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# Inequality

In Inequality problems, directions of these problems will be given. Each problem will have a statement followed by four probable conclusions. We can expect about 5 to 6 questions from the inequality topic. Inequality is the common topic in all competitive exams.  It is one of the topics in which you can gain full marks very easily. Below we are giving the clear explanation about this topic and I am sure that you can solve the five questions in just two minutes.

Symbols like >, < , ≥, ≤ and = are used. Before learning these shortcuts, Let us understand what these symbols actually mean. These symbols are usually learnt in lower classes and are familiar ones.

< – Less than

≤ – Less than or Equal to

> – Greater than

≥ – Greater than or Equal to

= – Equal to

Inequalities Points to remember:-

• Open Gate    >      Close Gate       –             Move from left to right
• Open Gate   >=    Open Gate         –            Move from left to right
• Open Gate         Open Gate        –            Move both direction
• Close Gate   <       Open Gate        –            Move from right to left
• Close Gate   =<     Open Gate        –            Move from right to left

Step 1:  Check which entity has a open gate in conclusion.

Step 2:  Then starts move from the entity which has a open gate to another entity.

Step 3:  If we have close gates between the entities while moving from source to destination in statement, then that conclusion fails.

## Case (I) : Closed Gate

If you have a closed gate in your statement to reach your destination entity then your conclusions will fail. You don’t need to check further.

Eg:- Statement     : A>B>C<D

Conclusion    :  i) A>D   ii) C>A

Here, From the 1st conclusion we can clearly see that “A” has a open gate. Here ‘A’ & ‘D’ are ‘source’ and ‘destination’ entity respectively. Next check the statement is also having the open gate to reach D from A. But, In this case we have closed gate while trying to move from C to D. So, We can say that the 1st conclusion fails by checking the gates alone.

From the 2nd conclusion we can clearly see that “C” has a open gate. Here ‘C’ & ‘A’ are ‘source’ and ‘destination’ entity respectively. Next check the statement is also having the open gate to reach A from C. But, In this case we have closed gate while trying to move from C to B. So, We can say that the 2nd conclusion fails by checking the gates alone.

## Case(Ii) : Open Gate

If you have open gate in the statement to reach your destination entity from source entity then you have to check the following rules:

Rule Box:

From the above tabular coloumn you can easily understand the rules for all the relational symbols. We will discuss all the applicable rules in details are given below:

Inequalities Rule Explanation:

“> – Greater than”

1. If you have this symbol in your conclusion, Then it will satisfies all the symbols but “>” this symbol must present in your statement at least in one instance.
2. In this case also “=”,“>=” symbol follows the condition but “>” this symbol must present in your statement at least in one instance between your entities given in the conclusion.

Eg:-   Statement:  A>=B>=C;F=E=B>D

Conclusion:

i) A>C       False

ii) A>E      False

iii) B>F      False

iv) A>D      True

“Greater than or Equals  >=”

1. If you have this symbol in your conclusion, then it only satisfies these two symbols “>=”,”=” which is shown in the above tabular column.
2. In this case, “=” symbol follows the condition but “>=” this symbol must present in your statement at least in one instance between your entities given in the conclusion.

Eg:-  Statement: A>=B>=C;F<B=E=G

Conclusion:

i)   A>=F    False

ii) A>=E     True

iii)  B>=G   False

iv) A>=C     True

“Equal to (=)”

1. If you have this symbol in your conclusion, Then it only satisfies the “=” symbol which is  present in your statement between the entities in the given conclusion.
2. If rest of the symbols are present in between your entities then your conclusion will fails.

Eg:-   Statement:  A=B>=C;E=B>F

Conclusion:

i) A=C         False

ii) A=F        False

iii) A=E        True

Now we are clear about the meanings of different symbols. Check the below table for relationships gives Conclusions for the Statements.

