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All questions of Inequalities for ACT Exam

Read the given statement and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions is true based on the given statement.
Statement:
S = Q > R ≤  U = M > B
Conclusions:
I. Q > U
II. R < S
  • a)
    Only I is true
  • b)
    Neither I nor II is true
  • c)
    Both I and II are true
  • d)
    Only II is true
Correct answer is option 'D'. Can you explain this answer?

Ayesha Joshi answered
Given statement: S = Q > R ≤  U = M > B
According to the given information:
​S is equal to Q, Q is greater than R, R is less than or equal to U, U is equal to M, and M is greater than B.
Conclusions:
I. Q > U → False( Q > R ≤ U; A definite relation cannot be established between Q and U)
II. R < S → True ( S = Q > R; S > R → R < S)
​Hence, the correct answer is "Only II is true".
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In the following question, a statement is being followed by two conclusions. Find which of the two conclusion is/are true according to the given statement.
Statement:
J = K ≥ L > X ≥ Y ;  J ≥ P ; P ≥ Z 
Conclusions:
I: J < L
II: P ≤ Y
  • a)
    Only I is true.
  • b)
    Both I and II are true.
  • c)
    Neither I nor II is true.
  • d)
    Only II is true.
Correct answer is option 'C'. Can you explain this answer?

Orion Classes answered
The given statement is: J = K ≥ L > X ≥ Y ;  J ≥ P ; P ≥ Z 
Combining the statement: Z < P < J = K ≥ L > X ≥ Y 
Here, Z is smaller than or equal to P which is Smaller than or equal to J which is equal to K which is greater than or equal to L which is greater than X and X is greater than or equal to Y.
Conclusions:
I: J < L → False (as J = K ≥ L gives as J = K and K ≥ L which implies that J ≥ L and J cannot be less than L . Thus, Conclusion I is false.).
II:  P ≤ Y → False (as Z < P < J = K ≥ L > X ≥ Y  → clear relationship between P and Y cannot be determined. Thus, Conclusion II is false.)
Hence, the "Option c" is the correct answer.

Given below is an inequality, which of the following condition is TRUE?
If A > B > C > D = F ≥ G < H
  • a)
    G > C
  • b)
    B > F
  • c)
    G > F
  • d)
    D < G
Correct answer is option 'B'. Can you explain this answer?

Orion Classes answered
From the given conditions, A > B > C > D = F ≥ G < H.
1. G > C → C > D = F ≥ G, C > G → False.
2. B > F → B > C > D = F, B > F → True.
3. G > F → D = F ≥ G < H, F ≥ G → False.
4. D < G → D = F ≥ G, D ≥ G → False.
So, B > F is the correct inequality.

Assuming the given statements to be true in the following question , find which of the given conclusion is/are definitely true and then give your answers accordingly.
Statements: 
V < Z ≥ Y ≥ X = A < B  < C < D
Conclusions:
I. X < B
II. Y > A
III. V < Y
  • a)
    Conclusions I and II are true
  • b)
    Conclusion I and III are true
  • c)
    Conclusion II and III are true
  • d)
    All conclusions are true
Correct answer is option 'A'. Can you explain this answer?

Ayesha Joshi answered
Given statements: V < Z ≥ Y ≥ X = A < B  < C < D
Conclusions:
I. X < B → True (As X = A < B  which implies X < B . So, Conclusion I is true.)
II. Y > A → True (As Y ≥ X = A  which implies Y > A . So, Conclusion II is true.)
III. V < Y → False (As V < Z ≥ Y  there is no direct relation between V and Y. So, Conclusion II is false)
Thus, Conclusions I or II are true.
Hence, "Option a" is the correct answer.

If W > X, X > Y and Y > Z, then which of the following conclusions is definitely wrong?
  • a)
    X > Z
  • b)
    W > Z
  • c)
    Z > W
  • d)
    W > Y
Correct answer is option 'C'. Can you explain this answer?

