Inequality Questions and Answers

Q: Statements K*H ,H%G ,G$I Conclusions I.H$I II.G#K III.K%I

Correct answer: All of the aboveTranslating the code: $@$ is “greater than”, $\#$ is “greater than or equal to”, $\%$ is “less than or equal to”, $*$ is “equal to” and $\$$ is “less than”. The statements read $ K = H $, $ H \le G $ and $ G < I $, which chain into the single line \[ K = H \le G < I. \] Conclusion I is $ H < I $: from $ H \le G < I $ it follows that $ H < I $ - true. Conclusion II is $ G \ge K $: since $ K = H $ and $ H \le G $, we have $ G \ge K $ - true. Conclusion III is $ K \le I $: $ K = H \le G < I $ gives $ K < I $, which satisfies $ K \le I $ - true. All three follow, so the answer is “All of the above”. As in the previous item, III is easy to reject by mistake: $ K \le I $ asks only that K is not greater than I, and $ K < I $ meets that.

Q: Which of the following order of letters in the blanks makes the expression D > E is definitely true? ___ < ___ ≤ ___ = ___ > ____

Correct answer: C, A, D, B, EFilling the blanks gives C < A ≤ D = B > E. The last two terms then read D = B and B > E, so D > E follows directly. What matters is that D is on the left of the equals and E on the right of the greater-than, so the two are joined without any reversal between them.

Q: Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the given expression in such a manner that makes the expressions ‘B>N’ as well as ‘D <= L’ definitely true? B__L__O__N__D

Correct answer: >, =, =, >=Substituting the keyed symbols into $ B \_ L \_ O \_ N \_ D $ gives \[ B > L = O = N \ge D. \] For $ B > N $: L, O and N are all equal, and B exceeds L, so $ B > N $ - definitely true. For $ D \le L $: the last link gives $ N \ge D $, and $ N = L $, so $ D \le L $ - definitely true. The other sets break one requirement or the other. The first, $ =,=,\ge,\ge $, makes $ B = L $, so $ B > N $ cannot be guaranteed. The third, $ >,<,=,\le $, points L below O and so leaves B and N unlinked. One caveat worth noticing: the set $ >,=,\ge,> $ yields $ B > L = O \ge N > D $, which also makes both required expressions true, so this item admits more than one working answer.

Q: Which of the following symbols should replace the question mark(?) in the given expression in order to make the expressions ‘P>A’ as well as ‘T<L’ definitely true? P > L ? A > = N = T

Correct answer: >The expression is $ P > L \;?\; A \ge N = T $, and both $ P > A $ and $ T < L $ must be definitely true. Test the keyed symbol first. With $\ge$ the line reads $ P > L \ge A \ge N = T $. That does give $ P > A $, but for the second requirement it gives only $ T = N \le A \le L $, that is $ T \le L $; taking $ L = A = N = T $ satisfies every statement while making $ T < L $ false, so $\ge$ fails. With $>$ the line reads $ P > L > A \ge N = T $. Now $ P > L > A $ gives $ P > A $, and $ T \le A < L $ gives $ T < L $ - both hold, and the answer is $>$. The remaining choices reverse the chain at L: $<$ or $\le$ leaves P and A unrelated, since both would then sit above L rather than in a single chain.

Q: Statements: M≤V; P<R; P=M; S>T; R≤T Conclusions: I. S>M II. V=P III. P<V IV. T<P

Correct answer: Either II or III and I are trueCombine the statements into what they jointly say. $ P = M $ ties two letters together, and $ P < R \le T < S $ runs from there up to S, while $ M \le V $ hangs V off the same point from below: \[ V \ge M = P < R \le T < S. \]I. $ S > M $: reading right from M gives $ M = P < R \le T < S $, strict twice. True.IV. $ T < P $: the same run shows $ T > P $, so this is definitely false.II and III are the pair to think about. All we know of V is $ V \ge P $, from $ M \le V $ and $ P = M $. II claims $ V = P $ and III claims $ P < V $; neither is forced, because V could equal P or stand above it. But those two cases exhaust $ V \ge P $, so exactly one of them holds.So I is definitely true, IV is definitely false, and either II or III is true - which is the keyed answer.

Important Instructions

91.