 S.No Statement Conclusion 1. A > B > C A > C 2. A > B ≥ C 3. A ≥ B > C 4. A = B > C 5. A > B = C 6. A C or A=C) 12. A = B ≥ C 13. A ≥ B = C 14. A ≤ B ≤ C A ≤ C (Either A < C or A = C) 15. A = B ≤ C 16. A ≤ B = C 17. A < B > C Either 1 or 2 follows if any of the following cases (a, b, c and d) are given as they form a complementary pair. a) 1. A > C         2. A ≤ C b) 1. A ≥ C         2. A < C c) 1. A < C         2. A ≥ C d) 1. A ≤ C         2. A > C 18. A ≤  B> C 19. A < B≥ C 20. A > B < C 21. A > B ≤ C 22. A ≥ B < C

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Inequalities Rule Explanation:

“> – Greater than”

1. If you have this symbol in your conclusion, Then it will satisfies all the symbols but “>” this symbol must present in your statement at least in one instance.
2. In this case also “=”,“>=” symbol follows the condition but “>” this symbol must present in your statement at least in one instance between your entities given in the conclusion.

Eg:-   Statement:  A>=B>=C;F=E=B>D

Conclusion:

1. i) A>C       False
2. ii) A>E      False

iii) B>F      False

1. iv) A>D      True

“Greater than or Equals  >=”

1. If you have this symbol in your conclusion, then it only satisfies these two symbols “>=”,”=” which is shown in the above tabular column.
2. In this case, “=” symbol follows the condition but “>=” this symbol must present in your statement at least in one instance between your entities given in the conclusion.

Eg:-  Statement: A>=B>=C;F<B=E=G

Conclusion:

1. i)   A>=F    False
2. ii) A>=E     True

iii)  B>=G   False

1. iv) A>=C     True

“Equal to (=)”

1. If you have this symbol in your conclusion, Then it only satisfies the “=” symbol which is  present in your statement between the entities in the given conclusion.
2. If rest of the symbols are present in between your entities then your conclusion will fails.

Eg:-   Statement:  A=B>=C;E=B>F

Conclusion:

1. i) A=C         False
2. ii) A=F        False

iii) A=E        True

Directions

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

Now our first work is to find what does the symbol means indirectly?

Lets solve it one by one.

P is not smaller than Q. So P can be either equal to Q or Greater than Q. Symbol is ≥.

P is neither smaller than nor equal to Q. So P is Greater than Q. Symbol is >.

P is neither greater than nor equal to Q. So P is lesser than Q. Symbol is <.

P is neither greater than nor smaller than Q. So P is equal to Q. Symbol is =.

P is not greater than Q. So P can be either equal to or lesser than Q. Symbol is ≤.

Now form a small table before solving this kind of problems.

Now we decoded the symbols and we clearly know what is the meaning of each symbol. Make sure that you form a table as above. The symbols >, ≥, and = should be in one row and the symbols <, ≤, and = should be in another row. Don’t memorize the table. Usually Symbols change. You have to decode it correctly and form a table.

Note:

———-> Forward Direction.

<———- Reverse Direction.

In Row 1, First Priority should go to >. Next ≥. Last =. Similarly,

In Row 2, First Priority should go to <. Next ≤. Last =.

Just remember these lines. You will understand it while solving problems.

Never make a mistake in forming this table. If you make a mistake then you loose all the 5 or 6 marks.

We will try to solve some problems based on the above directions.

Example 1

Statements: N ∂ B, B \$ W, W # H, H © M

Conclusions: (1) M @ W.  (2)  H @ N.  (3) W ∂ N.

Step 1 – Make a single statement.

N ∂ B \$ W # H © M

Step 2 – Analyze the conclusions one by one. Always compare with the Modified Statement.

1. M @ W

• If you draw a line from M to W, you will get a Reverse Line.
• Between M and W two symbols are there. One is © and other is #.  Both are in row 2. Highest Priority is #. Since the letters form a reverse line, we should note the symbol which is exactly above #. The symbol above to # is @. So M @ W is TRUE.

2. H @ N

• If you draw a line from H to N, you will get a Reverse line.
• Between H and N, the symbols are #, \$, and ∂.
• If you check these symbols with the table, # is in Row 2 and \$ is in Row 1. So Conclusion 2 is FALSE.

3. W @ N

• If you draw a line from W to N, you will get a Reverse line.
• Between W and N, the symbols are \$ and ∂. Between \$ and ∂, the higher priority goes to \$.
• The conclusion formed a reverse line. So we should note the symbol which is opposite to \$. That is ©. But given conclusion is W @ N. So it is FALSE

So Conclusion one alone Follows

Example 2

Statements – R © D, D \$ J, J # M, M @ K

Conclusions – 1. K # J.   2. D @ M   3. R # M.   4. D @ K

Step 1 – Modify the statement.

