Inequality Reasoning: Direct, Coded and Compound Conclusions

1. What does an inequality question ask?#

You are given comparisons between quantities and asked which conclusions must follow. Treat the statements as true and use only their information. A conclusion that happens to be true for one choice of values is not necessarily true for every allowed choice.

In this chapter, letters represent real numbers. They are not automatically positive, integer or distinct. Equality remains possible unless a strict comparison or another statement excludes it. The questions have four answer choices: only I, only II, both, or neither. In advanced questions a conclusion can itself contain an explicitly defined “or”; evaluate the complete conclusion.

2. Five signs and their meanings#

Sign	Meaning	What it permits
A > B	A is greater than B	Greater only
A < B	A is less than B	Less only
A = B	Equal values	Equality only
A ≥ B	A is greater than or equal to B	Greater or equal
A ≤ B	A is less than or equal to B	Less or equal

“Not less than” means ≥, not >. “Not greater than” means ≤, not <. “At least” gives an inclusive lower bound; “at most” an inclusive upper bound. “A is not equal to B” allows both A > B and A < B; it does not choose either direction.

If A > B, then A ≥ B is also true. The converse is unsafe: A ≥ B does not prove A > B because A = B is allowed. Similarly, A = B proves both A ≥ B and A ≤ B, but proves neither strict comparison.

3. Reverse a comparison correctly#

Read an inequality in either direction by reversing its sign: A > B becomes B < A; A ≥ B becomes B ≤ A. Equality stays equality. Reversing the letters without reversing the sign changes the claim.

This is a change of reading direction, not multiplication by a negative number. When doing algebra, multiplication or division by a negative quantity also reverses the sign. Multiplication by an unknown-sign quantity is unsafe without cases. Most comparison questions need only reordering and transitivity, not multiplication.

Example: P ≤ Q can be written Q ≥ P. It cannot be written Q > P unless another clue excludes equality.

4. Combining a chain in one direction#

Comparisons compose along a connected path. If every link points toward “greater”, the endpoint comparison is greater or equal; if at least one link is strict, the endpoint is strictly greater. The same rule applies toward “less”. Equality lets you substitute either equal quantity.

Given	Necessary endpoint relation
A > B > C	A > C
A ≥ B > C	A > C
A > B ≥ C	A > C
A ≥ B ≥ C	A ≥ C
A = B > C	A > C
A ≤ B < C	A < C

Example: A ≥ B = C > D gives A > D, B > D and C > D. It does not give A > B. The strict link C > D lies on the path to D, not on the separate comparison A versus B.

5. A change of direction can leave the endpoints unknown#

From A > B < C, both A and C exceed B, but their order is not fixed. Examples: A=3, B=1, C=2 gives A > C; A=2, B=1, C=3 gives A < C; A=C=2, B=1 gives equality. Therefore neither A > C nor A ≤ C individually follows.

Similarly, A < B > C has a common upper value but gives no definite comparison of A and C. Do not mechanically combine adjacent signs when their directions conflict. Search for another supplied path before deciding that the relation is unknown.

If an extra statement says A ≥ D > C, that independent path proves A > C. A broken chain does not cancel valid information elsewhere in the question.

6. Equality groups and multiple paths#

Merge equal quantities conceptually: if A = B and B = C, all three have one value. Any comparison involving one can be applied to the others. For A = B, B ≥ C and C > D, it follows that A > D.

Two weak comparisons in opposite directions can establish equality: A ≥ B and B ≥ A together imply A = B. Either statement alone is insufficient. Do not overlook this when a question has separate clauses instead of a single chain.

Worked example: A ≥ B; B ≥ A; B > C; C ≥ D. First A=B. Next B>C≥D makes B>D, and substituting A gives A>D. Conclusion I, A=B, follows. Conclusion II, A>D, also follows.

7. Check conclusions separately#

Never use conclusion I as a new statement while checking conclusion II. Both must be derived from the original statements. A conclusion being false or unproved does not automatically make the other true.

Example: A ≥ B > C = D. I: A > D. II: A > B. The path to D contains a strict link, so I follows. II is not necessary because A=B is allowed. Values A=B=2 and C=D=1 satisfy every statement but disprove II. The answer is only I.