Statements

K*H ,H%G ,G$I

Conclusions

I.H$I II.G#K III.K%I

A.only I is true
B.I and II is true
C.only III i true
D.None is true
E.All of the above

Answer: B

Correct answer: All of the above

Translating the code: $@$ is “greater than”, $\#$ is “greater than or equal to”, $\%$ is “less than or equal to”, $*$ is “equal to” and $\$$ is “less than”. The statements read $ K = H $, $ H \le G $ and $ G < I $, which chain into the single line \[ K = H \le G < I. \] Conclusion I is $ H < I $: from $ H \le G < I $ it follows that $ H < I $ - true. Conclusion II is $ G \ge K $: since $ K = H $ and $ H \le G $, we have $ G \ge K $ - true. Conclusion III is $ K \le I $: $ K = H \le G < I $ gives $ K < I $, which satisfies $ K \le I $ - true. All three follow, so the answer is “All of the above”. As in the previous item, III is easy to reject by mistake: $ K \le I $ asks only that K is not greater than I, and $ K < I $ meets that.

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92.

Which of the following order of letters in the blanks makes the expression D > E is definitely true?
___ < ___ ≤ ___ = ___ > ____

A.B, C, D, E, A
B.A, B, C, E, D
C.D, B, A, C, E
D.E, C, A, B, D
E.C, A, D, B, E

Answer: E

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

Filling the blanks gives C < A ≤ D = B > E. The last two terms then read D = B and B > E, so D > E follows directly. What matters is that D is on the left of the equals and E on the right of the greater-than, so the two are joined without any reversal between them.

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93.

Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the given expression in such a manner that makes the expressions ‘B>N’ as well as ‘D <= L’ definitely true?

B__L__O__N__D

A.=,=,>=,>=
B.>,>=,=,>
C.>,<,=,<=
D.>,=,=,>=
E.>,=,>=,>

Answer: D

Correct answer: >, =, =, >=

Substituting the keyed symbols into $ B \_ L \_ O \_ N \_ D $ gives \[ B > L = O = N \ge D. \] For $ B > N $: L, O and N are all equal, and B exceeds L, so $ B > N $ - definitely true. For $ D \le L $: the last link gives $ N \ge D $, and $ N = L $, so $ D \le L $ - definitely true. The other sets break one requirement or the other. The first, $ =,=,\ge,\ge $, makes $ B = L $, so $ B > N $ cannot be guaranteed. The third, $ >,<,=,\le $, points L below O and so leaves B and N unlinked. One caveat worth noticing: the set $ >,=,\ge,> $ yields $ B > L = O \ge N > D $, which also makes both required expressions true, so this item admits more than one working answer.

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94.

Which of the following symbols should replace the question mark(?) in the given expression in order to make the expressions ‘P>A’ as well as ‘T<L’ definitely true?

P > L ? A > = N = T

A.<=
B.>
C.<
D.>=
E.Either <=or<

Answer: D

Correct answer: >

The expression is $ P > L \;?\; A \ge N = T $, and both $ P > A $ and $ T < L $ must be definitely true. Test the keyed symbol first. With $\ge$ the line reads $ P > L \ge A \ge N = T $. That does give $ P > A $, but for the second requirement it gives only $ T = N \le A \le L $, that is $ T \le L $; taking $ L = A = N = T $ satisfies every statement while making $ T < L $ false, so $\ge$ fails. With $>$ the line reads $ P > L > A \ge N = T $. Now $ P > L > A $ gives $ P > A $, and $ T \le A < L $ gives $ T < L $ - both hold, and the answer is $>$. The remaining choices reverse the chain at L: $<$ or $\le$ leaves P and A unrelated, since both would then sit above L rather than in a single chain.

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95.

Statements: M≤V; P<R; P=M; S>T; R≤T
Conclusions:
I. S>M
II. V=P
III. P<V
IV. T<P

A.Only I is true
B.Only II is true
C.None is true
D.Either II or III and I are true
E.All are true

Answer: D

Correct answer: Either II or III and I are true

Combine the statements into what they jointly say. $ P = M $ ties two letters together, and $ P < R \le T < S $ runs from there up to S, while $ M \le V $ hangs V off the same point from below: \[ V \ge M = P < R \le T < S. \]

I. $ S > M $: reading right from M gives $ M = P < R \le T < S $, strict twice. True.

IV. $ T < P $: the same run shows $ T > P $, so this is definitely false.

II and III are the pair to think about. All we know of V is $ V \ge P $, from $ M \le V $ and $ P = M $. II claims $ V = P $ and III claims $ P < V $; neither is forced, because V could equal P or stand above it. But those two cases exhaust $ V \ge P $, so exactly one of them holds.

So I is definitely true, IV is definitely false, and either II or III is true - which is the keyed answer.

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