R © D \$ J # M @ K

Table

Step – 2 – Analyze the conclusion one by one.

1. K # J

• Between K and J – Reverse Line. Symbols are – @ and #.
• In table @ and # are in different rows.
• Conclusion 1 – FALSE

2. D @ M

• Between D and M – Forward Line. Symbols are \$ and #.
• In table \$ and # are in different rows.
• Conclusion 2 – False.

3. R # M

• Forward line. Symbols are ©, \$, and #.
• \$ and # are in different rows.
• Conclusion 3 – False.

4. D @ K

• Forward Line. Symbols are \$, # and @.
• Symbols are in different rows.
• Conclusion 4 – False.

NOTE: When more than one conclusion in false, check for merging concept. If the characters are same and both the statements are false, and while merging, if we get a meaningful symbol, then the statements can be merged. (To learn clearly about merging, check Syllogism Made Easy).

In the above problem, all the characters of the statements are different. So we can not merge it. So None is True.

Example 3

Statements: M \$ K, K @ N, N © R, R # W

Conclusions: 1) W @ K  2) M \$ R  3) K @ W  4) M @ N

Step 1: M \$ K @ N © R # W

(You don’t need to draw table again and again. I am pasting it for your convenience.)

Analyzing Conclusions

1) W @ K – Reverse line. Symbols are #, ©, and @. In table # and @ are in different Rows. False.

2) M \$ R – Forward Line. Symbols are \$, @, ©. They are in different rows, False.

3) K @ W – Forward Line. Symbols are @, ©, #. They are in different rows, False.

4) M @ N – Forward line. Symbols are \$ and @. Highest priority is for @. So M @ N is True.

Only Conclusion 4 is True.

Example 4

Statement: H @ T, T # F, F ∂ E, E © V.

Conclusions: 1) V \$ F.   2) E @ T.   3) H @ V.   4) T # V

Modified Statement – H @ T # F ∂ E © V

Conclusions

1) V \$ F – Reverse Line – Symbols are © and ∂. Both are in Row 2. Since it forms a reverse line, and highest priority is for ©, we should mark the symbol which is exactly opposite to ©. So the correct symbol is \$. Conclusion 1 is True.

2) E @ T – Reverse Line. Symbols are ∂ and #. Both are in Row 2. High priority is #. Symbol opposite to # is @. So E @ T is True.

3) H @ V – Forward line. Symbols are in different rows. So False.

4) T # V – Forward line. Symbols are #, ∂, and ©. All are in Row 2. High Priority is for #. So T # V is True.

Answer – Conclusion 1, 2, and 4 are True.

## Example of Coded Inequality in Reasoning

Directions: In the following questions, the symbols δ, @, ©, % and ⋆ are used with the following meaning as illustrated below.

‘A © B’ means ‘A is not smaller than B’.
‘A % B’ means ‘A is neither smaller than nor equal to B ’.
‘A ⋆ B’ means ‘A is neither greater than nor equal to B’.
‘A δ B’ means ‘A is not greater than B’.
‘A @ B’ means ‘A is neither greater than nor smaller than B’.

Now in each of the following questions assuming the given statements to be true, find which of the four conclusions I, II, II and IV given below them is / are definitely true and give your answer accordingly.

Statements:
P δ T, T @ R, R © O, O % K

Conclusions:
I. R @ P
II. R % P
III. K ⋆ T
IV. O δ T

1) Only either I or II is true
2) Only III and IV are true
3) Only either I or II and III are true
4) Only either I or II and IV are true
5) Only either I or II and III and IV are true

Follow the steps given below to simplify the process.

## Steps Involved in Solving Coded Inequality in Reasoning

Step 1:  Make Decoding Table.

The easiest method is to first make a table as shown below.

NOTE: Elements used in question are A and B so we have added A and B in table.

TIP: Sometimes, to make questions more complicated, reverse relations may be given as:
‘A * B’ means ‘B is not smaller than A’.
So here we will write B in the first row and A in the last row.

