Question

translate the verbal phrase and interpret, letting n represent the value(s)
at most eight people per room

translate the verbal phrase. choose the correct answer below.
a. n = 0,1,2,3, …,8
b. n = -8, -7, -6, -5, …,0
c. n = 1,2,3, …,8
d. n = 8,9,10,11, …

interpret the phrase. choose the correct answer below
a. there cannot be a negative number of people. there could be one, two, … or eight people. people cannot be represented with a fractional number.
b. there cannot be a negative number of people. there could be a minimum of eight people, and there is no limit on the maximum number of people. people cannot be represented with a fractional number.
c. there can be a negative number of people. there could be negative eight, negative seven, … or zero people. people cannot be represented with a fractional number.
d. there cannot be a negative number of people. there could be zero, one, two, … or eight people. people cannot be represented with a fractional number.

Explanation:

Step1: Analyze "At most eight people per room"

"At most eight" means \( n \leq 8 \), and \( n \) represents the number of people, so \( n \) is a non - negative integer (0, 1, 2, …, 8).

• Option A: \( n = 0,1,2,3,\dots,8 \) matches \( n\leq8 \) and non - negative integer.
• Option B: Negative numbers don't make sense for the number of people.
• Option C: Starts from 1, but 0 people (empty room) is possible.
• Option D: \( n\geq8 \) is "at least eight", not "at most eight".

Step2: Analyze the interpretation

• Option A: Says "one, two, … or eight", but 0 is also possible.
• Option B: "At least eight" is wrong, we need "at most eight".
• Option C: Negative number of people is impossible.
• Option D: Correctly states that there can't be negative people, and \( n \) can be 0, 1, 2, …, 8 (since you can't have a fraction of a person).

Answer:

For the translation: A. \( n = 0,1,2,3,\dots,8 \)
For the interpretation: D. There cannot be a negative number of people. There could be zero, one, two, … or eight people. People cannot be represented with a fractional number