INEQUALITY

Basic symbols and their meanings

S. No.	Symbol	Meaning	Example
1	>	Greater than	A > B
2	<	Less than	A < B
3	≥	Greater than or equal to	A ≥ B
4	≤	Less than or equal to	A ≤ B
5	=	Equal to	A = B
6	≠	Not equal to	A ≠ B

Coded Inequality (For Mains)

In coded inequalities, mathematical operators are replaced with special symbols such as @, #, $, % and &. The meaning of each symbol is provided in the question.

Symbol	Meaning
A @ B	A ≥ B
A # B	A < B
A $ B	A = B
A % B	A ≠ B
A & B	A ≤ B

Practice Questions

Q1. Statements: A > B ≥ C = D > E

Conclusions:
I. A > C
II. C > E

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (C) Both Conclusions I and II are true

Solution:
– A > B ≥ C, therefore A > C.
– C = D > E, therefore C > E.
Hence, both conclusions follow.

Q2. Statements: P ≤ Q < R = S ≤ T

Conclusions:
I. P < R
II. S > T

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (A) Only Conclusion I is true

Solution:
– P ≤ Q < R, therefore P < R.
– S ≤ T, therefore S > T is false.
Hence, only Conclusion I follows.

Q3. Statements: A < B = C ≤ D

Conclusions:
I. A = C
II. A < D

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (B) Only Conclusion II is true

Solution:
– A < B and B = C, therefore A < C. Hence, Conclusion I is false.
– Since C ≤ D and A < C, A < D.
Hence, only Conclusion II follows.

Q4. Statements: M ≥ N, N ≤ O

Conclusions:
I. M > O
II. M = O

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (D) Neither Conclusion I nor II is true

Solution:
The statements establish that M ≥ N and O ≥ N, but they do not establish a definite relationship between M and O.
– If M = 8 and O = 5, Conclusion I is true.
– If M = 5 and O = 5, Conclusion II is true.
– If M = 4 and O = 7, both conclusions are false.
Hence, neither conclusion follows definitely.

Q5. Statements: A ≥ B > C, C = D < E, E ≤ F

Conclusions:
I. A > D
II. D < F

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (C) Both Conclusions I and II are true

Solution:
Combining the statements:
A ≥ B > C = D < E ≤ F
– A ≥ B and B > D, therefore A > D.
– D < E and E ≤ F, therefore D < F.
Hence, both conclusions follow.

Q6. Statements: P > Q ≥ R, R < S ≤ T

Conclusions:
I. P > R
II. Q < T

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (A) Only Conclusion I is true

Solution:
– P > Q and Q ≥ R, therefore P > R.
– The relationship between Q and T cannot be determined from the given statements.
Hence, only Conclusion I follows.

Q7. Statements: A @ B, B # C, C $ D

Given Codes:

Symbol	Meaning
@	≥
#	<
$	=
%	≠
&	≤

Conclusions:
I. A > D
II. B < D

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (B) Only Conclusion II is true

Solution:
Decoding the statements:
A ≥ B < C = D
– A > D cannot be established.
– B < D is true because B < C and C = D.
Hence, only Conclusion II follows.

Q8. Statements: L ≤ M = N > O ≥ P

Conclusions:
I. M > P
II. L < O

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (A) Only Conclusion I is true

Solution:
– M = N > O ≥ P, therefore M > P.
– The relationship between L and O cannot be determined.
Hence, only Conclusion I follows.

Q9. Statements: X > Y = Z ≤ W

Conclusions:
I. X > Z
II. X > W

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (A) Only Conclusion I is true

Solution:
– X > Y and Y = Z, therefore X > Z.
– The relationship between X and W cannot be determined.
For example, X = 8, Z = 5 and W = 10 satisfy the statements, but X > W is false.
Hence, only Conclusion I follows.

Q10. Statements: M # N, N $ O, O @ P, P % Q

Decoded Form:
M > N ≥ O = P < Q

Conclusions:
I. M # P (M > P)
II. N $ P (N ≥ P)

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (C) Both Conclusions I and II are true

Solution:
Decoding the statements gives M > N ≥ O = P < Q.
From M > N ≥ O = P, we get M > P. Therefore, Conclusion I is true.
From N ≥ O = P, we get N ≥ P. Therefore, Conclusion II is true.
Hence, both Conclusions I and II are true.

Q11. Statements: A % B, B & C, C @ D, D # E

Decoded Form:
A < B ≤ C = D > E

Conclusions:
I. A % C (A < C)
II. B & D (B ≤ D)

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (C) Both Conclusions I and II are true

Solution:
Decoding the statements gives A < B ≤ C = D > E.
From A < B ≤ C, we get A < C. Therefore, Conclusion I is true.
From B ≤ C = D, we get B ≤ D. Therefore, Conclusion II is true.
Hence, both Conclusions I and II are true.

Q12. Statements: X $ Y, Y # Z, Z & W, W @ V

Decoded Form:
X ≥ Y > Z ≤ W = V

Conclusions:
I. X # Z (X > Z)
II. Z & V (Z ≤ V)

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (C) Both Conclusions I and II are true

Solution:
Decoding the statements gives X ≥ Y > Z ≤ W = V.
From X ≥ Y > Z, we get X > Z. Therefore, Conclusion I is true.
From Z ≤ W = V, we get Z ≤ V. Therefore, Conclusion II is true.
Hence, both Conclusions I and II are true.

Q13. Statements: H % I, I & J, J # K, K $ L

Decoded Form:
H < I ≤ J > K ≥ L

Conclusions:
I. H % J (H < J)
II. I # L (I > L)

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (A) Only Conclusion I is true

Solution:
Decoding the statements gives H < I ≤ J > K ≥ L.
From H < I ≤ J, we get H < J. Therefore, Conclusion I is true.
Between I and L, the relationship is I ≤ J > K ≥ L. Due to the change in inequality direction, no definite relationship can be established between I and L. Therefore, Conclusion II does not follow.
Hence, only Conclusion I is true.

Q14. Statements: S $ T, T % U, U & V, V # W

Decoded Form:
S ≥ T < U ≤ V > W

Conclusions:
I. S # U (S > U)
II. T # V (T > V)

Options:
(A) Only Conclusion I is true
(B) Only Conclusion II is true
(C) Both Conclusions I and II are true
(D) Neither Conclusion I nor II is true
(E) Either Conclusion I or II is true

Answer: (D) Neither Conclusion I nor II is true

Solution:
Decoding the statements gives S ≥ T < U ≤ V > W.
Between S and U, the relationship is S ≥ T < U. Since the inequality directions are opposite, no definite relationship can be established between S and U. Therefore, Conclusion I does not follow.
From T < U ≤ V, we get T < V. However, Conclusion II states T > V, which is incorrect.
Hence, neither Conclusion I nor Conclusion II follows.