Step 2:
Add Symbols to Table

Step 3: Start decoding symbols one by one. Then add decoded operator into the table.
Here symbols are:
© → not smaller than → means greater than or equal to → ‘≥’
% → neither smaller than nor equal to → means greater than → ‘>’
⋆ → neither greater than nor equal to → means smaller than → ‘<’
δ → not greater than → means smaller than or equal to → ‘≤’
@ → neither greater than nor smaller than → means equal to → ‘=’

So our decoding table becomes:

We will now use this decoding table to solve the actual questions.

Step 4: Decode Statements using Decoding Table.

Statements: P δ T, T @ R, R © O, O % K
Decoded statements: P ≤ T, T = R, R ≥ O, O > K

Step 5: Combine Decoded Statements
Combined statement will be: P ≤ T = R ≥ O > K

Step 6:
Conclude Individually
Look at conclusions one by one, decode each conclusion using the Decoding Table. Then check whether the conclusion follows or not.

Conclusion I: R @ P → R = P
Now from the combined statement we get, P ≤ T = R.
According to priority level we get, P ≤ R.
Thus R = P is false.

Conclusion II:  R % P → R > P
From the combined statement we get, P ≤ T = R.
Thus again we get P ≤ R.
So R > P is false.

But we know from the combined statement that P ≤ R. Hence either conclusion I or II has to be true as they form complementary pair.

TIP: Whenever there are 2 conclusions having the same 2 elements and both conclusions are false, always check for complementary pair.
Complementary means combining in such a way as to enhance or emphasize the qualities of each other or another.
Like, ‘<’, ‘>’ and ‘=’ form complementary pairs for any case. This is because there cannot be any relation other than these if these three cases are included.

Conclusion III
:  K ⋆ T → K < T
Now from the combined statement we get T = R ≥ O > K.
This can be shortened as T ≥ O > K.
Now according to priority, > has more priority than ≥. So the final relation between T and K will be T > K.
Thus the conclusion is true.

Conclusion IV: O δ T → O ≤ T
Now from the combined statement we get, T = R ≥ O.
According to priority, ≥ has more priority than =. So the final relation between T and O will be T ≥ O.
Thus the conclusion is true.

So our final answer will be, conclusion III, IV and either conclusion I or conclusion II follows.

# Practice Set for Inequality

Directions (1 – 5): In the following questions, the symbols %, @, #, \$ and & are used with the following meaning as illustrated below:
‘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 not greater than Q’.
‘P\$Q’ means ‘P is not smaller than Q’.
‘P&Q’ means ‘P is neither smaller than nor greater than Q’.
Now in each of the following questions, assuming the given statements to be true, find which of the four conclusions I, II, and III given below them is/are definitely true and give your answer accordingly.

1. Statements:  Q%C, C\$W, W&D, D@X
Conclusions:
I. C%D
II. W@Q
III. X%W
A) Only I
B) Only II
C) Both II and III
D)AllI, II and III
E) Only I and II

View Answer
Option C
Q > C ≥ W = D < X, so C ≥ D, Q> W and W < X
2. Statements: S&W, W#Q, Q%X, X\$V
Conclusions:
I. Q&S
II. X%W
III. V@Q
A) Only II
B) Only I
C)Only I and III
D) Only III
E) None of these

View Answer
Option D
S = W ≤ Q > X ≥ V, so S ≤ Q, no relation between X and W, Q> V
3. Statements:  A@W, W\$E, E#S, S&J
Conclusions:
I. A\$E
II. J&W
III. A@E
A) Either I or III
B) Only I
C) Only II and III
D) All I, II and III
E) Either I or III and II

View Answer
Option A
A < W ≥ E ≤ S = J, so no relation between W and J and no relation between E and A
In I and III we see subject predicate same and since no relation and I is A ≥ E and III is A < E so either or
4. Statements: A#B, X%B, R\$X, R@S
Conclusions:
I. R%B
II. R\$B
III. A@S
A)Both I and II
B) Both I and III
C) Both II and III
D) Either I or II and III
E) Only I

View Answer
Option B
A ≤ B < X ≤ R < S, so R > B, A< S
5. Statements:  X\$D, D%W, W&F, F@A
Conclusions:
I. A@W
II. X%F
III. D&A
A) Only III
B)Both I and II
C) All I, II and III
D) Both II and III
E) None of these

View Answer
Option E
X ≥ D > W = F< A, so A > W, X > F and no relation between D and A

Direction (6-10): Relationship between different elements is shown in the statements. Find if the conclusions also follow or not.

