In this chapter we are going to discuss the following topics with examples.

Introduction

Mathematical Inequality

Tips to solve mathematical inequality

Examples

### Introduction

In mathematics, an inequality is a relation that holds between two values when they are different.

## The worst form of inequality is to try to make unequal things equal.

Aristotle

### Tips to solve inequality

1.   A > B ≥ C → A > C

2.   A ≥ B > C → A > C

3.   A > B = C → A > C

4.   A = B > C → A > C

5.   A < B ≤ C = D → A < D and B ≤ D

6.   A < B ≤ C > D = E → A < C and C > E

In this case, the relations between AD, AE, BD and BE cannot be established.

For e.g. A < C and C > D so we get A < C > D. That means C is greater than both A and D. But we don’t know which is greater – A or D; or if they are both equal. Thus the relation between A and D cannot be established.

7.    A > B ≤ C !≥ D ≤ E → A > B ≤ C < D ≤ E
→B < E, C < E, B < D.
But the relations between AC, AD, and AE cannot be established.

8.    A < B = C < D > E, C > P < F
→ A < D, A < C, B <D, B > P, D > P
Relations between AE BE, CE, AP, AF, BF, CF, DF, EP and EF cannot be established.

9.     A !≤ B > C = D ≥ E, M ≥ B !<T → A > B > C = D ≥ E, M ≥ B ≥ T
→ A > C, A > D, A > E, B > D, B > E, C ≥ E , A > T, M > C, M > D, M > E
Relations between AM, CT, DT, ET cannot be established.

### Examples

1.  Statement: A≤B=C>D≥E
Conclusion: I) B> E      II) A≤D

a )If only conclusion I follows.
b )If only conclusion II follows.
c )If either conclusion I or conclusion II follows.
d )If neither conclusion I nor conclusion II follows.
e )If both conclusions I and II follow.

2. Statement:A>B>C, D≥E≥B
Conclusion: I) C = E     II)A≥E

a )If only conclusion I follows.
b )If only conclusion II follows.
c )If either conclusion I or conclusion II follows.
d )If neither conclusion I nor conclusion II follows.
e )If both conclusions I and II follow.

Answer -d)Neither conclusion I nor II follow

3. Statement:A>B=C≤E≤F≤G
Conclusion: I) G>A     II)A = G

a )If only conclusion I follows.
b )If only conclusion II follows.
c )If either conclusion I or conclusion II follows.
d )If neither conclusion I nor conclusion II follows.
e )If both conclusions I and II follow.

Answer -c )If either conclusion I or conclusion II follows.
Explanation:We cannot compare A and G

Directions :(Q4 to Q6)

‘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 equal to Q’
‘P & Q’ means ‘P is neither greater than nor smaller than Q’
‘P * Q’ means ‘P is not greater than Q’

4.Statements:N & B, B \$ W, W # H, H * M

Conclusions:
I.M @ W           II.H @ N            III.W & N         IV.W # N

a)Only I is true

b)Only III is true

c)Only IV is true

d)Only either III or IV are true

e)Only either III or IV and I are true

Answer :e)Only either III or IV and I are true

5. Statements:H @ T, T # F, F& E, E * V

Conclusions:
I.V \$ F      II.E @ T      III.H @ V       IV.T # V

a) Only I II and III are true

b) Only I II and IV are true

c) Only II III and IV are true

d) Only I III and IV are true

e) All I II III and IV are true

Answer : b) Only I II and IV are true

6.Statements:N & B, B \$ W, W # H, H * M

Conclusions:
I.M @ W           II.H @ N            III.W & N         IV.W # N

a)Only I is true

b)Only III is true

c)Only IV is true

d)Only either III or IV are true

e)Only either III or IV and I are true

Answer :e)Only either III or IV and I are true

Thus in this post we learned about the concepts of mathematical inequality and tips to find the relations between two variables with examples.

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