Inequality Questions and Answers

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.

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.

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.

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.

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.