6. Statements: W > A ≤ G, S ≤ A, I ≤ B = S
Conclusions:
G ≥ I
G = I
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option A
G ≥ A ≥ S ≥ I.
So G ≥ I
7. Statements: Q< W ≤ B, C> B < K, W ≥ A
Conclusions:
I. K > W
II. C> A
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option E
K > B ≥ W , so K > W
C > B ≥ W ≥ Aso C > A
8. Statements: A > E ≥ F, P = E, S > A
Conclusions:
S> P
F < S
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option E
S > A > E = P, so S> P
F ≤ E < A < S, so F < S
9. Statements: S ≥ W > K, W > F < A, P > S
Conclusions:
I. P >A
II. S> F
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option B
P > S ≥ W > F < A, so relation cant be determined between P and A
S ≥ W > F, so S> F
10. Statements: R ≥ X ≥ S, S< C, C > W
Conclusions:
W> X
X ≥ W
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option C
W < C> S ≤ X, so relation cant be determined between W and X, but is for sure that either W is greater than X or is less than equal to X, so either or between the two

• Statements: A ≥ B; C > G; A ≥ H; B ≥ C; I = B
Conclusions:
I. C > H
II. H > B
III. B > G
IV. I > A
A. Only I is true
B. Only II is true
C. Either I or II true
D. Neither I nor II is true
E. Only III is true

Answer & Explanation
Answer – E. Only III is true
• Statements: A=B; C≤F; G>C; B<F
Conclusions:
I. F < B
II. F > G
III. A > G
IV. A > C
A. Only II is true
B. None is true
C. Only I and II are true
D. Only II and III are true
E. Only IV is true

Answer & Explanation
Answer – B. None is true
• Statements: A≤H, G≥H; G>M; O≤M
Conclusions:
I. G≥A
II. G≥O
III. H>M
IV. H≤G
A. Only I, II and III are true
B. Only II is true
C. Only IV is true
D. Only I and IV are true
E. None is true

Answer & Explanation
Answer – D. Only I and IV are true
• Statement : P ≥ Q > R < S ≤ T
Conclusion:
I. T > R
II. T > Q
III. R < P
IV. Q > P
A. Only I is true
B. Only II is true
C. Only I and III are true
D. Only I and IV are true
E. All I, II, III and IV are true

Answer & Explanation
Answer – C. Only I and III are true
• Statement : P < Q ≥ R > S ≤ T
Conclusion:
I. T ≥ R
II. P < R
III. Q > S
IV. S < P
A. Only I is true
B. Only III is true
C. Only II is true
D. Only IV is true
E. Both I and II are true

Answer & Explanation
Answer – B. Only III is true
• Statements: M<N≤O=P; N = Q Conclusions: i. P>M, ii. Q≤O
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – E.Both I and II are true
• Statements: A>N=I≥W<O≤P; O>S; T<N
Conclusions: i. P>S, ii. A<W
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – A.Only I is true
• Statements: M≥N≥O>P≤Q≤R
Conclusions: i. M>Q, ii. N≤R
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – D.Neither I nor II is true
• Statements: M<P≤G = F≥B; H≥F<I; J≥P
Conclusions: P≤B, M<H
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – B.Only II is true
• Statements: M<P≤G = F≥B; H≥F<I; J≥P
Conclusions: H≥J, B<I
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – B.Only II is true
• Direction(1-5): Study the following information to answer the given questions
A\$B means A is not smaller than B
A@B means A is neither smaller than nor equal to B
A#B means A is neither greater than nor equal to B
A&B means A is neither greater than nor smaller than B
A*B means A is not greater than B

1. Statements: O & A, A \$ R, R # S, S * Q
Conclusions:
I. Q @ R
II. S @ O
III. R & O
IV. R # O

A. Only I is true
B. Only III is true
C. Only IV is true
D. Either III or IV is true
E. Either III or IV and I are true

Answer
Answer – E. Either III or IV and I are true
2. Statements: A * E, E \$ F, F # O, O @ L
Conclusions:
I. L # F
II. E @ O
III. A # O
IV. E @ L