“Neither follows” means neither conclusion is forced by the statements. It does not necessarily mean both conclusions are always false. Either can hold in some assignments. Distinguish a false statement from an undetermined one.

8. Counterexamples prove non-necessity#

To disprove “must follow”, find one assignment satisfying all premises but violating the conclusion. For A ≥ B, the assignment A=B=4 disproves the strict conclusion A>B. It does not disprove the original statement.

A counterexample that violates one given comparison is invalid. With A ≥ B > C, using A=B=C=1 is not allowed because B>C is strict. Use A=B=2, C=1 to test whether A>B is necessary.

One working example proves possibility, not necessity. To prove necessity, use valid chain reasoning, eliminate every contrary order, or examine all order cases. The answer explanations here give concrete counterexamples whenever a conclusion is not forced.

9. Coded inequalities: decode before solving#

Symbols have no universal meaning across tests. A question may define @ as > today and as ≤ elsewhere. Read its key every time. In this chapter's coded questions:

Code	Ordinary relation
X @ Y	X > Y
X # Y	X < Y
X $ Y	X ≥ Y
X % Y	X ≤ Y
X & Y	X = Y

Example: A $ B; B & C; C @ D decodes to A ≥ B; B=C; C>D. Thus A>D follows. A>B is not definite because A=B is possible. Do not confuse the symbol $ with an arithmetic operation or import a code key from a previous question.

Translate a verbal key exactly. If @ means “neither smaller nor equal”, @ means >. If @ means “not greater”, it means ≤. Remove the verbal negation before linking the comparisons.

10. Either–or: the condition that actually matters#

For real numbers, exactly one of A>B, A=B or A<B holds. If A≥B is known, A>B and A=B cover the remaining possibilities. Neither alone need follow, but A>B or A=B does follow.

A strict complementary pair A>B and A≤B is always exhaustive and mutually exclusive for real numbers. So is A<B versus A≥B. By contrast, A>B versus A<B leaves equality uncovered unless the statements rule it out. Two conclusions being individually unproved is not enough for an either–or answer.

Worked example: A≥B. I: A>B. II: A=B. In a conventional test offering “either I or II”, that option is correct because exactly one must hold. If a question offers only the four individual-necessity choices, neither I nor II is individually necessary. Always read what the answer choices are asking.

Our advanced questions avoid this option-scheme ambiguity: they write the disjunction inside a conclusion and explicitly define “or” as inclusive, meaning at least one comparison is true. For example, I: A>B or A=B is one complete conclusion, equivalent to A≥B. Evaluate that complete statement, then evaluate II separately.

11. Possibility, necessity and contradiction#

“Can be true” requires at least one consistent assignment. “Must be true” requires all consistent assignments. “Cannot be true” means no consistent assignment permits the claim. These are different tasks even when the same comparisons are used.

With A≥B≥C, A=C is possible if all three are equal. It is not necessary because A=3, B=2, C=1 is also allowed. If instead A>B≥C, A=C is impossible: the strict path proves A>C.

Contradictory premises, such as A>B and B≥A, admit no ordinary numerical arrangement. For exam practice, identify that inconsistency rather than inventing values. Every question in this chapter has at least one valid assignment; contradictions are discussed as a checking skill, not hidden in the bank.

12. A full mixed example#

Statements: A ≥ B; C = B; D < C. Conclusions: I. D < A. II. A = C.

First rewrite D<C as C>D, and substitute B for C: A≥B=C>D. The path from A to D is strict, so A>D, equivalent to D<A. I follows. A=C is permitted, but it is not forced. A=3, B=C=2, D=1 satisfies all statements and makes A=C false. Therefore only I follows.

If II were “A>C or A=C”, it would follow because the premises establish A≥C. This illustrates why the whole conclusion, including “or”, matters.

13. Revision checklist and practice routine#

• Decode symbols first; do not rely on a remembered key.
• Reverse both the order and the sign when rewriting a comparison.
• Join only connected paths with compatible directions.
• A strict link makes a compatible endpoint chain strict.
• An all-weak chain does not justify strict inequality.
• Equality allows substitution; opposing weak bounds can force equality.
• Check I and II independently from the original statements.
• A valid counterexample disproves necessity.
• For either–or, test coverage and the question's option convention.
• Do not assume positivity, integrality or distinctness.

