Inequalities: Rules explained with examples

Inequalities is an important topic of reasoning and its questions appear frequently in many competitive exams. It is a scoring topic and questions based on this concept should not be left in the exam. Once you grab the concept, solving any question from this topic becomes easy.
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To understand the concept of inequalities, we must keep in mind that equality means equal and inequality means there are five possibilities between terms/objects. Let us understand these types of possibilities through a table.
Signs Meaning
> Greater than
< Less than
Greater than or Equal to
Less than or Equal to
= Equal to
Let us consider two variables X and Y.
Signs Meaning
X > Y X is Greater than Y
X < Y X is Less than Y
X≥Y X is Greater than or Equal to Y
X≤Y X is Less than or Equal to Y
X=Y X is Equal to Y
Rules of Inequalities:
The relation between two inequalities can be established if they have a common term. For example,
a) A > B, B > C : By looking at this, we can easily define the relation that A > B > C which means A > C or C < A.
b) A < B, B < C : By looking at this, we can easily define that A < B < C, which means A > C or C < A.
c) A ≥ B, B ≥ C : This implies that A ≥ B ≥ C, which means A ≥ C or C ≤ A.
A relation cannot be defined if they don't have a common term. For example,
a) A > B, C > B : This implies that we cannot define a specific relation between A and C as both of them are greater than B.
b) B > A, D < B : This implies that a relation cannot be established between A and D as both are lesser than B.
c) A ≥ B, B ≤ C : This implies that a relation cannot be established between A and C as both are greater than or equal to B.
Complementary Pairs: (Either & or)-Either and or cases only takes place in complementary pairs. We cannot combine two elements with common elements in which no relation is established. For example, B ≤ C, A ≥ B. here we can conclude that A ≥ C or A < C.
To understand the concept properly let us try doing some questions.
Solved Questions
Question 1: Statements: a) A > B b) B > C
Conclusions: a) A > C b) C > A
Solution: On combining both the statements, we get: A > B > C
So, we can easily say that the conclusion a) follows i.e. A > C
Question 2: Statements: a) A > B b) B < C
Conclusions: a) A > C b) A < C
Solution: Here, we can see that both A and C are greater than B but we cannot derive any relation between A and C.
So the answer to the above questions will be "nothing can be concluded".
Question 3: Statement: A > B > C < D≥E
Conclusion: a) A > D
b) D > B
Solution: Here, nothing can be concluded because there is no definite relation between A & D and D & B. we cannot say that which one is greater, equal or lesser.
Question 4: Statement: P < Q ≤ R < S > T
Conclusions: a) T < R
b) S > P
c) R > T
d) Q < S
Solution: Here, we can see that R is either greater or lesser or equal to T because no specific relation between them is defined. So, we cannot conclude anything about part "a)". But in part "b)", we can see that S > P. So, it is true.
Talking about part "c)", we cannot say that R > T as no specific relation is defined between them in the statement.
When we look at the statement, we can easily say that Q < S. Hence, part "d)" is true.
Question 5: Statements: a) M ≤ N≤ O< P b) K = L ≥ O > C
Conclusion: a) M < O
b) P ≥ M
c) P ≥ K
d) M ≥ D
e) C = P
f) O ≥ M
Solution: a) From the statements, we can see that O ≥ M. So, we cannot say that M < O.
b) Here, P > M and not greater than or equal to. Hence, this conclusion is also incorrect.
c) We can see here that P > N and K ≥ O. But, we cannot find any specific relation between P and K. So, we cannot say that P ≥ K.
d) D is not mentioned anywhere in the statement. So, this cannot be concluded at all.
e) From statement a), we can see that P > O and from statement b), we can say that O > C. On, combining both, we can say that P > O > C. So, we can say that P > C but we cannot say that P = C. Hence, the conclusion given in the question is incorrect.
f) We can see that O ≥ N ≥ M. Therefore, we can easily say that O ≥ M. hence, the conclusion is true.
Question 6: Statements: a) A > B = C ≥ D b) P ≥ Q = R ≥ D
Conclusions: a) C = P
b) A ≥ Q
c) A < Q
Solution: a) Looking at both the statements, we cannot conclude the relation between C and P.
b) We cannot say that A ≥ Q as no relation or combination is there between the two terms.
c) Also, we cannot say that A < Q.
But, we can say that A < Q or A ≥ Q. Therefore, either conclusion b) or conclusion c) is true.
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Key Learning:
In this article, we have learnt ways of finding a specific relation between two or more given terms.