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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