` or `=`	`<` (and so on for less than)
`>`	`<`	No Conclusion / Blocked Path

Core Techniques

Combine Statements: Find the common variable between statements to merge them into a single continuous chain. E.g., A > B and B ≥ C becomes A > B ≥ C.
Check the Path: To check a conclusion like `A > C`, trace the path from A to C in the combined statement.
Apply Priority: The relationship between the endpoints is determined by the lowest priority sign in the path (e.g., `>` has higher priority than `≥`). If signs are opposing (`>` and `<`), the path is blocked, and no definite conclusion can be drawn.
The 'Either/Or' Case: This is a crucial and special condition. It applies in two scenarios:
- Scenario 1 (Blocked Path): The path between the two elements in the conclusion is blocked (e.g., A > B < C). If the two conclusions cover all three possible relationships (>, <, =), then the answer is "Either/Or". Example Conclusions: A ≥ C and A < C.
- Scenario 2 (Uncertain Relation): The established relationship is uncertain (e.g., A ≥ B). If the conclusions are the individual components of this relationship, the answer is "Either/Or". Example Conclusions: A > B and A = B.

Types of Questions Asked

Direct Inequalities: The statements and conclusions are given with standard mathematical symbols.
Coded Inequalities: The inequality symbols are replaced with other symbols like @, #, $, %, etc. The first step is to decode the symbols into their mathematical equivalents.

2. Questions, Answers, and Explanations

Type 1: Direct Inequalities (Questions 1-25)

Question 1:
Statements: P > Q ≥ R = S
Conclusions:
I. P > S
II. Q > S

A) Only I is true
B) Only II is true
C) Both I and II are true
D) Neither I nor II is true

Answer: A) Only I is true

Diagram (Combined Statement)
P > Q ≥ R = S

Explanation
Technique: Path Tracing and Priority Rule.
Conclusion I (P > S): The path from P to S is P > Q ≥ R = S. The signs are > and ≥ and =. The highest priority sign is >. So, P > S is true.
Conclusion II (Q > S): The path from Q to S is Q ≥ R = S. The signs are ≥ and =. The highest priority sign is ≥. The definite conclusion is Q ≥ S, not Q > S. So, Conclusion II is false.
Question 2:
Statements: A < B < C ≤ D = E
Conclusions:
I. A < E
II. B ≤ E

A) Only I is true
B) Only II is true
C) Both I and II are true
D) Neither I nor II is true

Answer: A) Only I is true

Diagram (Combined Statement)
A < B < C ≤ D = E

Explanation
Conclusion I (A < E): The path from A to E is A < B < C ≤ D = E. The highest priority sign is <. So, A < E is true.
Conclusion II (B ≤ E): The path from B to E is B < C ≤ D = E. The highest priority sign is <. The definite conclusion is B < E, not B ≤ E. So, Conclusion II is false.
Question 3:
Statements: M ≥ N > O, P < O = Q
Conclusions:
I. M > P
II. N > Q

A) Only I is true
B) Only II is true
C) Both I and II are true
D) Neither I nor II is true

Answer: A) Only I is true

Diagram (Combined Statement)
M ≥ N > O > P and M ≥ N > O = Q

Explanation
Technique: Combine statements using the common variable 'O'.
From P < O, we can write O > P. Combining with the first statement: M ≥ N > O > P.
Also, O = Q. Combining with the first statement: M ≥ N > O = Q.
Conclusion I (M > P): The path is M ≥ N > O > P. The highest priority sign is >. So, M > P is true.
Conclusion II (N > Q): The path is N > O = Q. The highest priority sign is >. So, N > Q is true. Wait, let me re-check. Yes, N > O and O = Q implies N > Q. So Conclusion II is also true.
Corrected Answer: C) Both I and II are true
Question 4:
Statements: X > Y, Y < Z, Z = W
Conclusions:
I. X > Z
II. X < Z

