BASIC CONCEPT OF INEQUALITY

Inequality

In this chapter, we are going to understand the problems related to this important topic. Firstly, we will have to understand what an inequality term means.

We are also aware of this term in mathematics.

An inequality is a mathematical expression that shows the relationship between two values that are not equal, using symbols to indicate whether one value is greater than, less than, greater than or equal to, or less than or equal to the other. The most common inequality symbols are:

(a) < = Less than
(b) > = Greater than
(c) = = Equal to
(d) ≤ = Less than or equal to
(e) ≥ = Greater than or equal to

Example:

A is less than B → A < B
A is greater than B → A > B
A is equal to B → A = B
A is greater than or equal to B → A ≥ B
A is less than or equal to B → A ≤ B

Combining Inequalities

If A > B and B > C then we can say A > B > C → combined form

So, A ≥ B ≥ C → combined form

If A ≥ B and B ≥ C

Solved Example:

(i) C ≥ D > A

We can say C > A
We can say C ≥ A

(ii) X ≤ Y ≤ Z

We can say X ≤ Z

(iii) a < b ≤ c

We can say a < c

(iv) A > B ≥ C

We can say A > C

(v) X ≥ Y ≥ Z

We can say X ≥ Z

Note: If the direction of the sign changes, then no relationship can be established.

Example:

Similarly, if X > Y ≥ Z ≥ P = Q, then the relationship between X and Z can be inferred.

X & Z → no relation (> <)
X & P → no relation (> ≥)
Y & P → no relation (> ≥)
Y & Q → no relation (≥ =)
Z & Q → no relation (≥ =)

Either or Case:

If X and Y, then conclusion:

(a) X > Y is false (Because X < Y, not X > Y)
(b) X = Y is false (Because X ≥ Y, not X = Y)

Complicated Case of Either or:

Statement: K < L ≤ M = D

Find the relationship between K and D.

Conclusion:
(i) K < D
(ii) K ≥ D

In the above statement, we cannot find a relation between K and D, but there are three possibilities for the relationship between K and D.

(i) K > D
(ii) K = D
(iii) K < D

So, this case is ‘Either or’.

Basically, we can say when we have all three signs that are ‘>’, ‘<‘ and ‘=’ present in the conclusion with two terms only.

K ≥ D or K < D

Some Examples of Direct Inequalities

Statement 1

X = Y ≥ Z < P = Q ≤ R

Conclusion:
(i) R > Y
(ii) Y ≥ P

Solution:

When we compare R & Y, we can see the direction of the sign is opposite. As well as between Y & P.

So, both conclusions do not follow.

Statement 2

A > B > C, C > D > E

Conclusion:
(i) E < A
(ii) B ≥ D

Solution:

If we combine both statements, we get A > B > C > D > E. Then, conclusion (i) E < A follows.

Conclusion (ii) B ≥ D

We clearly see there is not any equal to sign, so it does not follow.

Statement 3

A > B, B ≥ C, C = D, D < E

Conclusion:
(i) B > D
(ii) B = D

Solution:

Combining the above statements, we get A > B ≥ C = D < E.

We get B ≥ D.

It means B is greater than or equal to D, so conclusions (i) and (ii) both do not follow, but it is a case of ‘either-or’.

Coded Inequalities

In this type of inequality, some symbols are used to define inequality with different meanings as follows:

‘A © B’ means ‘A’ is smaller than ‘B’.

‘A # B’ means ‘A’ is greater than ‘B’.

‘A % B’ means ‘A’ is either smaller than or equal to ‘B’.

‘A $ B’ means ‘A’ is either greater than or equal to ‘B’.

‘A @ B’ means ‘A’ is neither smaller nor greater than ‘B’.

Questions

1.

Statement: V#S, S©L, L©J

Conclusion:
I. V@L
II. S©J

2.

Statement: M#R, R©J, J#H

Conclusion:
I. M#H
II. R©H

3.

Statement: M$F, F@G, G%M

Conclusion:
I. F$H
II. F@M

4.

Statement: T#L, L%W, W@V

Conclusion:
I. L%V
II. T%W

5.

