Inequality Questions and Answers

Answer: B

Correct answer: Only conclusion b follows

The statements are \( A > B \), \( B \le C \), \( D = E \) and \( C > D \). Written as one line they read \[ A > B \le C > D = E, \] which reverses direction at B.

Conclusion II claims \( C > E \). Both lie on the right of the reversal: \( C > D = E \) gives it in one step. Follows.

Conclusion I claims \( A > E \). A sits on the left of the reversal and E on the right, so nothing links them. Concretely, with \( B = 5 \) and \( C = 100 \) we may take \( A = 6 \) and \( D = E = 99 \), which satisfies every statement and makes \( A < E \). Not determined.

Only the second conclusion follows. The trap is that A looks like the largest element because the chain opens with it; but \( A > B \) caps nothing beyond B, and B is the bottom of the chain.

Answer: D

Correct answer: Neither conclusion a nor b follows

The statements are \( A \ge B \), \( C = D \), \( E > F \), \( E \ge D \) and \( B > C \). Assembling them gives \( A \ge B > C = D \), with E standing at or above D and F below E: \[ E \ge D = C, \qquad E > F. \]

Both conclusions compare something in the C–D pair with F, and both fail for the same reason: D and F are attached to E from the same side, D at or below it and F strictly below it, which fixes no order between them.

I. \( D > F \): with \( E = 10 \), take \( D = 9, F = 1 \) and the conclusion holds; take \( D = 1, F = 9 \) and it fails. Both satisfy every statement. Not determined.

II. \( C \ge F \): C equals D, so this is the same comparison in weak form, and the same two cases defeat it. Not determined.

Neither conclusion follows.

Answer: B

Correct answer: Both conclusions a and b follow

The statements are \( A > B \), \( D > E \), \( C = D \), \( E > F \) and \( B \ge C \). They assemble into a single descending chain with no reversal anywhere: \[ A > B \ge C = D > E > F. \]

I. \( B \ge E \): reading from B, \( B \ge C = D > E \) contains a strict link, so in fact \( B > E \) - more than the conclusion asks. Follows.

II. \( B > F \): continue one link further, \( B \ge C = D > E > F \), strict twice over. Follows.

Both follow. When the statements line up into one unbroken chain like this, every comparison along it is decidable, and the only work is checking whether at least one strict link falls between the two elements named.

Answer: D

Correct answer: Neither conclusion a nor b follows

The statements are \( P \le Q \), \( Q > R \), \( T \ge U \), \( S > R \) and \( T = S \). Merging T with S, the picture is R sitting below both Q and S, with P at or below Q and U at or below S: \[ P \le Q > R < S = T \ge U. \] There are two reversals here, at Q and at R.

I. \( R \ge U \): R sits below S and U sits at or below S, so both hang beneath the same element from the same side. With \( S = 10 \), \( R = 9, U = 1 \) and \( R = 1, U = 9 \) both satisfy the statements. Not determined.

II. \( P \le U \): P is capped by Q, U is capped by S, and the two caps are joined only through R, which is below both. Nothing carries across. Not determined.

Neither conclusion follows. Every element here is bounded from above and nothing is bounded from below, which is exactly the arrangement in which almost no comparison can be made.

Answer: C

Correct answer: Either Conclusion a or b follows

The statements give \( C = D \), \( D \le E \le B \) and \( E = F \). Both conclusions compare B with D, so assemble that run: \[ D \le E \le B. \] Every link is weak, so the most that follows is \( B \ge D \).

Conclusion I claims \( B > D \) and conclusion II claims \( B = D \). Neither is forced - taking \( D = E = B \) satisfies the statements and makes II true, while \( D = 1, E = 2, B = 3 \) makes I true - but \( B \ge D \) admits no third possibility, so one of them must hold.

That is the “either…or” answer: not a hedge, but the precise consequence of a purely weak chain between the two elements the conclusions name. The statement in this record had lost a symbol to a stray < in the stored HTML; it is restored above as the relation the keyed answer requires.

Answer: A

Correct answer: A < E ≥ B > D > F = G < C

Reading this option, E ≥ B > D > F = G gives E > G, and G < C with G = F gives C > F. Both required relations therefore follow from unbroken chains. Each of the other options breaks at least one of them at a reversal of direction.

Answer: D

Correct answer: A > E, though the statement it depends on is missing

The statement this question refers to has been lost from the record - what survives reads only “D ≤ CE”, with a symbol missing between C and E and the earlier part of the chain gone entirely. The five options between them mention A, B, C, D, E and F, so the original statement linked at least six letters, and none of those links can be recovered from what remains. The keyed answer is \( A > E \), which tells us the original chain placed A above E; but with the chain itself absent, that cannot be demonstrated here, and the four rejected options cannot be shown to fail. The item needs its original statement restored before it can be worked.

Answer: B

Correct answer: >

The chain is C = B ? A ≥ D, and C > D needs a strict step somewhere between C and D. The link A ≥ D is not strict and C = B is not either, so the strictness must be supplied at the question mark: B > A gives C = B > A ≥ D and therefore C > D.

Answer: C

Correct answer: D,B,A,C,E

Filling the blanks gives D < B ≤ A = C > E, from which D < B ≤ A = C yields D < C. The requirement is that D sits to the left of the strict less-than and C to the right of the equals, so the chain runs upward from D to C without a reversal.

Answer: A

Correct answer: A>E

The chain A ≥ B = C > D = E runs without reversal and contains one strict step at C > D, so A > E follows. The other options fail at the reversal E ≤ F: nothing above E can be compared with F or G, so neither B < F nor C > G can be established.