A) Only I is true
B) Only II is true
C) Either I or II is true
D) Neither I nor II is true

Answer: D) Neither I nor II is true

Diagram (Combined Statement)
X > Y < Z = W

Explanation
Technique: Check for blocked paths.
The path between X and Z is X > Y < Z. The signs > and < are opposite. This is a blocked path.
When the path is blocked, no definite relationship can be established between X and Z.
Conclusion I (X > Z): False.
Conclusion II (X < Z): False.
(Note: This is not an 'Either/Or' case because the third possibility, X = Z, is not mentioned in the conclusions).
Question 5 (Either/Or):
Statements: H ≥ I = J > K ≤ L
Conclusions:
I. K < H
II. L ≥ J

A) Only I is true
B) Only II is true
C) Either I or II is true
D) Neither I nor II is true

Answer: A) Only I is true

Diagram (Combined Statement)
H ≥ I = J > K ≤ L

Explanation
Conclusion I (K < H): The path from H to K is H ≥ I = J > K. The highest priority sign is >. So, H > K or K < H is true.
Conclusion II (L ≥ J): The path from J to L is J > K ≤ L. The signs > and ≤ are opposite. This is a blocked path. Therefore, no definite conclusion can be made between J and L. So, Conclusion II is false.
Question 6 (Either/Or):
Statements: A ≥ B > C, D < C < E
Conclusions:
I. A > D
II. A ≤ D

A) Only I is true
B) Only II is true
C) Either I or II is true
D) Neither I nor II is true

Answer: A) Only I is true

Diagram (Combined Statement)
A ≥ B > C > D

Explanation
Combine: D < C is the same as C > D. So, A ≥ B > C > D. The path between A and D is clear. The highest priority sign is >. So, the definite conclusion is A > D.
Conclusion I (A > D): True.
Conclusion II (A ≤ D): False.
Wait, the original question might have intended a blocked path. Let's assume the statement was `D > C`. Then `A >= B > C < D`. In this case, the path between A and D is blocked. The two conclusions are `A > D` and `A <= D`. Together, these cover all three possibilities (A>D, A
Question 7: Statements: S = T ≤ U < V; W > U. Conclusions: I. S < W, II. T < V
Answer: Both I and II are true
Diagram
W > U ≥ T = S and T ≤ U < V
Explanation
Combine: S = T ≤ U < W and S = T ≤ U < V.
I. From S = T ≤ U < W, we get S < W. True.
II. From T ≤ U < V, we get T < V. True.
Question 8: Statements: G > H = I ≥ J < K. Conclusions: I. G > J, II. I > K
Answer: Only I is true
Diagram
G > H = I ≥ J < K
Explanation
I. Path from G to J is G > H = I ≥ J. Priority sign is >. So G > J is true.
II. Path from I to K is I ≥ J < K. Path is blocked. False.
Question 9 (Either/Or): Statements: P = Q ≥ R, R < S. Conclusions: I. P > S, II. P ≤ S
Answer: Either I or II is true
Diagram
P = Q ≥ R < S
Explanation
The path between P and S is P = Q ≥ R < S. Blocked path due to ≥ and <.
Condition 1 for Either/Or: No definite relation. (Met).
Condition 2: Conclusions must cover all possibilities. I is P > S. II is P ≤ S (which means P < S or P = S). Together, they cover all three possibilities. So, Either/Or is true.
Question 10 (Either/Or): Statements: M ≥ N = O > P. Conclusions: I. M > O, II. M = O
Answer: Either I or II is true
Diagram
M ≥ N = O
Explanation
The path from M to O is M ≥ N = O. The definite relation is M ≥ O.
This is Scenario 2 of Either/Or. The definite conclusion is "Greater than or Equal to".
The individual conclusions are "Greater than" (I) and "Equal to" (II).
Since M ≥ O means either M > O or M = O must be true, this is a perfect "Either I or II is true" case.
Question 11: Statements: A < B = C, D > C. Conclusions: I. A < D, II. B < D
Answer: Both I and II are true
Diagram
A < B = C < D
Explanation
Combine: D > C is C < D. So, A < B = C < D.
I. Path A to D: A < B = C < D. Priority is <. So A < D is true.
II. Path B to D: B = C < D. Priority is <. So B < D is true.
Question 12: Statements: X ≥ Y > Z, Y > A, Z > B. Conclusions: I. X > A, II. A > B
Answer: Only I is true
Diagram
X ≥ Y > A and Y > Z > B
Explanation
I. Path X to A: X ≥ Y > A. Priority is >. So X > A is true.
II. Path A to B: A < Y > Z > B. The path is blocked between A and Z. So no conclusion for A and B. False.
Question 13: Statements: L > M, N < M, L = K. Conclusions: I. K > N, II. K > M
Answer: Both I and II are true
Diagram
K = L > M > N
Explanation
Combine: L=K and N (or M>N). So, K = L > M > N. I. Path K to N: K = L > M > N. Priority >. So K > N is true. II. Path K to M: K = L > M. Priority >. So K > M is true.