Statement: K%L, L%T, I$T

Conclusion:
I. L%I
II. L#I

Solutions

Symbol meanings:

= >

© = <
@ = =
% = ≤
$ = ≥

Or:

© = <

= >

% = ≤
$ = ≥
@ = =

Answers:

1. V > S < L < J
   (i) False
   (ii) True

2. M > R < J > H
   (i) False
   (ii) False

3. H ≥ F = G ≤ M
   (i) False
   (ii) False

4. T > L ≤ W = V
   (i) True
   (ii) False

5. K ≤ L ≤ I ≥ T
   (i) True
   (ii) False

Some Common Examples Asked in Mains Exam

Directions (1–3): In the following questions, the symbols @, #, $, % and & are used with the following meanings as illustrated below.

‘M@N’ means ‘N is smaller than or equal to M’.

‘M#N’ means ‘N is greater than or equal to M’.

‘M$N’ means ‘N is equal to M’.

‘M%N’ means ‘N is greater than M’.

‘M&N’ means ‘N is smaller than M’.

Now, in each of the following questions, assuming the given statements to be true, find which of the three conclusions I, II and III given below them is/are definitely true and give your answer accordingly.

Example 1

Statements:
U&V @ W%X;
P#W $ Q;
R&X

Conclusions:
I. P%U
II. R&W
III. Q&X

(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Both III and II are true
(e) All are true

Solution (c):

I. P%U (True)
II. R&W (True)
III. Q&X (False)

Example 2

Statements:
A#B$C&D;
E%B@F;
G&D&X

Conclusions:
I. F#A
II. G&B
III. B&X

(a) Only III is true
(b) Only II is true
(c) Only I is true
(d) Both I and II are true
(e) All are true

Solution (a):

I. F#A (False)
II. G&B (False)
III. B&X (True)

Example 3

Statements:
F$G#S&T;
Q@S#R;
B$T

Conclusions:
I. F#Q
II. G%R
III. B%G

(a) Only I is true
(b) Only II is true
(c) Only III is true
(d) Both I and II are true
(e) All are true

Solution (a):

I. F#Q (True)
II. G%R (False)
III. B%G (False)

Directions (4–6)

In the following questions, the symbols @, &, $, and © are used with the following meanings as illustrated below.

‘P&Q’ means ‘Q is smaller than or equal to P’.

‘P$Q’ means ‘Q is greater than or equal to P’.

‘PµQ’ means ‘Q is equal to P’.

‘P@Q’ means ‘Q is greater than P’.

‘P©Q’ means ‘Q is smaller than P’.

Now, in each of the following questions, assuming the given statements to be true, find which of the three conclusions I, II and III given below them is/are definitely true and give your answer accordingly.

Example 4

Statements:
A©B&CµD;
F&CµE;
G&D

Conclusions:
I. B&E
II. G$B
III. E$G

(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Both I and III are true
(e) All are true

Solution (d):

I. B&E (True)
II. G$B (False)
III. E$G (True)

Example 5

Statements:
I&J©K;
L@KµM;
N©J

Conclusions:
I. N©K
II. I©L
III. L@J

(a) Only I is true
(b) Only II is true
(c) Only III is true
(d) Both I and II are true
(e) All are true

Solution (e):

I. N©K (True)
II. I©L (True)
III. L@J (True)

Example 6

Statements:
U©V&W@X;
Y@VµZ;
X©Y

Conclusions:
I. Y@W
II. W©Y
III. U©W

(a) Only I is true
(b) Only II is true
(c) Only III is true
(d) Both I and II are true
(e) All are true

Solution (c):

I. Y@W (False)
II. W©Y (False)
III. U©W (True)

Points to Remember

(1) In the chapter of Inequalities, you should give proper attention to the signs of inequalities.

(2) Be careful about the “either-or” case, which has been clearly described in the concept part.

(3) In the case of coded inequalities, you should read the directions and decode them properly.

(4) Use the tabular form for coded inequalities, which will help you to solve questions in less time.

(5) Relate the conclusions from the given statements carefully.

Practice MCQs for BANKING/SSC/RAILWAY

In this question, the relationship between different elements is shown in the statement. The statement is followed by two conclusions. Study the conclusions based on the given statement and select the appropriate answer.

