Inequalities

Inequalities

Questions based on inequalities are repeated almost every year in CAT exam. Inequalities for CAT is one of the easiest to solve and scoring topics in the paper and question related to inequalities should not be left unattempted.
The questions related to inequalities are mainly of two types i.e.

1. Direct Inequalities
2. Indirect Inequalities

The detailed explanations for both these types are given below along with several solved examples to help the CAT aspirants understand the topic in a better way.

Some Specific Rules to Solve Inequalities:

1. A relation can be defined only if the statements of inequalities have a common term. For example, if A > B and B < C, it can be concluded that A > C.
2. A relation is considered undefined if the inequalities do not have a common term. For example, if A > B and C > D, the relation between A and D is undefined.
3. In case of complementary pairs, only the complementary relation is defined. For example, if A ≥ B and B ≥ C, the relation between A and C can only be A ≥ C and not A > C.

Direct Inequalities:

The questions related to direct inequalities only includes basic symbols of inequalities like <, >, =, ≥, etc. In these questions, the candidates are given a set of sentences followed by a set of conclusions. The candidates are required to analyze the conclusion and answer the question accordingly.

Examples of Direct Inequality Reasoning Questions :

Few sentences are given along with two conclusions. Read the statements and check which conclusion is true.

Direct Inequality Question- 1:

Statements:

A > B = C

G < C > D

Conclusions:

A < D

A = G

Answer Options:

1. Only conclusion I holds true.
2. Only conclusion II holds true.
3. Either conclusion I or II holds true.
4. Neither conclusion I nor II holds true.
5. Both conclusions I and II hold true.

Solution:

Option (4)

Reason:

From the given statements, it can be concluded that A must be greater than D as D < C < A.

Also, as C > G, A can never be equal to G.

Direct Inequality Question- 2:

Statement:

• I ≥ U > T ≤ R
• U ≥ V = W > C

Conclusions:

• W > T
• C > I

Answer Options:

1. Only conclusion I holds true.
2. Only conclusion II holds true.
3. Either conclusion I or II holds true.
4. Neither conclusion I nor II holds true.
5. Both conclusions I and II hold true.

Solution:

Option (4)

Conclusion 1 does not holds true as T < U while U ≥ W. Similarly, conclusion 2 also does not hold since I ≥ U and U ≥ C.

Coded Inequalities:

The questions related to coded inequalities involve codes like @, #, &, etc. to define a specific relation between different entities. So, in these type of questions, the candidates need to analyze (or write down) the coded relations in standard form and then check the conclusions to answer the question. They are also called symbol operations and are repeated often in CAT exams.

Examples of Coded Inequality Reasoning Questions :

Few sentences are given along with two conclusions. Read the statements and check which conclusion is true.

Coded Inequality Question- 1:

In the given statements, the relations are represented as follows:

• ‘P # Q‘ means ‘P is not smaller than Q’.
• ‘P $ Q‘ means ‘P is neither smaller than nor equal to Q’.
• ‘P % Q‘ means ‘P is neither greater than nor smaller than Q’.
• ‘P ^ Q’ means ‘P is not greater than Q’.
• ‘P & Q’ means ‘P is neither greater than nor equal to Q’.

Statements:

• X # Y
• Y $ Z
• A & Z

Conclusions:

• Z & X
• A $ Y

Answer Options:

1. Only conclusion I holds true.
2. Only conclusion II holds true.
3. Either conclusion I or II holds true.
4. Neither conclusion I nor II holds true.
5. Both conclusions I and II hold true.

Solution:

Option (1)

At first, the code must be interpreted in the general form. Here the symbols can be interpreted as :

• # means ≥
• $ means > and
• & means <

Representing the statements in general form gives:

• X ≥ Y
• Y > Z
• A < Z

Similarly, the conclusions can be interpreted as, Z < X and A > Y.

Thus, from the given conclusions, Z < X holds true. But as A < Z and Z < Y, the conclusion A > Y is false. So, only the first conclusion follows.