A. None is true
B. Only I is true
C. Only II is true
D. Only III is true
E. Only IV is true

Answer
Answer – A. None is true
3. Statements: B @ Q, Q # A, A & L, L * N
Conclusions:
I. N \$ A
II. L @ Q
III. B @ N
IV. Q # N

A. I, II and III are true
B. I, II and IV are true
C. I, III and IV are true
D. I, III and IV are true
E. All are true

Answer
Answer – B. I, II and IV are true
4. Statements:  E # M, M * N, N @ O, O \$ P
Conclusions:
I. P # M
II. P # N
III. M # O
IV. N @ E

A. II and III are true
B. II and IV are true
C. III and IV are true
D. I, and IV are true
E. All are true

Answer
Answer – D. II, and IV are true
5. Statements: A \$ E, E @ F, F * G, G # H
Conclusions:
I. H @ E
II. A \$ G
III. E @ H
IV. A @ F

A. None is true
B. Only I is true
C. Only II is true
D. Only III is true
E. Only IV is true

Answer
Answer – E. Only IV is true

Directions (Q.No – 6-10) In these questions, relationships between different elements is shown in the statements. These statements are followed by two conclusions.

Give Answer
A. If only Conclusion I follows
B. If only Conclusion II follows
C. If either Conclusion I or II follows
D. If neither Conclusion I nor II follows
E. If both Conclusions I or II follow

1. Statement:  B≥E<N<Q<R=S
Conclusions:
I. S>E
II.Q<B

Answer
Answer – A. If only Conclusion I follows
2. Statement:  P≥Q>R<E=G>N
Conclusions:
I. P>G
II. R>N

Answer
Answer – D. If neither Conclusion I nor II follows
3. Statement:  A>S>P>O=E
Conclusions:
I. P≥E
II. S>E

Answer
Answer – B. If only Conclusion II follows
4. Statement:  A=B≥C, D<C
Conclusions:
I. A≥D
II. B>D

Answer
Answer – B. If only Conclusion II follows
5. Statement:  P≥R<Q=D>E>O
Conclusions:
I. P>E
II. Q>O

Answer
Answer – B. If only Conclusion II follows
• In which of these expressions ‘U > W’ be definitely false?
A. U>P≥Q=G≥R>W
B. P<A≤U≤T;W≥O>T
C. W≤A≤L=R<U
D. U>C>=F≤H; W<F
E. U>T=O≥P; W<J=P

Answer & Explanation
Answer – B. P<A≤U≤T;W≥O>T
• Which of the following symbols should be placed in the blank spaces respectively(in the same order from left to right) in order to complete the given expression in such a manner that both ‘N≥Q’ as well as ‘Q≤M’ definitely holds true? M _ N _ P _ Q _ R
A. >, ≥, <, =
B. >, >, ≥, <
C. ≥, ≥, ≤,≤
D. ≥, =, ≥,<
E. Other than those given as options

Answer & Explanation
Answer – D. ≥, =, ≥,<
• In Which of the following expressions does the expression ‘D=V’ to definitely hold true?
A. K ≥ D ≤ R = P < S ≤ V
B. U ≥ V ≥ M = F ≤ A ≥ D
C. D ≥ C > Q ≥ B = N ≤ V
D. G ≥ D = A < B ≤ S ≤ V
E. V ≥ E = G ≥ W = Y ≥ D

Answer & Explanation
Answer – E. V ≥ E = G ≥ W = Y ≥ D
• Which of the following expressions is true if the expression P<T<=B>S>M>=A is definitely true?
A. A ≤ P
B. S < P
C. M > P
D. A < B
E. T ≤ M

Answer & Explanation
Answer – D. A < B
• In which of these expressions ‘S > V’ and ‘V > B’ be definitely false?
A. S>P≥Q=G≥R>V>B
B. P<A≤S≤T;V≥O>T<B
C. B>V≤A≤L=R<S
D. S>C>=F≤H; B>V<F
E. S>T=O≥P; B<V<J=P

