Important notes on Inequality

Inequality

Meaning of different symbols

Before getting deep into the inequality, let us try to understand the meaning of the basic operations used in equality –
(1) ‘>’ symbol: This symbol indicates that variable on the left side is definitely greater than the variable on the right side of the symbol.
For example: A>B means A is definitely greater than B.
(2) ‘<’ symbol: This symbol indicates that variable on the left is definitely smaller than the variable on the right side of the symbol.
For example: A<B means A is definitely smaller than B.
(3) ‘=’ symbol: This symbol indicates that variable on the left side is equal to the variable on the right side of the symbol.
For example: A=B means A is definitely equal to B.
(4) ‘≥’ symbol: This symbol indicates that variable on the left side is either greater than or equal to the variable on the right side of the symbol.
For example: A≥B means A is either greater than B or equal to B.
(5) ‘≤’ symbol: This symbol indicates that variable on the left side is either smaller than or equal to the variable on the right side of the symbol.
For example: A≤B means A is either smaller than B or equal to B.

Process to solve inequality problems

First of all, we make the outcomes using the given statements.
Outcome(s) is/are equation(s) which we conclude on the basis of given statements. We write the outcomes so that we can easily find out the relationship between the variables asked.

• All >,=,≥ come in a single group of outcome.
• All <,=,≤ come in another group of outcome.

For example:
Statement: A>B=C<D is given
The outcome will be:
A>B=C
B=C<D
Now we can easily use these two outcomes to find out the relationship between the variables.
After getting the outcomes, we use those to check whether the given conclusions follow or not.

Some basic statements and their conclusions

Types of questions

Inequality questions can be asked in two different ways:

1. The relationship between the variables is directly given.
2. The relationship between the variables is not directly given.

Both of these types can be solved by using the same method. All we have to do is analyze the statement given, write the outcomes in least possible groups and then check whether the conclusions follow or not.
Let us understand both of the types in detail one by one –

1. The relationship between the variables is directly given:

Most of the times, we can directly see the relationship between the variables which enables us in writing the outcomes very quickly and we can check whether the given conclusions follow or not on the basis of obtained outcomes.
Let us see few examples on type I.
Direction: In these questions, relationship between different elements is shown in the statements. These statements are followed by two conclusions.
Mark answer:

1. If only conclusion I follows.
2. If only conclusion II follows.
3. If either conclusion I or II follows.
4. If neither conclusion I nor II follows.
5. If both conclusions I and II follow.

Q.1) Statements: J < R, W ≤ R < A, M > R.
Conclusions:
I. J ≥ W
II.  J < M
Conclusion:
Joining given statements to get least possible groups of outcomes –

1. M > R > J
2. A > R ≥ W

We can’t establish relation between J & W because they are in different outcome group. So conclusion I don’t follow.
On the basis of outcome 1, M will always be greater than J. So conclusion II follows.
So only conclusion II follows.
Q.2) Statements: B ≥ Y, H < D ≤B, K = Y
Conclusions:
I. B > K
II. H < K
Solution:
Joining given statements to get outcomes –

1. B ≥ Y = K
2. B ≥ D > H

From outcome 1, we get B ≥ K. Which doesn’t give the surety that B > K (B is either greater than K or equal to K). So conclusion I doesn’t follow.
K and H are in different outcome groups, so we can’t establish any relation between K & H. So conclusion II doesn’t follow.
So neither conclusion I nor II follows.
Q.3) Statements: A>B=C≤D
Conclusions:
I. A>C
II. A>D
Solution:
Finding outcomes –

1. A>B=C
2. D≥C=B

From outcome 1, we can clearly see that A>C. So conclusion I follows.
A & D are not in the same outcome group, so we can’t establish any relationship between A & D. So conclusion II doesn’t follow.
So only conclusion I follows.
Direction for Q.4) What should come at the place of question mark(?) to make both the conclusions true.
Q.4) Statements: A>B=C=D?E=F≥G
Conclusions:
I. D>G
II. B>E
Solution:
Conclusion I gives us the idea that we can use either ‘>’ or ‘=’ at the place of question mark.
But conclusion II eliminates the possibility of ‘=’.
So ‘>’ will come at the place of question mark.

2. The relationship between the variables is not directly given –

Sometimes the relationship between the variables is given in form of symbols and the meaning of each of the symbol is predefined in the given directions. So one extra step is needed to be done in these types of questions i.e. we have to decode the statements given.
Let us see few examples on the same:
Directions (5-7): In the following questions, the symbols @, $, &, % and © are used with meaning as illustrated below:
'P @ Q' means P is neither smaller than nor equal to Q.
'P $ Q' means 'P is not smaller than Q'.
'P & Q' means P is neither greater than nor equal to Q'.
'P © Q' means 'P is not greater than Q'.
'P % Q' means 'P is neither greater than nor smaller than Q'.
Now in each of the following questions, assuming the given statements to be true, find which of the two conclusions I and II given below them is/are definitely true.
Q.5) Statements: R © S @ T, U % T $ V, W & R% Y
Conclusions:
I. R @ U
II. W & R

1. Only conclusion I is true.
2. Only conclusion II is true.
3. Either conclusion I or II is true.
4. Neither conclusion I nor II is true.
5. Both conclusions I and II are true.

@means >       & means <      $ means >       © means <      % means =
Solution:
Decoding the statements:
R © S @ T → R ≤ S > T
U % T $ V → U = T ≥ V
W & R % Y → W < R = Y
Combining statements to get groups of outcomes:
1. S>T=U ≥ V
2. S ≥ R=Y>W
R and U are in different groups. So R and U cannot be compared. So conclusion I (R>U) doesn’t follow.
From outcome 2, R>W. So conclusion II (W<R) follows.
So only conclusion II follows.
Q.6) Statements: K $ M @ O, R © Q & O, G % M
Conclusions:
I. R & O
II. K @ Q

1. Only conclusion I is true.
2. Only conclusion II is true.
3. Either conclusion I or II is true.
4. Neither conclusion I nor II is true.
5. Both conclusions I and II are true.

Solution:
Decoding the statements:
 K $ M @ O → K ≥ M > O
 R © Q & O → R ≤ Q < O
 G % M → G= M
Combining the statements to get the outcomes:
1. K ≥ M = G > O > Q ≥ R
From the outcome, O>R so the conclusion I (R<O) follows.
Similarly, K>Q also follows.
So both the conclusions follow.
Q.7) Statements: A % B & C, D @ E © C, F & G © B
Conclusions:
I. C @ F
II. D $ A

1. Only conclusion I is true.
2. Only conclusion II is true.
3. Either conclusion I or II is true.
4. Neither conclusion I nor II is true.
5. Both conclusions I and II are true.

Solution:
Decoding the statements:
A % B & C→ A = B < C
D @ E © C → D > E ≤ C
F & G © B → F < G ≤ B
Combining the statements to get the outcomes:
1. F < G ≤ A = B < C
2. E ≤ C
3. E < D
From outcome 1, we can clearly see that F < C. So conclusion I follow.
D and A are not in one group of outcome, so we cannot compare D and A. Thus conclusion II doesn’t follow.

Key points related to Inequality:

• First step is to decode the statements (if the problem is on coded inequality)
• Second step is to write the least possible group of outcomes.
• Last step is to check whether the given conclusions follow or not on the basis of outcomes.
• All >, =, ≥ come in a single group of outcome.
• All <, =, ≤ come in another group of outcome.
• If any variable is not in both of the groups then we can’t define the relation between those variables.