Solving Inequalities Problems in Reasoning

What is Inequality in Reasoning?
Consider this example: We know that the result of multiplication between 5 and 3 and number 15 are equal. Since they are equal it is an equality.
In the same way, 5 × 5 ? 15. Here the product of 5 and 5 is not equal to the number 15. And since they are not equal, it is an inequality.

Two types of inequalities problems:
Direct inequalities
Coded inequalities

Tips: Learn all the symbols related to inequalities - e.g. greater than >, less than <...

Priority of Symbols in Inequality

1. > ? =
For example: If A>K?M=O
Then, A> M and T>O

2. < ? =
For example: If PThen, P

3. > < (No relation)
For example: If Q>KThen there will be no relation between Q and L.

4. > ? (No relation)
For example: If O>J?H
Then there will be no relation between O and H.

5. < > (No relation)
For example: If FQ
Then there will be no relation between F and Q.

6. < ? (No relation)
For example: If DThen there will be no relation between D and Z.

Either- or case
In equality it is very important condition. Mostly students make mistakes in this condition. For clear concept we are giving example of “either-or”

1st condition for “either-or” is both conclusions should be wrong.
2nd condition is that variables of both conclusions should be same.

Example: Statement: P?Q=R
Conclusion: (a) P > R (b) P = R

In above example, relation between P and R is P?R. But both the conclusions are wrong and both have same variables. And by combining both conclusions you will get the actual relation between A and C which comes from statement.

2. Statement: P=Q?R?S=T
Conclusion I: (a)P>T (b)P=T
From the above statement it is clear that P is either equals to T or P is greater than T ,So individually both the conclusions are wrong but by combining them we will get that P is either greater than or equals to T (P?T).
Conclusion II: (a) Q>S (b) Q=S
Similarly from the above statement for conclusion II we can see that there is an either / or case between Q and S, So Q either will be greater than or equals to S.

Complicated case of “Either-or”
Statement: H?M?V=K
Conclusions: (1) H
In the above statement we cannot find the relation between H and K. There may be three possibilities of relation between H and K.
i.e. (a) H>K (b) HAnd we are getting all possibilities by combining both conclusions. So, this is also one case of “Either-or”.

Statement: FS,M?T
Conclusions: I.M?S II. S>M
In the above question by combining the statements together we get S
 

Statement: L?KF?B
Conclusions: I.LThis is another example showing that no direct relation is found between B and L and all the three possible conditions as L>B, L
There are two types of questions are given in exams. one is direct relation between two objects or elements and another coded form. In coded form relation would be given as symbolic form, you have to solve the problems with the help of those symbolic form.
 

Types of Inequalities:

1. Common sign
1. If only conclusion i is true.
2. If only conclusion ii is true.
3. If either conclusion i or ii is true
4. If neither conclusion i nor ii is true.
5. If both conclusion i and ii are true.

Statement:
P ? D = M > T ? L = O
Conclusion:
i) P ? T
ii) D ? L
Explanation : In conclusion (i) From P to T present ?, = and  > sign , we can ignore = sign and between ? and > common sign is > but in conclusion given ? sign. So, its a False. In Conclusion (ii) also same. So, its False.
 

2. Opposite sign
1. If only conclusion i is true.
2. If only conclusion ii is true.
3. If either conclusion i or ii is true
4. If neither conclusion i nor ii is true.
5. If both conclusion i and ii are true.

Statement:
T > A < M = L


Conclusion:
i) T > M
ii) T < M
Explanation : In case of opposite sign we can’t compare. So, conclusion is always wrong.
 

3. Coded form
A ? B means, A is not smaller than B.
A $ B means, A is neither greater than nor smaller than B.
A * B means, A is neither equal to nor smaller than B.
A £ B means, A is neither greater than to nor equal than B.
A @ B means, A is not greater than B.

Statement:
 

S £ T, M ? T, Q $ N
 

Conclusion:
i) M $ Q (False) M = Q


ii) S? N (False) S ? N


iii) Q * M (False) Q > M
Explanation:
S < T, M ? T, Q = N
M ? T > S, Q = N
Only (i) is true.
Only (ii) is true.
Either (i) or (ii) is true.
Only (iii) is true.
None of these. (Answer)