Inequality • Direct and coded inequalities
Inequality: Full-Length Notes
(Direct and Coded Inequalities with Examples)
1. Introduction
Inequality reasoning is a core topic in competitive exams, testing your ability to deduce relationships between variables using mathematical or coded symbols. Questions are typically of two types:
Direct Inequality: Uses standard symbols (<, >, =, ≤, ≥).
Coded Inequality: Uses new or coded symbols to represent the above relations, which must be decoded before solving123.
2. Direct Inequality
A. Symbols and Meanings
> : Greater than
< : Less than
= : Equal to
≥ : Greater than or equal to
≤ : Less than or equal to
≠ : Not equal to
B. How to Approach Direct Inequality Questions
Combine Statements Using Common Terms
If A > B and B > C, then A > C.
If A ≥ B and B = C, then A ≥ C.
Undefined Relations
If statements have no common term (e.g., A > B and C > D), the relation between A and D is undefined2.
Complementary Pairs
Only the direct relation is defined. For example, if A ≥ B and B ≥ C, the only certain relation is A ≥ C, not A > C2.
C. Examples
Example 1:
Statements: A > B = C; G < C > D
Conclusions:
I. A < D
II. A = G
Solution:
From A > B = C and C > D, A > C > D ⇒ A > D, so I is false.
C > G, so A cannot be equal to G.
Answer: Neither conclusion follows2.
Example 2:
Statements: I ≥ U > T ≤ R; U ≥ V = W > C
Conclusions:
I. W > T
II. C > I
Solution:
U ≥ V = W > C, so W > C.
I ≥ U > T, so I > T.
No direct link between W and T, or C and I.
Answer: Neither conclusion follows2.
Example 3:
Statements: P < S < R < T > Q
Conclusions:
P < R (True: P < S < R)
S < T (True: S < R < T)
No relation between P & Q (True: No direct link)
P < T (True: P < S < R < T)
Answer: Only "No relation between P & Q" is incorrect1.
3. Coded Inequality
A. What is Coded Inequality?
In coded inequality, each mathematical symbol is replaced by a new symbol or code. You must first decode the meaning of each symbol.
Example of Codes:
: Equal to (=)
: Greater than or equal to (≥)
% : Less than (<)
& : Greater than (>)
$ : Less than or equal to (≤)
@ : Greater than (>)
(Note: Codes may vary by question; always check the legend)13.
B. How to Approach Coded Inequality Questions
Decode Each Symbol
Write what each symbol means based on the legend.
Combine Statements
Arrange all relations into a single chain using decoded symbols.
Analyze Conclusions
Check if the conclusion can be definitely established from the combined chain.
C. Examples
Example 1:
Symbols:
P @ Q: P is neither less nor equal to Q (i.e., P > Q)
P # Q: P is neither greater nor equal to Q (i.e., P < Q)
P $ Q: P is neither less nor greater than Q (i.e., P = Q)
P & Q: P is not greater than Q (i.e., P ≤ Q)
P % Q: P is not less than Q (i.e., P ≥ Q)3
Statements: Y # X & Z # N; X % R $ T @ W
Decoded:
Y < X ≤ Z < N; X ≥ R = T > W
Conclusions:
I. N @ W → N > W (True: N > Z ≥ X ≥ R = T > W, so N > W)
II. T % Y → T ≥ Y (No direct relation between T and Y)
III. Z % R → Z ≥ R (Z ≥ X ≥ R, so Z ≥ R)3
Answer: Only I and III follow.
Example 2:
Symbols:
A + B: A is neither smaller nor equal to B (A > B)
A - B: A is not greater than B (A ≤ B)
A × B: A is not smaller than B (A ≥ B)
A ÷ B: A is neither greater nor equal to B (A < B)
A ± B: A is neither smaller nor greater than B (A = B)3
Statements: P × Q, P - T, T ÷ R, R ± S
Decoded: P ≥ Q, P ≤ T, T < R, R = S
So, S = R > T ≥ P ≥ Q
Conclusions:
I. Q ± T (Q = T): Not necessarily true
II. S + Q (S > Q): True
Answer: Only II follows3.
Example 3:
Symbols:
X * Y: X is neither less than nor greater than Y (=)
X # Y: X is either greater than or equal to Y (≥)
X % Y: X is less than Y (<)
X & Y: X is neither less than nor equal to Y (>)
X $ Y: X is not greater than Y (≤)3
Statements: A % B, C & D, F * E # C, D % A
Decoded: A < B, C > D, F = E ≥ C, D < A
Combined: F = E ≥ C > D < A < B
Conclusions:
I. D % B (D < B): True
II. E & A (E > A): Can't be determined
III. F & D (F > D): True
Answer: I and III follow3.
4. Key Rules and Tips
Transitive Property:
If A > B and B > C, then A > C.
If A ≥ B and B ≥ C, then A ≥ C.
Undefined Relation:
If there is no common term, the relation is undefined2.
Equalities:
If A = B and B > C, then A > C.
Coded Symbols:
Always decode symbols before solving.