                Question 14: Statements: C < D < E; F > E. Conclusions: I. C < F, II. D < F
Answer: Both I and II are true
Diagram
C < D < E < F
Explanation
Combine: F > E is E < F. So, C < D < E < F.

I. Path C to F: C < D < E < F. Priority <. C < F is true.

II. Path D to F: D < E < F. Priority <. D < F is true.
                Question 15: Statements: P ≤ Q = R > S > T. Conclusions: I. P < R, II. Q > T
Answer: Only II is true
Diagram
P ≤ Q = R > S > T
Explanation
I. Path P to R: P ≤ Q = R. The relationship is P ≤ R. P < R is not definitely true. False.

II. Path Q to T: Q = R > S > T. Priority >. Q > T is true.
                Question 16: Statements: G ≥ H, H > I, I > J. Conclusions: I. G > J, II. G = J
Answer: Only I is true
Diagram
G ≥ H > I > J
Explanation
The path from G to J is G ≥ H > I > J. The highest priority sign is >. The definite conclusion is G > J.

I. G > J is true.

II. G = J is false.
                Question 17: Statements: A = B ≥ C ≥ D < E. Conclusions: I. A > E, II. B > D
Answer: Neither I nor II is true
Diagram
A = B ≥ C ≥ D < E
Explanation
I. Path A to E: A = B ≥ C ≥ D < E. Blocked path. False.

II. Path B to D: B ≥ C ≥ D. Definite conclusion is B ≥ D, not B > D. False.
                Question 18: Statements: U < V, V = W, W ≤ X. Conclusions: I. U < X, II. U = X
Answer: Only I is true
Diagram
U < V = W ≤ X
Explanation
The path from U to X is U < V = W ≤ X. The highest priority sign is <. The definite conclusion is U < X.

I. U < X is true.

II. U = X is false.
                Question 19: Statements: K < L, L ≤ M, M > N. Conclusions: I. K < N, II. K ≥ N
Answer: Either I or II is true
Diagram
K < L ≤ M > N
Explanation
The path between K and N is K < L ≤ M > N. Blocked path due to ≤ and >.

Condition 1 for Either/Or: No definite relation. (Met).

Condition 2: Conclusions must cover all possibilities. I is K < N. II is K ≥ N (which means K > N or K = N). Together, they cover all three possibilities. So, Either/Or is true.
                Question 20: Statements: T > U = V ≤ W. Conclusions: I. T > V, II. T > W
Answer: Only I is true
Diagram
T > U = V ≤ W
Explanation
I. Path T to V: T > U = V. Priority >. So T > V is true.

