Inequalities: Understanding "Definitely True"

Inequalities questions test your ability to determine relationships between different elements based on given symbolic expressions. The key is to find a definite path from one element to another that establishes the required relationship without any ambiguity. An expression is "definitely true" if it holds true under all possible interpretations of the given statements.

Analyzing the Problem Statement

The question asks us to identify the option where the expression \($P < M$\) is definitely true. This means we need to trace the relationship between P and M in each given option and see if a clear, unambiguous "P is less than M" connection can be established.

Step-by-Step Analysis of Each Option

1. Option Analysis: M < R > P > S

• The given expression is \($M < R > P > S$\).
• We need to find the relationship between M and P.
• From the expression, we have \($M < R$\) and \($R > P$\).
• Both M and P are shown to be "less than" R. However, there is no direct relationship or connection established between M and P themselves.
• Consider possible scenarios:
  • If R = 10, M = 5, P = 7, then \($M < P$\) (5 < 7).
  • If R = 10, M = 7, P = 5, then \($M > P$\) (7 > 5).
  • If R = 10, M = 5, P = 5, then \($M = P$\) (5 = 5).
• Since M could be less than, greater than, or equal to P, the expression \($P < M$\) is not definitely true.

2. Option Analysis: M > S < P < F

• The given expression is \($M > S < P < F$\).
• We need to find the relationship between M and P.
• From the expression, we have \($M > S$\) and \($S < P$\).
• Both M and P are related to S, but in opposite directions (M is greater than S, P is greater than S). There is a "common element" S, but the relation "opens" away from S for both M and P.
• Consider possible scenarios:
  • If S = 5, M = 10, P = 7, then \($M > P$\) (10 > 7).
  • If S = 5, M = 7, P = 10, then \($M < P$\) (7 < 10).
  • If S = 5, M = 7, P = 7, then \($M = P$\) (7 = 7).
• Since M could be less than, greater than, or equal to P, the expression \($P < M$\) is not definitely true.

3. Option Analysis: Q < M < F = P

• The given expression is \($Q < M < F = P$\).
• We need to find the relationship between M and P.
• From the expression, we have \($M < F$\) and \($F = P$\).
• Combining these, if \($M < F$\) and \($F = P$\), then it definitively means \($M < P$\).
• The question asks for \($P < M$\) to be definitely true. Here, we found \($M < P$\), which is the opposite.
• Therefore, \($P < M$\) is definitely false in this option.

4. Option Analysis: P = A < R < M

• The given expression is \($P = A < R < M$\).
• We need to find the relationship between P and M.
• Let's break down the relationships:
  • \($P = A$\) (P is equal to A)
  • \($A < R$\) (A is less than R)
  • \($R < M$\) (R is less than M)
• Combining \($P = A$\) and \($A < R$\), we can infer \($P < R$\).
• Now, combining \($P < R$\) and \($R < M$\), we can definitively infer \($P < M$\).
• Since all the inequality signs point in the same direction from P towards M (or from M away from P), a definite relationship is established.
• Therefore, the expression \($P < M$\) is definitely true in this option.

Conclusion

Based on the detailed analysis of all options, only in the expression \($P = A < R < M$\) can we definitively conclude that \($P < M$\).