Test: Inequalities - 2 - ACT MCQ

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”.

Given: A ≤ P = B; C = P; F < B

On combining: A ≤ P = B = C

I. B < A → False as A ≤ B.

II. F = P → False as F < P.

Hence both conclusions are false.

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.

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.

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.

Given statements: Q > J ≥ M > L = K < S

I. Q > L → True (as Q > J ≥ M > L)

II. J > K → True (as J ≥ M > L = K)

III. L < S → True (as L = K < S)

Hence, the answer is ‘All follow’. 

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.

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.

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".

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.

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.

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 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.

Given the logic, M<S (Conclusion I) follows from the statements. Therefore: Only  I follows.

Given: A > B, B > C and C > D

After combining the operators,

we get: A > B > C > D

Now, checking the options:

Option A: A > D → True (As, A > B > C > D gives A > D)

Option B: A > C → True (As, A > B > C > D gives A > C)

Option C: D > A → False (As, A > B > C > D gives A > D)

Option D: B > D → True (As, A > B > C > D gives B > D)

So, Option C, D > A is definitely wrong.

Hence, the correct answer is "Option c".