Ayesha Joshi answered
Given: W > X, X > Y and Y > Z
On combining: W > X > Y > Z
I) X > Z → True
II) W > Z → True
III) Z > W → False (as W > Z)
IV) W > Y → True
Hence, Z > W is definitely wrong.

A statement is given followed by two conclusions. Find which conclusion is true based on the given statement.
Statement:
C < A ≤ T = D ≥ O; G ≥ D < S; T ≥ A
Conclusions:
I: A ≤ O
II: C < G
  • a)
    Either I or II
  • b)
    Only I
  • c)
    Only II
  • d)
    Neither I nor II
Correct answer is option 'C'. Can you explain this answer?

Ayesha Joshi answered
The given statement is: C < A ≤ T = D ≥ O; G ≥ D < S; T ≥ A
Combine the statement: C < A ≤ T = D ≥ O; G ≥ D < S
Conclusions:
I: A ≤ O → False (as A ≤ T = D ≥ O gives; as A ≤ D and D ≥ O → clear relationship between A and O cannot be determined).
II: C < G → True (as C < A ≤ T = D ≥ O; G ≥ D < S gives; as by combinatin C < A ≤ T = D ≤ G so C < G True).
Hence, the correct answer is “Only II”.

Directions: In the following question assuming the given statements to be true, find out which of the following two conclusions definitely true.
Statements:
A ≤ P = B; C = P; F < B
Conclusions:
I. B < A
II. F = P
  • a)
    Only conclusions I is true.
  • b)
    Only conclusions II is true.
  • c)
    Either conclusions I or II is true.
  • d)
    Neither conclusions I nor II is true.
Correct answer is option 'D'. Can you explain this answer?

Nora Barnes answered
Given Statements:
- A ≤ P = B
- C = P
- F < B

Conclusions:
I. B < A
II. F = P

Explanation:
- From the statements given, we know that A is less than or equal to P, which is equal to B. This implies that B is greater than or equal to A, so conclusion I (B < a)="" is="" not="" definitely="" />
- We are also given that F is less than B and C is equal to P. However, there is no direct relationship given between F and P, so we cannot definitively conclude that F equals P. Therefore, conclusion II (F = P) is not definitely true.
Therefore, neither conclusion I nor II is definitely true based on the given statements. Hence, the correct answer is option 'D'.

Below is a statement followed by some conclusions. Find which of the given conclusions logically follow(s) from the given statement.
Statement: 
P > Q ≤ C ≤ B = M > D
Conclusions:
I. M > Q
II. D ≤ Q
III. M = Q
IV. C < D
  • a)
    Only either I or III is true.
  • b)
    Only I is true.
  • c)
    None is true.
  • d)
    Only either II or IV is true.
Correct answer is option 'A'. Can you explain this answer?

Ayesha Joshi answered
I. M > Q → False (as Q ≤ C ≤ B = M → M gives either M = Q or M > Q).
II. D ≤ Q → False (as Q ≤ C ≤ B = M > D)
III. M = Q → False (as Q ≤ C ≤ B = M → M gives either M = Q or M > Q).
IV. C < D → False (as C ≤ B = M > D).
P > Q ≤ C ≤ B = M > D is given in statement. Therefore but Conclusion I and III forms a complementary pair for either.
Thus, the correct answer is option 1.

In the following question, Few statements are given which are followed by some conclusions. According to the given statements, which conclusion is/are definitely true?
Statements:
M ≤ N = O ; M ≤ P ; O < Q
Conclusions:
I. Q > N
II. M ≥ O
  • a)
    Only conclusion II is true.
  • b)
    Only conclusion I is true.
  • c)
    Both conclusions I and II are true.
  • d)
    None of the conclusions are true.
Correct answer is option 'B'. Can you explain this answer?

