A party room holds \(\text{at}~\text{most}~18~\text{people}\), and \(13\) are already there. Let \(g\) be the whole-number count of extra guests; solving gives \(g~\le~5\). Evaluate every statement below in context and report the complete set of valid labels.
A. You can invite \(5~\text{or}~\text{fewer}~\text{more}~\text{guests}\).
B. The number of extra guests must be \(\text{less}~\text{than}~5\).
C. Any whole number of extra guests \(\text{from}~0~\text{through}~5\) works.
D. \(g\) could be \(5.5\) because \(5.5\) is \(\text{less}~\text{than}~\text{or}~\text{equal}~\text{to}~5\).
E. There must be \(\text{exactly}~5~\text{more}~\text{guests}\).
F. The room can hold \(\text{no}~\text{more}~\text{than}~5~\text{additional}~\text{people}\).
What this problem is really about
Use one reference meaning for every statement: g can be any nonnegative whole-number guest count at or below the inclusive bound. Test each wording for four features—direction, boundary inclusion, whole-number validity, and whether it describes the full set rather than one value. Collect every statement that preserves all four features, even when equivalent ideas use different words.