Practice:
More practice increases speed and accuracy, especially for coded inequalities1.
5. Summary Table: Common Coded Inequality Symbols
| Symbol | Meaning |
|---|---|
| * | = |
| # | ≥ |
| % | < |
| & | > |
| $ | ≤ |
| @ | > |
| ± | = |
(Always check the code legend in each question!)
6. Conclusion
Direct inequalities use standard mathematical symbols and can be solved by chaining statements with common terms.
Coded inequalities require decoding symbols, forming a logical chain, and then checking each conclusion.
Undefined relations occur when a direct link cannot be established.
Practice is essential for mastering both types, especially coded inequalities, which are now common in exams123.
References:
1: BYJU’S Reasoning Inequality
2: BYJU’S Inequalities Concepts and Examples
3: Testbook Coded Inequalities MCQ
Direct Inequality Questions (1–25)
If A > B, B > C, then which of the following is true?
If X ≥ Y, Y = Z, then which of the following is true?
If P < Q, Q ≤ R, then what is the relation between P and R?
If M = N, N > O, then what is the relation between M and O?
If S ≥ T, T > U, then what is the relation between S and U?
If W < X, X < Y, then what is the relation between W and Y?
If A ≥ B, B ≥ C, then what is the relation between A and C?
If D > E, E = F, then what is the relation between D and F?
If G ≤ H, H < I, then what is the relation between G and I?
If P = Q, Q = R, then what is the relation between P and R?
If A > B, B ≥ C, then what is the relation between A and C?
If X < Y, Y ≤ Z, then what is the relation between X and Z?
If M > N, N > O, O > P, then what is the relation between M and P?
If S ≥ T, T = U, U > V, then what is the relation between S and V?
If A = B, B < C, then what is the relation between A and C?
If D > E, E ≤ F, then what is the relation between D and F?
If G ≥ H, H = I, I < J, then what is the relation between G and J?
If P < Q, Q = R, R < S, then what is the relation between P and S?
If A > B, B > C, C > D, then what is the relation between A and D?
If X ≤ Y, Y ≤ Z, Z ≤ W, then what is the relation between X and W?
If M = N, N ≥ O, then what is the relation between M and O?
If S > T, T = U, U ≥ V, then what is the relation between S and V?
If A ≥ B, B > C, then what is the relation between A and C?
If D < E, E < F, F ≤ G, then what is the relation between D and G?
If H = I, I = J, J = K, then what is the relation between H and K?
Coded Inequality Questions (26–50)
Legend for Questions 26–35:
@ means ‘greater than’ (>)
means ‘less than’ (<)
$ means ‘equal to’ (=)
% means ‘greater than or equal to’ (≥)
& means ‘less than or equal to’ (≤)
If A @ B $ C, which of the following is true?
If X # Y % Z, which of the following is true?
If P $ Q & R, which of the following is true?
If M % N @ O, which of the following is true?
If S & T # U, which of the following is true?
If A @ B & C, which of the following is true?
If D $ E % F, which of the following is true?
If G # H $ I, which of the following is true?
If J % K & L, which of the following is true?
If M & N @ O, which of the following is true?
Legend for Questions 36–50:
means ‘greater than’ (>)
means ‘less than’ (<)
= means ‘equal to’ (=)
× means ‘greater than or equal to’ (≥)
÷ means ‘less than or equal to’ (≤)
If A + B = C, which of the following is true?
If X - Y × Z, which of the following is true?
If P × Q ÷ R, which of the following is true?
If M = N + O, which of the following is true?
If S ÷ T - U, which of the following is true?
If A × B - C, which of the following is true?
If D + E ÷ F, which of the following is true?
If G = H × I, which of the following is true?
If J ÷ K = L, which of the following is true?
If M + N - O, which of the following is true?
If P × Q = R, which of the following is true?
If S - T × U, which of the following is true?
If V = W ÷ X, which of the following is true?
If Y + Z × A, which of the following is true?
If B ÷ C - D, which of the following is true?
Answers
Direct Inequality (1–25)
A > C
X ≥ Z
P < R
M > O
S > U
W < Y
A ≥ C
D > F
G < I
P = R
A > C
X < Z
M > P
S > V
A < C
D > F or D = F (cannot be determined exactly)
G > J
P < S
A > D
X ≤ W
M ≥ O
S > V
A > C or A = C (A ≥ C)
D < G
H = K
Coded Inequality (26–35)
A > B = C
X < Y ≥ Z
P = Q ≤ R
M ≥ N > O
S ≤ T < U
A > B ≤ C
D = E ≥ F
G < H = I
J ≥ K ≤ L
M ≤ N > O
Coded Inequality (36–50)
A > B = C
X < Y ≥ Z
P ≥ Q ≤ R
M = N > O
S ≤ T < U
A ≥ B < C
D > E ≤ F
G = H ≥ I
J ≤ K = L
M > N < O
P ≥ Q = R
S < T ≥ U
V = W ≤ X
Y > Z ≥ A
B ≤ C < d
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