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Inequalities

Concept of Inequalities:

Inequality is one of the most important chapters for any competitive exam. In  the prelims and mains exam of any banking and insurance exam, you can expect 3 to 5 marks from this chapter.

Inequality is the comparison between two or more than two variables.

In this topic, we have to compare two or more variables and we have to conclude them.

In general, we use symbols.

Inequalities are of two types.

1.  Direct Inequalities

2.  Coded Inequalities

We use few symbols for inequalities.

(i) Greater than → > → A > B ⇒ A is greater than B.

(ii) Less than →  < → A < B ⇒ A is less than B.

(iii) Equal to → = → A = B ⇒ A is equal to B.

(iv) Greater than or equal to →  ≥  → A  ≥  B ⇒ A is either greater or equal to B.

(v) Less than or equal to →  ≤   → A ≤ B ⇒ A is either less or equal to B.

Now we will see the indirect representation of symbols

(i) A ≱ B means A is neither equal nor greater than B which implies A is less than B.

So, A ≱ B ⇒ A < B

(ii) A ≰ B means ⇒ A > B

(iii) A ≯ B means A is not greater than B which implies A ≤ B

(iv) A ≮ B means ⇒ A ≥ B

(v) A ≠ B means ⇒ A > B or A < B

Combining Inequalities

If A> B and B > C then 

We can conclude A > C

A > B > C → is combining inequalities

If A ≥ B and B ≥ C then

We can conclude that A ≥ C

A ≥ B ≥ C → is combining inequalities.

> and < symbols are definite symbol

≥ and ≤ symbols are possible inequality symbol

Now →

(i)  A > B > C → Here no possibility so we can definitely conclude this → A > C

(ii) A > B ≥ C → Here two possible condition 

A > B = C → Concludes A > C

A > B > C → Concludes A > C

from this we can conclude A > C

(iii) A ≥ B > C → Here two possible conditions

A > B > C → concludes A > C

A = B > C → concludes A > C

from this we can conclude A > C

(iv)  A< B ≤ C → Here two possible condition

A < B < C → concludes A < C

A < B = C → Concludes A < C

So, from this we can conclude A < C

(v) A ≤ B < C → Here two possible conditions are

A < B < C → concludes A < C

A = B < C → concludes A < C

from this we can conclude A < C

(vi) A ≥ B ≥ C → concludes A ≥ C

* If the direction of the sign changes then no relationships can be established

Now Short Trick

# Whenever definite symbols are there in the same direction we will take definite symbols for the relationships

A > B ≥ C ≥ D

Here between A and D, there are symbols (> ≥ ≥) So, we can conclude A > D

#  Whenever possible symbols (≥ ≤) are there with ‘=’ sign we will conclude with the possible symbol.

Like A ≥ B ≥ C = D ≥ E

Between A and E relationships is A ≥ E

Either – OR Case

If A ≥ B then the conclusions

(i) A > B is false because we cannot definitely say that A is greater than B

(ii) A = B is also false because we cannot definitely say that A is equal to B

But we can say that

Either A > B or A = B is definitely true

It is the same as A ≥ B 

Example:

1. A > B ≥ C > D ≤ E = F ≤ G

(i) A > D

(ii) D = G

a) Only (i) follow

b) Only (ii) follow

c) Either (i) or (ii) follow

d) Both (i) nor (ii) follow

e) None of these

Solution: 

Between  A and D symbols are >, ≥, >

So, we can conclude that A > D

Between D and G symbols are ≤, =, ≤

So, we can conclude that D ≤ G

So, from this, we can conclude only (i) follow.

So, option (a) is correct.

2. A > B ≥ C, B ≥ D > E, E > F < G

(i) A > G

(ii) B > F

a) Only (i) follow

b) Only (ii) follow

c) Either (i) or (ii) follow

d) Both (i) nor (ii) follow

e) None of these

Solution:

Between A and G symbols are >, ≥, >, >, <

Opposite directions are there

So, we cannot make a relation between D and G

Between B and F symbols are ≥, >, >

So, we can conclude B > F

So, conclusion (ii) follows.

So, option (b) is correct.

3. G ≤ T ≤ K ≤ L = M, F ≥ D ≥ C = U ≥ M

(i) D > T

(ii) D = T

a) Only (i) follow

b) Only (ii) follow

c) Both (i) or (ii) follow

d) Either (i) nor (ii) follow

e) None follow

Solution:

By combining we get

G ≤ T ≤ K ≤ L = M ≤ U = C ≤ D ≤F

Between D and T relation is;

D ≥ T

It meant either D > T or D = T

So, option (d) is correct.