Given the inequality chain T ≥ U ≥ M = N, which of the following relations must be definitely true according to the expression?

Introduction / Context:
This question checks the ability to read a chained inequality correctly and derive relations that must always hold. You are given a relation involving four symbols T, U, M and N and asked to identify which statement is definitely true in every valid case.

Given Data / Assumptions:

• The expression T ≥ U ≥ M = N is given as true.
• Symbol ≥ means greater than or equal to.
• Symbol = means equality.
• No extra information about the actual values of T, U, M and N is provided.
• We must pick the relation that always holds, not just sometimes holds.

Concept / Approach:
A chain like T ≥ U ≥ M = N can be split into smaller pieces. From T ≥ U and U ≥ M and M = N we can substitute and combine these to get new relations. Any option that contradicts the chain or is only sometimes true cannot be chosen. Only a relation that follows in every allowed arrangement of the four symbols is definitely true.

Step-by-Step Solution:
Step 1: Write the chain as three parts: T ≥ U, U ≥ M and M = N. Step 2: From M = N we know M and N have exactly the same value. Step 3: From U ≥ M and M = N, replace M by N to get U ≥ N. Step 4: From T ≥ U and U ≥ N, we get T ≥ N. Thus T can never be less than N in any valid case. Step 5: Any option that claims T < U, N > U or U < M directly contradicts the chain and must be false.

Verification / Alternative check:
Take an example that satisfies the chain. Let N = 3, M = 3, U = 4 and T = 6. Then T ≥ U ≥ M = N becomes 6 ≥ 4 ≥ 3 = 3, which is true. Here T ≥ N becomes 6 ≥ 3, which is also true. Try another example with equal values, for instance N = M = U = T = 5. The chain still holds and T ≥ N is again true. No valid example can violate T ≥ N, which confirms that it is definitely true.

Why Other Options Are Wrong:
Any option that says T < U or N > U or U < M conflicts with the directions of ≥ in the original chain. The option claiming M > T also contradicts T ≥ U ≥ M. There is no way to assign values so that the chain is true and those relations are simultaneously true, so they cannot be definitely true statements.

Common Pitfalls:
A common mistake is to ignore the equality part M = N or to treat ≥ as if it always meant strictly greater than. Another error is to rely on a single example instead of checking what must happen in all valid examples. The safest method is to break the chain into pieces, substitute step by step and then derive the final relation.

Final Answer:
The relation that must always hold is T ≥ N.