Practise by writing a one-line proof for each necessary conclusion and a valid counterexample for each unforced one. The bank uses only comparisons between four variables. Any arrangement of four real values can be replaced by ranks 0–3 while preserving less/equal/greater relations, so checking every such rank assignment covers every possible order pattern, including equalities. This is a verification method; during an exam, use chains and counterexamples to solve quickly.

Practice set: 30 questions#

Practice 01#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B < C; C ≤ D; D < A. Conclusions: I. D > B. II. D > C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 02#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: A = B; B ≤ D; D = C. Conclusions: I. D > B. II. A = B.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 03#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B ≥ D; D = C; C > A. Conclusions: I. A < D. II. B ≥ A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 04#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: A = D; D ≤ B; B ≤ C. Conclusions: I. C ≤ D. II. B < C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 05#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: A ≥ D; D ≥ C; C = B. Conclusions: I. D ≥ A. II. A < D.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 06#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B ≤ C; C = A; A ≤ D. Conclusions: I. D ≤ B. II. C > A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 07#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B = A; A < D; D = C. Conclusions: I. C ≤ A. II. C = B.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 08#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: C > B; B > A; A = D. Conclusions: I. C = B. II. C < D.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 09#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: A ≥ D; D > C; C < B; B < D. Conclusions: I. C < B. II. D ≤ C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 10#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: D = C; C < A; A = B; B = A. Conclusions: I. D ≥ B. II. D < A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 11#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B = D; D ≤ A; A > C; C > B. Conclusions: I. A ≥ D. II. C > B.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 12#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: A ≤ D; D ≥ B; B ≥ C; C < A. Conclusions: I. C > B. II. B ≥ D.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 13#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B = A; A ≤ D; D > C; D ≥ A. Conclusions: I. D ≤ B. II. A > B.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 14#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: B ≤ A; A > C; C ≥ D; C = B. Conclusions: I. A = B. II. C ≥ A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 15#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: D < B; B > A; A ≥ C; C ≤ D. Conclusions: I. A = B. II. D = B.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 16#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Statements: C > B; B ≤ A; A > D; B < D. Conclusions: I. C < B. II. D > C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 17#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: B @ A; A & C; C % D. Conclusions: I. A = C. II. B ≤ D.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 18#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: A @ C; C # B; B @ D. Conclusions: I. D > C. II. C < A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 19#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: D @ C; C $ B; B @ A. Conclusions: I. D > C. II. B > A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 20#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: C $ A; A & B; B % D. Conclusions: I. A = D. II. C ≤ A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 21#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: D & B; B % C; C & A. Conclusions: I. D > B. II. A > C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 22#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: C # A; A # D; D @ B. Conclusions: I. A < B. II. C ≥ A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 23#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: A # C; C $ D; D # B. Conclusions: I. B = A. II. B ≤ A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 24#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Codes: X @ Y means X > Y; X # Y means X < Y; X $ Y means X ≥ Y; X % Y means X ≤ Y; X & Y means X = Y. Decode the statements; conclusions use ordinary signs. Statements: C # A; A $ B; B % D. Conclusions: I. B ≥ C. II. C ≥ B.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 25#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Within a conclusion, "or" is inclusive: at least one comparison must be true. Evaluate I and II separately. Statements: D > C; C ≥ B; B = A; B = C. Conclusions: I. B > A or B ≤ A. II. A > D or A = D.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 26#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Within a conclusion, "or" is inclusive: at least one comparison must be true. Evaluate I and II separately. Statements: C = D; D > B; B < A; B < A. Conclusions: I. C > A or C = A. II. B < A or B = A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 27#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Within a conclusion, "or" is inclusive: at least one comparison must be true. Evaluate I and II separately. Statements: D = C; C < A; A ≥ B; B > C. Conclusions: I. D > A or D ≤ A. II. A > C or A = C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 28#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Within a conclusion, "or" is inclusive: at least one comparison must be true. Evaluate I and II separately. Statements: A = C; C ≤ D; D > B; A ≥ C. Conclusions: I. D < B or D = B. II. B < A or B = A.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 29#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Within a conclusion, "or" is inclusive: at least one comparison must be true. Evaluate I and II separately. Statements: D < B; B > C; C ≤ A; B < A. Conclusions: I. A < C or A = C. II. B < D or B = D.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Practice 30#

Treat A, B, C and D as real numbers. Decide which conclusions are necessarily true in every assignment satisfying the statements. Within a conclusion, "or" is inclusive: at least one comparison must be true. Evaluate I and II separately. Statements: D ≤ A; A < C; C = B; C ≤ B. Conclusions: I. A < D or A = D. II. A > C or A = C.