Answer & Explanation
Answer – B. P<A≤S≤T;V≥O>T<B
• Which of the following symbols should be placed in the blank spaces respectively(in the same order from left to right) in order to complete the given expression in such a manner that both ‘B>S’ as well as ‘E≤F’ definitely holds true? B _ A _ S _ E _ D _ F _ G
A. >, ≥, <, =, <, <
B. >, >, ≥, <, >, =
C. ≥, ≥, ≥, ≤, >, >
D. >, =, ≥, =, ≤, =
E. Other than those given as options

Answer & Explanation
Answer – D. >, =, ≥, =, ≤, =
• In Which of the following expressions does the expression ‘L=T’ and “E≥W” to definitely hold true?
A. E ≥ W ≤ R = P < S ≤ T
B. U ≥ T ≥ M = W ≤ E ≥ L
C. L ≥ C > E ≥ W = N ≤ T
D. E ≥ W = A < B ≤ S ≤ T
E. T ≥ E = G ≥ W = Y ≥ L

Answer & Explanation
Answer – E. T ≥ E = G ≥ W = Y ≥ L
• Which of the following expressions is true if the expression P<T<=Q>= R ≥ S>M>=W>A = R is definitely true?
A. W ≤ P
B. S < P
C. M < R
D. W > Q
E. T ≤ M

Answer & Explanation
Answer – C. M < R
• In which of these expressions ‘P > R’ and ‘P = R’ be definitely true?
A. S>P≥Q=G≥R>V
B. P<A≤S≤T<R;V≥O>T
C. V≤A≤L=R<S=P
D. P>S>C>=F≤H; V<F<R
E. S>T=O≥P; V<J=P>R

Answer & Explanation
Answer – A. S>P≥Q=G≥R>V
• In which of these expressions ‘T > P’ and ‘T = P’ be definitely false?
A. T≥S≥P≥Q=G≥R>V
B. P<A≤S≤T;V≥O>T
C. V≤A≤L=R<S
D. S>C>=F≤H=P≤Q=T; V<F
E. S>T=O≥P; V<J=P

Answer & Explanation
Answer – B. P<A≤S≤T;V≥O>T
• In which of these expressions ‘I > K’ be definitely false?
A. I>P≥Q=G≥R>K
B. P<A≤I≤T;K≥O>T
C. K≤A≤L=R<I
D. I>C>=F≤H; K<F
E. I>T=O≥P; K<J=P

Answer & Explanation
Answer – B. P<A≤I≤T;K≥O>T
• Which of the following symbols should be placed in the blank spaces respectively(in the same order from left to right) in order to complete the given expression in such a manner that both ‘F>N’ as well as ‘N≤B’ definitely holds true? B _ A _ N _ E _ F
A. >, ≥, <, =
B. >, >, ≥, <
C. ≥, ≥, ≥,≤
D. ≥, =, ≤,<
E. Other than those given as options

Answer & Explanation
Answer – D. ≥, =, ≤,<
• In Which of the following expressions does the expression ‘I≥D’ to definitely hold true?
A. K ≥ I ≤ R = P < S ≤ D
B. U ≥ D ≥ M = F ≤ A ≥ I
C. I ≥ C ≥ Q ≥ B = N ≥ D
D. G ≥ I = A < B ≤ S ≤ D
E. D ≥ E = G ≥ W = Y ≥ I

Answer & Explanation
Answer – C. I ≥ C ≥ Q ≥ B = N ≥ D
• Which of the following expressions is true if the expression P<T≤B>S>M≥E is definitely true?
A. E ≤ P
B. S < P
C. M > P
D. E < S
E. T ≤ M

Answer & Explanation
Answer – D. E < S
• Statements: Y ≤ K < D = S; D < B < O; A ≥ D < Z
Conclusions: i. A > B, ii. Y < Z
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – B.Only II is true
• Statements: H<L≤I=K; L = B Conclusions: i. K>H, ii. B≤I
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – E.Both I and II are true
• Statements: A>Z=R≥N<J≤E; J>F; K<Z
Conclusions: i. E>F, ii. A<N
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – A.Only I is true
• Statements: U≥J≥S>C≤B≤M
Conclusions: i. U>B, ii. J≤M
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – D.Neither I nor II is true
• Statements: G<S≤A = N≥B; F≥N<O; D≥S
Conclusions: S≤B, G<F
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – B. Only II is true
• Statements: C<L≤A = N≥G; R≥N<S; F≥L
Conclusions: R≥F, G<S
A.Only I is true
B.Only II is true
C.Either I or II true
D.Neither I nor II is true
E.Both I and II are true