(a) If only conclusion I is true
(b) If only conclusion II is true
(c) If either conclusion I or II is true
(d) If neither conclusion I nor II is true
(e) If both conclusions I and II are true

1. Statement:

A > F ≥ C < W < E < H ≥ Z

Conclusions:
I. A ≤ W
II. W < A

2. Statement:

M ≤ L < X < P > R > S < N ≥ U

Conclusions:
I. P ≥ M
II. P > S

3. Statement:

P > O = R ≤ E < K > L ≤ U < A

Conclusions:
I. A < R
II. O ≤ A

4. Statement:

U < T ≥ R < M ≤ P < A > Q < W

Conclusions:
I. M < W
II. Q ≤ U

5. Statement:

N < H ≤ O > P ≤ W = Q < V

Conclusions:
I. N > Q
II. N ≤ W

6. Statement:

H ≥ F = L > C < A < P ≤ Z > R < K

Conclusions:
I. Z > C
II. H < A

7. Statement:

X > B ≥ M = T < L < C = D < A

Conclusions:
I. B < C
II. D > T

8. Statement:

A ≤ C < D > F < I < K < M > O

Conclusions:
I. A > F
II. I < M

9. Statement:

M ≥ H > R = Y > W ≥ D < X = A > Z

Conclusions:
I. R > A
II. Y ≤ X

10. Statement:

O > W = T ≥ I < Y = C ≤ S

Conclusions:
I. W < C
II. S > I

11. Statement:

M < N ≥ O = P > Q > R ≤ S

Conclusions:
I. M ≥ P
II. M < P

12. Statement:

W ≥ S > P < R ≥ T = Q ≤ U < V

Conclusions:
I. S > Q
II. S ≤ T

13. Statement:

C ≥ V > K > L < W ≤ O > P = F < S

Conclusions:
I. V > O
II. P ≤ C

14. Statement:

X > Y > K > L < M ≤ N < O

Conclusions:
I. Y > M
II. K < O

15. Statement:

A > E > F < G < I = K > M > H

Conclusions:
I. E < M
II. G ≥ H

16. Statement:

U > I ≤ M = P > L > E ≤ W < K

Conclusions:
I. M > K
II. K ≤ P

17. Statement:

M > P = O ≤ R > T > E ≤ W = S

Conclusions:
I. P > W
II. S ≥ O

18. Statement:

A < B ≤ T > U ≥ N > J < H ≥ F

Conclusions:
I. A < J
II. F ≥ B

19. Statement:

T > G ≥ H > Y < E = J ≥ I

Conclusions:
I. E > T
II. T ≥ J

20. Statement:

U ≤ P = O > R ≤ T > E < W ≥ A = M

Conclusions:
I. P ≥ M
II. O < A

21. Statement:

$ < ¥ < ₹ ≥ P = € > £ > £

Conclusions:
I. $ < €
II. ₹ > £

22. Statement:

Russia > America > India < Bangladesh ≥ Canada = Mexico

Conclusions:
I. America < Mexico
II. America ≥ Canada

23. Statement:

75 < 61 < 59 = 67 < 12 > 35 ≤ 19 ≤ 21

Conclusions:
I. 75 < 67
II. 75 = 12

24. Statement:

4 > 8 ≥ 9 > 3 ≤ 2 < 5 > 6

Conclusions:
I. 4 > 5
II. 6 > 8

25. Statement:

5 < 3 ≤ 6 > 9 < 1 ≤ 4 > 7 = 8

Conclusions:
I. 3 > 7
II. 3 ≤ 8

26. Statement:

7 > 8 ≥ 5 < 4 < 3 ≥ 9 = 2

Conclusions:
I. 7 > 2
II. 9 ≥ 7

27. Statement:

8 < 7 ≤ 4 > 5 > 2 ≤ 3 ≤ 1

Conclusions:
I. 8 < 2
II. 1 > 7

28. Statements:

2 < 7 ≤ 4 < 9 > 0; 6 > 1 ≥ 5 < 7 > 8

Conclusions:
I. 9 > 8
II. 6 < 2

29. Statement:

8 < 7 ≤ 6 > 5 = 3 < 9 ≤ 2 ≤ 4

Conclusions:
I. 3 ≤ 8
II. 8 < 5

30. Statement:

8 > 0 ≥ 3 < 7 ≤ 2 < 9 > 6 = 1

Conclusions:
I. 1 ≤ 3
II. 6 > 3