• A. Only conclusion I follows
• B. Only conclusion II follows
• C. Both conclusions I and II follow
• D. Neither conclusion follows

Answers and explanations#

Answer 01#

A. Only conclusion I follows

Statements: B < C; C ≤ D; D < A. Conclusion 1: the possible comparisons under the statements are D > B. Every one is covered by this conclusion, so it must follow. Conclusion 2 does not necessarily follow. Counterexample A=2, B=0, C=1, D=1 satisfies every statement but makes this conclusion false.

Answer 02#

B. Only conclusion II follows

Statements: A = B; B ≤ D; D = C. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2: the possible comparisons under the statements are A = B. Every one is covered by this conclusion, so it must follow.

Answer 03#

C. Both conclusions I and II follow

Statements: B ≥ D; D = C; C > A. Conclusion 1: the possible comparisons under the statements are A < D. Every one is covered by this conclusion, so it must follow. Conclusion 2: the possible comparisons under the statements are B > A. Every one is covered by this conclusion, so it must follow.

Answer 04#

D. Neither conclusion follows

Statements: A = D; D ≤ B; B ≤ C. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=1, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 05#

D. Neither conclusion follows

Statements: A ≥ D; D ≥ C; C = B. Conclusion 1 does not necessarily follow. Counterexample A=1, B=0, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 06#

D. Neither conclusion follows

Statements: B ≤ C; C = A; A ≤ D. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=0, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 07#

D. Neither conclusion follows

Statements: B = A; A < D; D = C. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=1, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=1, D=1 satisfies every statement but makes this conclusion false.

Answer 08#

D. Neither conclusion follows

Statements: C > B; B > A; A = D. Conclusion 1 does not necessarily follow. Counterexample A=0, B=1, C=2, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=1, C=2, D=0 satisfies every statement but makes this conclusion false.

Answer 09#

A. Only conclusion I follows

Statements: A ≥ D; D > C; C < B; B < D. Conclusion 1: the possible comparisons under the statements are C < B. Every one is covered by this conclusion, so it must follow. Conclusion 2 does not necessarily follow. Counterexample A=2, B=1, C=0, D=2 satisfies every statement but makes this conclusion false.

Answer 10#

B. Only conclusion II follows

Statements: D = C; C < A; A = B; B = A. Conclusion 1 does not necessarily follow. Counterexample A=1, B=1, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2: the possible comparisons under the statements are D < A. Every one is covered by this conclusion, so it must follow.

Answer 11#

C. Both conclusions I and II follow

Statements: B = D; D ≤ A; A > C; C > B. Conclusion 1: the possible comparisons under the statements are A > D. Every one is covered by this conclusion, so it must follow. Conclusion 2: the possible comparisons under the statements are C > B. Every one is covered by this conclusion, so it must follow.

Answer 12#

D. Neither conclusion follows

Statements: A ≤ D; D ≥ B; B ≥ C; C < A. Conclusion 1 does not necessarily follow. Counterexample A=1, B=0, C=0, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=1, B=0, C=0, D=1 satisfies every statement but makes this conclusion false.

Answer 13#

D. Neither conclusion follows

Statements: B = A; A ≤ D; D > C; D ≥ A. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=0, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=0, D=1 satisfies every statement but makes this conclusion false.

Answer 14#

D. Neither conclusion follows

Statements: B ≤ A; A > C; C ≥ D; C = B. Conclusion 1 does not necessarily follow. Counterexample A=1, B=0, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=1, B=0, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 15#

D. Neither conclusion follows

Statements: D < B; B > A; A ≥ C; C ≤ D. Conclusion 1 does not necessarily follow. Counterexample A=0, B=1, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=1, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 16#

D. Neither conclusion follows

Statements: C > B; B ≤ A; A > D; B < D. Conclusion 1 does not necessarily follow. Counterexample A=2, B=0, C=1, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=2, B=0, C=1, D=1 satisfies every statement but makes this conclusion false.