Answer & Explanation
Answer – B.Only II is true
1. Statements: U > G = F ≥ K = R < E;  F > P ≤ S;  O = P ≤ J > M
Conclusions:
I. U > S,
II. E > M
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option D
Solution:
U > G = F > P ≤ S
E > P = K ≤ F > P ≤ J > M
2. Statements: W > J = F ≥ L ≤ B > T;  W < Q = T ≤ H;  D > I  ≥ Q
Conclusions:
I. L < H
II. I > B
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option A
Solution:
L ≤ F = J < W < Q ≤ H
I ≥ Q = T < B
3. Statements: D = J ≥ S > L < P = O;   O > M  ≥ D < A;   R = F ≤ S
Conclusions:
I.  P > D,
II. R ≤ M
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option E
Solution:
D = J ≥ S > L < P gives that there can be any relation between P and D. BUT P = O > M  ≥ D gives P > D
R = F ≤ S ≤ J = D ≤ M, gives R ≤ M
4. Statements: R > G < O ≥ L = E > H;   D = H > B;   S > B = N ≥ W
Conclusions:
I. O > S,
II. O ≤ S
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option C
Solution:
O ≥ L = E > H > B < S, so no relationship can be determined, but it will surely be either of O > S or O ≤ S OR either of O < S or O ≥ S
5. Statements: H < G ≤ F < D = T > S;  F > L ≥ V = E; V ≥ P = Q
Conclusions:
I. Q < S,
II. D > P
A) only I follows
B) only II follows
C) either I or II follows
D) neither I nor II follow
E) both I and II follow

View Answer
Option B
Solution:
Q = P ≤ V ≤ L < F < D = T > S
D > F > L ≥ V ≥ P

1. In which of the following expressions does the expression ‘I > D’ and ‘A ≤ G’ definitely hold true?
A) A ≥ I ≥ G = K > S > D
B) A ≤ D ≥ M = F ≤ G < I
C) I ≥ C > Q ≥ A = G ≥ D
D) G ≥ D = A < B ≤ S ≤ I
E) D ≥ E = G ≥ W = A < I

View Answer
Option D
Solution:
G ≥ D = A and D = A < B ≤ S ≤ I
2. In which of these expressions ‘A > P’ and ‘A = P’ be definitely false?
A) W < S ≥ P ≥ Q = A ≥ R > V
B) X < A ≤ S ≤ T;  X ≥ L > P
C) M ≤ A ≤ L = P < S
D) S > F > = C ≤ H = P ≤ Q = T;  F < A
E) None of these

View Answer
Option E
Solution:
In A) – A ≤ P, so A can be = P, so cannot say that ‘A = P’ is definitely false
In B) – A > P
In C) – A ≤ P
In D) – Cannot be determined
3. Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the given expression in such a manner that both ‘A > S’ as well as ‘S ≤ B’ definitely holds true? B _ M _ S _ Q _ A
A) >, >, ≥, <
B) ≥, =, ≤,<
C) ≥, ≥, ≥,≤
D) >, ≥, <, =
E) Other than those given as options

View Answer
Option B
Solution:
T > S > C ≥ R = P ≤ M
4. In which of these expressions ‘Z > X’ and ‘X < B’ be definitely false?
A) Z > P ≥ Q < A ≥ R > X > B
B) P < K ≤ Z ≤ T;  X ≥ O > T = B
C) B < X ≤ A ≤ K = R ≤ Z
D) Z > C > F ≤ Q;   B > X < F
E) Z > T = O ≥ P;   B < X < J = N

View Answer
Option B
5. Which of the following expressions is true if the expression N < C < B > S > W ≥ D is definitely true?
A) D ≤ N
B) S < N
C) D < B
D) W > N
E) C ≤ W

View Answer
Option C

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• Jaskaran Saluja

thanks for soo exhaustive coverage.. ur students will surely perform well

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