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.

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.

Let us consider two variables X and Y.

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.

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.

In this article, we have learnt ways of finding a specific relation between two or more given terms.