Orion Classes answered
According to given information:
Statements: M ≤ N = O ; M ≤ P ; O < Q
Combining the statement: P > M ≤ N = O < Q
Here, P is greater than or equal to M which is less than or equal to N equal to O which is less than Q.
Conclusions:
I.  Q > N → True (In the given statement, P > M ≤ N = O < Q ,It clearly mentioned Q is greater than O which is equal to N .So, Q is greater than N too. Thus, the conclusion I is true).
II. M ≥ O → False (In the given statement, M ≤ N = O, It is clearly mentioned that M is less than or equal to N which is equal to O. So, M can never be greater than O. Thus, conclusion II is a false).
Hence, "Option b" is the correct answer.

Given below is an inequality, which of the following condition is TRUE?
If P > Q ≤ R = S ≥ T
  • a)
    Q > S
  • b)
    Q = T
  • c)
    R < T
  • d)
    None
Correct answer is option 'D'. Can you explain this answer?

Orion Classes answered
Given inequality is, P > Q ≤ R = S ≥ T
1. Q > S → Q ≤ R = S, Q < S → False.
2. Q = T → Q ≤ R = S ≥ T, data is not sufficient to determine the relation → False.
3. R < T → R = S ≥ T, R ≥ T → False.
So, none of the options are correct.

Directions: In the following question assuming the given statements to be true, find which of the given conclusions is definitely true.
Statement:
C < O > K; Z < R < N = C
  • a)
    R < C
  • b)
    Z > N
  • c)
    O = R
  • d)
    C < K
Correct answer is option 'A'. Can you explain this answer?

Ayesha Joshi answered
Given:
C < O > K; Z < R < N = C
On Combining:
Z < R < N = C < O > K
Conclusions:
1. R < C → True (as R < N = C)
2. Z > N → False (as Z < R < N; therefore Z < N)
3. O = R → False (as R < N = C < O; therefore R < O)
4. C < K → False (as C < O > K; thus, a clear relation between C and K cannot be established)
Hence, “R < C” is definitely true among the given options.

If A + B = C + D and A + D > B + C, then which one of the following is definitely wrong?
  • a)
    A > B
  • b)
    A > C
  • c)
    C < D
  • d)
    B > D
Correct answer is option 'D'. Can you explain this answer?

Ayesha Joshi answered
Given : A + B = C + D
⇒ A - D = C - B ----- (i)
Also, it is given that A + D > B + C ----- (ii)
Subtracting (i) from (ii), we get :
D > B
Therefore, B > D is definitely wrong.
Hence, option 4 is the correct answer.

In the following question assuming the given statements to be True, find which of the conclusion among given conclusions is/are definitely true and then give your answers accordingly.
Statements:
P < Q < R; R = S ≤ T
Conclusions:
I. Q > T
II. S > P
  • a)
    None is true
  • b)
    Both I and II are true
  • c)
    Only II is true
  • d)
    Only I is true
Correct answer is option 'C'. Can you explain this answer?

Ayesha Joshi answered
Given statements: P < Q < R; R = S ≤ T
On combining: P < Q < R = S ≤ T
Conclusions:
I. Q > T → False (as Q < R and R = S ≤ T → Q < R = S ≤ T → Q ≤ T)
II. S > P → True (as P < Q < R; R = S → P < Q < R = S → P < S)
Hence, only conclusion II is true.

In the following question assuming the given statements to be true, find which of the conclusion among given conclusions is/are definitely true and give your answers accordingly.
Statements:
B > C; C < D; D > E; E > F
Conclusions:
I. C < F
II. B > D
  • a)
    Both I and II follow
  • b)
    Only I follows
  • c)
    Only II follows
  • d)
    None follows
Correct answer is option 'D'. Can you explain this answer?

Ayesha Joshi answered
Given statements: B > C; C E; E > F
On combining: B > C < D > E < F
Conclusions:
I. C < F → False (as C < D > E < F → clear relationship between F and D cannot be determined)
II. B > D → False (as B > C < D → clear relation between B and D cannot be determined)
Hence, none of the conclusions is true.