Answer 17#

A. Only conclusion I follows

Decoded statements: B > A; A = C; C ≤ D. Conclusion 1: the possible comparisons under the statements are A = C. Every one is covered by this conclusion, so it must follow. Conclusion 2 does not necessarily follow. Counterexample A=0, B=1, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 18#

B. Only conclusion II follows

Decoded statements: A > C; C < B; B > D. Conclusion 1 does not necessarily follow. Counterexample A=1, B=1, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2: the possible comparisons under the statements are C < A. Every one is covered by this conclusion, so it must follow.

Answer 19#

C. Both conclusions I and II follow

Decoded statements: D > C; C ≥ B; B > A. Conclusion 1: the possible comparisons under the statements are D > C. Every one is covered by this conclusion, so it must follow. Conclusion 2: the possible comparisons under the statements are B > A. Every one is covered by this conclusion, so it must follow.

Answer 20#

D. Neither conclusion follows

Decoded statements: C ≥ A; A = B; B ≤ D. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=0, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=1, D=0 satisfies every statement but makes this conclusion false.

Answer 21#

D. Neither conclusion follows

Decoded statements: D = B; B ≤ C; C = A. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 22#

D. Neither conclusion follows

Decoded statements: C < A; A < D; D > B. Conclusion 1 does not necessarily follow. Counterexample A=1, B=0, C=0, D=2 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=1, B=0, C=0, D=2 satisfies every statement but makes this conclusion false.

Answer 23#

D. Neither conclusion follows

Decoded statements: A < C; C ≥ D; D < B. Conclusion 1 does not necessarily follow. Counterexample A=0, B=1, C=1, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=1, C=1, D=0 satisfies every statement but makes this conclusion false.

Answer 24#

D. Neither conclusion follows

Decoded statements: C < A; A ≥ B; B ≤ D. Conclusion 1 does not necessarily follow. Counterexample A=2, B=0, C=1, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=1, B=1, C=0, D=1 satisfies every statement but makes this conclusion false.

Answer 25#

A. Only conclusion I follows

Statements: D > C; C ≥ B; B = A; B = C. Conclusion 1: the possible comparisons under the statements are B = A. Every one is covered by this conclusion, so it must follow. Conclusion 2 does not necessarily follow. Counterexample A=0, B=0, C=0, D=1 satisfies every statement but makes this conclusion false.

Answer 26#

B. Only conclusion II follows

Statements: C = D; D > B; B < A; B < A. Conclusion 1 does not necessarily follow. Counterexample A=2, B=0, C=1, D=1 satisfies every statement but makes this conclusion false. Conclusion 2: the possible comparisons under the statements are B < A. Every one is covered by this conclusion, so it must follow.

Answer 27#

C. Both conclusions I and II follow

Statements: D = C; C < A; A ≥ B; B > C. Conclusion 1: the possible comparisons under the statements are D < A. Every one is covered by this conclusion, so it must follow. Conclusion 2: the possible comparisons under the statements are A > C. Every one is covered by this conclusion, so it must follow.

Answer 28#

D. Neither conclusion follows

Statements: A = C; C ≤ D; D > B; A ≥ C. Conclusion 1 does not necessarily follow. Counterexample A=0, B=0, C=0, D=1 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=1, C=0, D=2 satisfies every statement but makes this conclusion false.

Answer 29#

D. Neither conclusion follows

Statements: D < B; B > C; C ≤ A; B < A. Conclusion 1 does not necessarily follow. Counterexample A=2, B=1, C=0, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=2, B=1, C=0, D=0 satisfies every statement but makes this conclusion false.

Answer 30#

D. Neither conclusion follows

Statements: D ≤ A; A < C; C = B; C ≤ B. Conclusion 1 does not necessarily follow. Counterexample A=1, B=2, C=2, D=0 satisfies every statement but makes this conclusion false. Conclusion 2 does not necessarily follow. Counterexample A=0, B=1, C=1, D=0 satisfies every statement but makes this conclusion false.

Online exams#

Four tests contain 20 questions each: direct chains, mixed relations, coded inequalities and compound conclusions. The complete test contains all 80 questions. Decode signs before comparing and evaluate each conclusion separately.

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