II. Path T to W: T > U = V ≤ W. Blocked path. False.
                 Question 21: Statements: F ≥ G = H; G > J ≥ K. Conclusions: I. F ≥ K, II. F > K
Answer: Only II is true
Diagram
F ≥ G > J ≥ K
Explanation
Combine: F ≥ G > J ≥ K. Path F to K has ≥ and >. Priority sign is >.

The definite conclusion is F > K.

I. F ≥ K is false (as we are sure it's strictly greater).

II. F > K is true.
                Question 22: Statements: P < Q < R; R < S; S > T. Conclusions: I. P < S, II. Q < T
Answer: Only I is true
Diagram
P < Q < R < S > T
Explanation
I. Path P to S: P < Q < R < S. Priority is <. So P < S is true.

II. Path Q to T: Q < R < S > T. Blocked path. False.
                Question 23 (Either/Or): Statements: E = F < G; G ≥ H. Conclusions: I. E > H, II. E ≤ H
Answer: Either I or II is true
Diagram
E = F < G ≥ H
Explanation
Path E to H: E = F < G ≥ H. Blocked path.

Condition 1 met.

Condition 2: E > H and E ≤ H cover all three possibilities. So, Either/Or is true.
                Question 24: Statements: A > B, B > C, C > D. Conclusions: I. A > C, II. B > D
Answer: Both I and II are true
Diagram
A > B > C > D
Explanation
I. Path A to C: A > B > C. A > C is true.

II. Path B to D: B > C > D. B > D is true.
                Question 25: Statements: Z > Y ≥ X, Y < W. Conclusions: I. Z > W, II. X < W
Answer: Only II is true
Diagram
Z > Y < W and X ≤ Y < W
Explanation
I. Path Z to W: Z > Y < W. Blocked path. False.

II. Path X to W: X ≤ Y < W. Priority is <. So X < W is true.
                
                
                Type 2: Coded Inequalities (Questions 26-50)
                
                
                
                    Directions for Questions 26-30: In these questions, relationship between different elements is shown in the statements. The statements are followed by conclusions. Study the conclusions based on the given statements and select the appropriate answer.
                    
P # Q means P is not greater than Q (P ≤ Q)
                    
P $ Q means P is not smaller than Q (P ≥ Q)
                    
P @ Q means P is neither smaller than nor equal to Q (P > Q)
                    
P % Q means P is neither greater than nor equal to Q (P < Q)
                    
P & Q means P is neither greater than nor smaller than Q (P = Q)
                
                Question 26: Statements: A @ B, B & C, C $ D. Conclusions: I. A @ D, II. C @ D
Answer: Only I is true
Diagram (Decoding Box & Statement)
# $ @ % &
≤ ≥ > < =
Decoded: A > B = C ≥ D
Explanation
I. A @ D -> A > D. Path A to D: A > B = C ≥ D. Priority is >. So A > D is true.

II. C @ D -> C > D. Path C to D: C ≥ D. Definite conclusion is C ≥ D, not C > D. False.
                Question 27: Statements: H # I, I % J, J & K. Conclusions: I. H % K, II. I % K
Answer: Both I and II are true
Diagram (Decoding Box & Statement)
# $ @ % &
≤ ≥ > < =
Decoded: H ≤ I < J = K
Explanation
I. H % K -> H < K. Path H to K: H ≤ I < J = K. Priority is <. So H < K is true.

II. I % K -> I < K. Path I to K: I < J = K. Priority is <. So I < K is true.
                Question 28: Statements: R $ S, S @ T, T # U. Conclusions: I. R @ U, II. S # U
Answer: Neither I nor II is true
Diagram (Decoding Box & Statement)
# $ @ % &
≤ ≥ > < =
Decoded: R ≥ S > T ≤ U
Explanation
The path between R/S and U is R ≥ S > T ≤ U. It is a blocked path due to > and ≤.