In the following question, Few statements are given which are followed by some conclusions. According to the given statements, which conclusion is/are definitely true?
Statements:
S = R ≥ Q ≤ P ; N ≤ P ; M < N
Conclusions:
I. M < S
II. M ≥ S 
  • a)
    Either I or II follows. 
  • b)
    Only II follows.
  • c)
    Both I and II follows.
  • d)
    Only I follows.
Correct answer is option 'D'. Can you explain this answer?

Ayesha Joshi answered
According to given information:
Statements: S = R ≥ Q ≤ P ; N ≤ P ; M < N
Combine the statement: S = R ≥ Q ≤ P  ≥ N > M
Here, S is equal to R which is greater than or equal to Q which is less than or equal to P which is greater than or equal to N which is greater than M.
Conclusions:
I. M < S → False (As S = R ≥ Q ≤ P ; N ≤ P ; M < N → there is no direct relation between M and S. So, conclusion I is false)
II. M ≥ S → False (As S = R ≥ Q ≤ P ; N ≤ P ; M < N → there is no direct relation between M and S. So, conclusion II is false)
Here, both conclusions I and II form a complementary pair as both have the same element i.e. M and S and all the three symbols (<, =, >) are present between M and S.
Hence, Either I or II follows.
Hence, "Option d" is the correct answer.

Solve problems using substitution (Mathematical Operations): If A > B, B > C and C > D, then which of the following conclusions is definitely wrong?
  • a)
    A > D
  • b)
    A > C
  • c)
    D > A
  • d)
    B > D
Correct answer is option 'C'. Can you explain this answer?

Joseph Simmons answered
If A = 4 and B = 2, find the value of 3A - 2B.

Substituting the values of A and B into the equation, we have:

3(4) - 2(2)
= 12 - 4
= 8

Therefore, the value of 3A - 2B is 8.

Given below is an inequality, which of the following condition is TRUE?
If P ≤ Q ≥ R; E ≥ Q; L < Q = D
  • a)
    L < P
  • b)
    E ≥ R
  • c)
    E < D
  • d)
    Q < R
Correct answer is option 'B'. Can you explain this answer?

Ayesha Joshi answered
Given inequality is: P ≤ Q ≥ R; E ≥ Q; L < Q = D
1. L < P → P ≤ Q, L < Q, data is not sufficient to determine the relation → False
2. E ≥ R → E ≥ Q ≥ R, E ≥ R → True.
3. E < D → E ≥ Q, L < Q = D, E ≥ D → False.
4. Q < R → Q ≥ R → False.
Hence, E ≥ R, is the correct inequality.

In the given question, relationship between different elements is shown in the statement. The statement is followed by two conclusions. Study the conclusions based on the given statements and select the correct answer.
Statements:
U > H = G ≥ N; E < N > W ≥ O
Conclusion:
I. G > E
II. H > N
  • a)
    Either conclusion I or II is true
  • b)
    Only conclusion I is true
  • c)
    Neither conclusion I nor conclusion II is true
  • d)
    Only conclusion II is true
Correct answer is option 'B'. Can you explain this answer?

Ayesha Joshi answered
Statements:
U > H = G ≥ N; E < N > W ≥ O
Conclusion:
I. G > E → True → According to the statement G ≥ N > E
II. H > N → False → According to the statement U > H = G ≥ N
Hence, Option (b) is correct.
Mistake Points:
As from the equation H = G ≥ N.
There is a difference in (≥, >) both the sign.
When H ≥ N means H is greater or equal to N.
When H > N means H is greater than N.
As in conclusion 2, it is said H = G ≥ N from this we cannot say H is greater than N.
Thus conclusion 2 is false.

Chapter doubts & questions for Inequalities - Mathematics for ACT 2025 is part of ACT exam preparation. The chapters have been prepared according to the ACT exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for ACT 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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