2018 AMC 12A Problem 4

Problem 4 of 25EasierAlgebra

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, “We are at least

6

miles away,” Bob replied, “We are at most

5

miles away.” Charlie then remarked, “Actually the nearest town is at most

4

miles away.” It turned out that none of the three statements was true. Let

d

be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of

d?

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Solution

Negating the three false statements gives

d \lt 6,

d \gt 5,

and

d \gt 4.

The intersection of these conditions is

5 \lt d \lt 6,

that is, the interval

(5, 6).

Thus, the correct answer is D.