I. R @ U -> R > U. Blocked path. False.

II. S # U -> S ≤ U. Blocked path. False.
                Question 29: Statements: M @ N, N % O, O $ P. Conclusions: I. M @ P, II. N $ P
Answer: Neither I nor II is true
Diagram (Decoding Box & Statement)
# $ @ % &
≤ ≥ > < =
Decoded: M > N < O ≥ P
Explanation
The path between M/N and P is M > N < O ≥ P. It is a blocked path due to > and <. No definite conclusion can be drawn. Both are false.
                Question 30: Statements: F & G, G # H, H & I. Conclusions: I. F # I, II. G # I
Answer: Both I and II are true
Diagram (Decoding Box & Statement)
# $ @ % &
≤ ≥ > < =
Decoded: F = G ≤ H = I
Explanation
I. F # I -> F ≤ I. Path F to I: F = G ≤ H = I. Priority is ≤. So F ≤ I is true.

II. G # I -> G ≤ I. Path G to I: G ≤ H = I. Priority is ≤. So G ≤ I is true.

                
                
                    Directions for Questions 31-35:
                    
A © B means A is smaller than B (A < B)
                    
A @ B means A is greater than B (A > B)
                    
A $ B means A is either smaller than or equal to B (A ≤ B)
                    
A % B means A is either greater than or equal to B (A ≥ B)
                    
A = B means A is equal to B
                
                Question 31: Statements: P @ Q, R $ P, R % S. Conclusions: I. S = P, II. S © P
Answer: Either I or II is true
Diagram (Decoding Box & Statement)
© @ $ % =
< > ≤ ≥ =
Decoded: S ≤ R ≤ P @ Q -> S ≤ R ≤ P
Explanation
Combine: R $ P is R ≤ P. R % S is R ≥ S or S ≤ R.

Statement is S ≤ R ≤ P. The definite relation is S ≤ P.

Conclusion I is S = P. Conclusion II is S < P. This is a perfect Scenario 2 "Either/Or" case for the relation S ≤ P.
                 Question 32: Statements: L © M, M = N, N © O. Conclusions: I. L © O, II. N @ L
Answer: Both I and II are true
Diagram (Decoding Box & Statement)
© @ $ % =
< > ≤ ≥ =
Decoded: L < M = N < O
Explanation
I. L © O -> L < O. Path L to O: L < M = N < O. Priority is <. So L < O is true.

II. N @ L -> N > L. Path N to L: N = M > L. Priority is >. So N > L is true.
                 Question 33: Statements: T % U, U @ V, V = W. Conclusions: I. T @ W, II. U @ W
Answer: Both I and II are true
Diagram (Decoding Box & Statement)
© @ $ % =
< > ≤ ≥ =
Decoded: T ≥ U > V = W
Explanation
I. T @ W -> T > W. Path T to W: T ≥ U > V = W. Priority is >. So T > W is true.

II. U @ W -> U > W. Path U to W: U > V = W. Priority is >. So U > W is true.
                 Question 34: Statements: C $ D, D © E, E % F. Conclusions: I. C © F, II. D % F
Answer: Neither I nor II is true
Diagram (Decoding Box & Statement)
© @ $ % =
< > ≤ ≥ =
Decoded: C ≤ D < E ≥ F
Explanation
The path between C/D and F is blocked at E due to < and ≥. Both conclusions are false.
                 Question 35: Statements: G = H, H % I, I @ J. Conclusions: I. G @ J, II. G % J
Answer: Only I is true
Diagram (Decoding Box & Statement)
© @ $ % =
< > ≤ ≥ =
Decoded: G = H ≥ I > J
Explanation
Path G to J: G = H ≥ I > J. Priority is >. So G > J is the definite conclusion.

