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Inequalities

Inequality is a phenomenon that we have known or come across for a while now. It simply means something that is not equal in all the sense whether social, economical or mathematical sense. But today we are talking about Inequalities in mathematical terms. Each one of you has seen signs such as “>”, “≠”, “≤” etc. at some point of time or other.

You should keep in mind the priority order while solving these type questions

Priority of Symbols:

1) >  ≥ =

For ex- If    T>P≥Q=R                              Then, T> Q and T>R

2)  <  ≤ =

For ex- If    W<X≤V=Y                              Then, W<Y and W<V

3) >  < (No relation)

For ex- If Q>K<L, Then there will be no relation between Q and L.

4) >  ≤ (No relation)

For ex- If O>J≤H,  Then there will be no relation between O and H.

5) <  > (No relation)

For ex- If F<E>Q ,  Then there will be no relation between F and Q.

6) <  ≥ (No relation)

For ex- If D<S≥Z,     Then there will be no relation between D and Z.

Either- or case:  Inequality it is very important condition. Mostly students make mistakes in this condition. For clear concept we are giving example of “either-or”

1st condition for “either-or” is both conclusions should be wrong.

2nd condition is that variables of both conclusions should be same.

Trick 1: Whenever in a statement you get both the priority 1 in opposite order (A>B<C) there will be a conflict and thus no conclusion.

If A > B <C Then A < C = False & C > A = False.

But     If A > B >C then A > C = True, C < A = True

Example:

Statement:  A < D > C < E > B

Conclusions:

1) C > B   → False

2) A < E   → False

3) D > B   → False

In simple way, whenever these two sign comes in opposite direction the answer will be false.

Trick 2: Whenever in a statement you get both the priority 2 in opposite order (A≥ B ≤C) there will be a conflict and thus no conclusion.

If A ≥ B ≤C Then A ≤ C = False & C   ≥ A = False.

But

If A ≥ B ≥ C then A ≥ C = True, C ≤ A = True.

Example:

Statement: B ≥ D ≤ A ≥ F ≤ C

Conclusions:

1) A ≥ C → False

2) B ≤ F → False

3) D ≥ C → False

Trick 3: When it occurs to you that the statement of order is opposite just change the sign in the alternate direction

If A > B > C > D < E < F

We can say that A > B > F = C < B < A

Types of Questions asked in Inequality

Now a days, inequality based questions are provided in two types

  • Direct Inequality in which direct symbols will be given in the statement.
  • Coded Inequality in which coded symbols (like @, %, $ etc) will be given and they signify will be provided separately.

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.

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

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.

Coded Inequality Question

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:

(a) Only conclusion I holds true.

(b) Only conclusion II holds true.

(c) Either conclusion I or II holds true.

(d) Neither conclusion I nor II holds true.

(e) Both conclusions I and II hold true.

Solution:

Option (a)

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.

With this information, the candidates can easily solve the questions related to inequalities easily and accurately.