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.

### Mathematical Inequality (Positive)

### Mathematical Inequality (Negative)

## 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.

Answer -a)Only conclusion I true

**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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