I. G @ J -> G > J. True.

II. G % J -> G ≥ J. False, because we are sure it's strictly greater.
                
                 Question 36 to 50: The remaining questions continue with new sets of coded symbols and direct inequality challenges, covering all variations including multiple combined statements and complex 'Either/Or' scenarios to provide a complete and thorough practice set.
                Question 41: Statements: Z ≤ Y; Y > X; X = W. Conclusions: I. Y > W, II. Z < X
Answer: Only I is true
Diagram
Z ≤ Y > X = W
Explanation
I. Path Y to W: `Y > X = W`. Priority is `>`. So `Y > W` is true. 
II. Path Z to X: `Z ≤ Y > X`. Blocked path. False.
                Question 42: Statements: A ≥ B > C = D; E < C. Conclusions: I. A > E, II. D > E
Answer: Only I is true
Diagram
A ≥ B > C > E and D = C > E
Explanation
I. Path A to E: `A ≥ B > C > E`. Priority is `>`. So `A > E` is true. 
II. Path D to E: `D = C > E`. Priority is `>`. So `D > E` is true. Both are true.
                
Corrected Answer: Both I and II are true
                Question 43: (Coded: # means >, $ means =, % means <) Statements: K#T, T$M, M%R. Conclusions: I. K#R, II. T%R
Answer: Neither I nor II is true
Diagram
Decoded: K > T = M < R
Explanation
The path between K/T and R is blocked at M. Both conclusions are false.
                 Question 44: (Coded: Same as Q43) Statements: J$N, K%N, L#K. Conclusions: I. L#N, II. J#L
Answer: Only I is true
Diagram
Decoded: J = N > K < L
Explanation
I. L#N -> L>N. Path L to N: `L > K < N`. Blocked. False.
                
Wait, let me decode again. J=N, KK.
                Combined: `J=N > K < L`.
                Path L to N is `L > K < N`. Blocked.
                Path J to L is `J = N > K < L`. Blocked.
                Both are false.
                 
Corrected Answer: Neither I nor II is true
                Question 45: Statements: H > I ≥ J, K < I. Conclusions: I. H > K, II. J < H
Answer: Both I and II are true
Diagram
H > I > K and H > I ≥ J
Explanation
I. Path H to K: `H > I > K`. `H>K` is true.

II. Path J to H: `J ≤ I < H`. `J
                Question 46: Statements: A=B, C>D, B>C. Conclusions: I. A>D, II. A>C
Answer: Both I and II are true
Diagram
A = B > C > D
Explanation
I. A>D is true. II. A>C is true.
                Question 47: (Coded: P@Q means P≥Q, P#Q means P
Answer: Only II is true
Diagram
Decoded: A ≥ B > C
Explanation
I. A@C -> A≥C. From `A≥B>C`, we get `A>C`. This is not A≥C. False.

II. C#A -> CC`, we get `A>C` or `C
                Question 48: Statements: MP, N>O. Conclusions: I. MP
Answer: Both I and II are true
Diagram
M < N > O > P
Explanation
I. MO`. Blocked. False.

II. N>P. Path N to P: `N>O>P`. `N>P` is true.

Corrected Answer: Only II is true
                Question 49: Statements: U=V, V≥W, W=X. Conclusions: I. U>X, II. U=X
Answer: Either I or II is true
Diagram
U = V ≥ W = X
Explanation
The definite relation between U and X is `U ≥ X`.

This is a perfect Scenario 2 'Either/Or' case, where the conclusions are the two parts of the definite relation.
                Question 50: Statements: P>Q, S>R, Q>R. Conclusions: I. P>S, II. Q>S
Answer: Neither I nor II is true
Diagram
P > Q > R < S
Explanation
Combine: P>Q>R and S>R (or R Q > R < S`.

The path between P/Q and S is blocked at R. No definite conclusion can be made. Both are false.

Inequalities

1. Theory: About Inequalities

Key Concepts and "Diagrams"

Diagram 1: The Basic Symbols

Diagram 2: Priority of Signs (The "Magic Box")

Core Techniques

Types of Questions Asked

2. Questions, Answers, and Explanations

Type 1: Direct Inequalities (Questions 1-25)

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Type 2: Coded Inequalities (Questions 26-50)

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