Inequality Questions and Answers

Answer: B

Correct answer: only conclusion II is true.

This set’s code reads \(\#\) as “smaller than”, \(©\) as “greater than”, \(\pi\) as \(\le\), \(\$\) as \(\ge\), and \(\%\) as “neither smaller nor greater”, that is \(=\). The statement becomes \[ L \ge M = Q > S < T \le P. \] S is a low point, so the chain reverses there.

II. \( L > S \): both elements sit on the left of the pivot, joined by \( L \ge M = Q > S \), which contains a strict link. True.

I. \( T > L \): T lies on the right of the pivot and L on the left, so no chain runs between them. With \( S = 1 \), \( L = 9, T = 2 \) and \( L = 2, T = 9 \) both satisfy every statement. Not determined.

Only II follows.

Answer: B

Correct answer: both conclusions I and II are true.

This set’s code reads \(\#\) as “smaller than”, \(©\) as “greater than”, \(\pi\) as \(\le\), \(\$\) as \(\ge\), and \(\%\) as “neither smaller nor greater”, that is \(=\). The statement becomes \[ W < H \le I = C \ge H > S. \] Note that H is named twice, which is what ties the two halves together.

I. \( C \ge W \): the run \( W < H \le I = C \) points one way throughout and opens strict, so \( C > W \) - which certainly satisfies the weaker \( C \ge W \) that the conclusion states. True.

II. \( C > S \): reading from C, \( C \ge H > S \), strict at the last link. True.

Both follow. Conclusion I is the one candidates most often reject, on the grounds that the chain gives a strict relation while the conclusion asks for a weak one; but \( \ge \) requires only that C is not smaller than W, and \( C > W \) delivers that. A strict result always satisfies the corresponding weak claim - never the reverse.

Answer: E

Correct answer: both conclusions I and II are true.

This set’s code reads \(\#\) as “smaller than”, \(©\) as “greater than”, \(\pi\) as \(\le\), \(\$\) as \(\ge\), and \(\%\) as “neither smaller nor greater”, that is \(=\). The statement becomes \[ W = X > Y \le Z > R = L. \] Y is a low point, so the chain reverses there - but neither conclusion needs to cross it.

I. \( Z > L \): both lie on the right of the pivot, and \( Z > R = L \) reads straight off. True.

II. \( Y < W \): both lie on the left, and \( W = X > Y \) gives it directly. True.

Both follow. The comparison this chain would refuse is W against Z, or W against L - anything spanning the reversal at Y - and the question asks for neither.

Answer: D

Correct answer: only conclusion I is true.

This set’s code reads \(\#\) as \(<\), \(©\) as \(>\), \(\pi\) as \(\le\), \(\$\) as \(\ge\) and \(\%\) as \(=\). The statement becomes \[ A < N = O \le T \le H < E. \] This one has no reversal at all - it climbs from A to E in a single direction - so every comparison along it can be settled, and the only question is whether the result is strict.

I. \( E \ge O \): the run \( O \le T \le H < E \) contains a strict link at the end, so in fact \( E > O \), which satisfies the weaker claim the conclusion makes. True.

II. \( H > N \): the run \( N = O \le T \le H \) is weak at every link, so it gives only \( H \ge N \). The conclusion asks for the strict relation, and equality is not excluded - taking N, O, T and H all equal satisfies every statement. Not determined.

Only I follows. The contrast between the two is the whole lesson: a chain with one strict link proves the strict conclusion and the weak one; a chain of purely weak links proves only the weak one.

Answer: A

Correct answer: only conclusion I is true.

This set’s code reads \(\#\) as \(<\), \(©\) as \(>\), \(\pi\) as \(\le\), \(\$\) as \(\ge\) and \(\%\) as \(=\). The statement becomes \[ P > V = T < D < M \ge F. \] The chain falls to T and then climbs away, so T is a low point.

I. \( M > V \): both are reached from T on the same side of nothing - \( V = T < D < M \) runs one way from V up to M, with strict links. True.

II. \( P < F \): P sits above the low point on the left, F at or below M on the right, and no chain crosses between them. With \( T = 1 \), \( M = 10 \), F may be anything from 1 upward and P anything above 1, so either order is possible. Not determined.

Only I follows.

Answer: A

Correct answer: Only conclusion I is true.

Read the chain from B rightwards: \( B = Q \le P < J \le Y \). A single strict link anywhere in a chain makes the whole relation strict, and there is one at \( P < J \), so \( B < Y \) - conclusion I is true.

Conclusion II claims \( X \ge J \). Here the chain works against us. We know \( J \le Y \) and \( Y > X \), so both J and X sit at or below Y, but the two are approached from the same side and nothing fixes their order relative to each other. Take \( Y = 10 \): then \( J \) could be 9 and \( X \) 2, making \( X < J \); or \( J \) could be 2 and \( X \) 9, making \( X > J \). Both are consistent with every statement, so conclusion II does not follow.

Only the first conclusion holds. The general rule this illustrates: a conclusion follows only when an unbroken chain of relations runs between its two elements all pointing the same way; two elements that both hang below a common third are never ordered by that alone. The statement in this record had been corrupted by a stray < in the stored HTML, which swallowed the middle of the chain; it is restored above.

Answer: E

I. Z ≥ A ≥ B = Q [FALSE]

II. Z ≥ A ≥ B = Q [FALSE]

Answer: D

Correct answer: Both conclusions I and conclusion II are true.

The statements give \( G < R \), \( R = A \), \( A \le S \) and \( R > T \).

Conclusion I claims \( G < S \). Chain it: \( G < R = A \le S \). The strict link at the left end carries through the equality and the weak link, giving \( G < S \). True.

Conclusion II claims \( S > T \). Chain it the other way: \( S \ge A = R > T \). Again a single strict link makes the whole relation strict, so \( S > T \). True.

Both follow, so the answer is that both conclusions are true. Note how R does the work in each case - it is the element every other one is tied to, and finding that pivot is usually the quickest route through a question of this shape. The statements in this record had been destroyed by a stray < in the stored HTML; they are restored above from the relations the surviving working refers to.

Answer: B

Correct answer: Neither conclusion I nor II is true.

The statements give P = U < M > N, with D ≥ P and I ≥ C. Conclusion II asks whether N > U, but the chain reverses at M and both U and N lie below it with no link between them. Conclusion I involves C, which is connected only to I and to nothing else in the problem, so it cannot be reached at all.

Answer: C

Correct answer: Only conclusion II is true, though the statements this depends on are damaged

The statements in this record have been corrupted. What survives reads “P = U < M> N; D ≥ P; I ≥ C”, where a stray < in the stored HTML has swallowed the text between M and N, and conclusion I speaks about an element K that appears nowhere in the statements at all - so at least one further relation, the one introducing K, has been lost with it.

What can still be said is limited. From \( D \ge P \) and \( P = U \) we know D stands at or above both P and U, but nothing connects D to K, so conclusion I cannot be established from what remains - consistent with the key, which treats it as not following. Conclusion II, \( I > P \), must have followed through the missing link: \( I \ge C \) is the only statement mentioning I, so the lost text has to have tied C to P strictly enough to give \( I > P \).

The item needs its original statements restored before the reasoning can be shown in full; unlike the two questions beside it, the damage here removed a relation rather than merely obscuring one, and it cannot be inferred from the surviving text.