2018 AMC 12A Problem #4
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 were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?
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1 solution
Pretty straightforward. Since none are true, we can take the reverse of each statement.
We are less than 6 miles away, so d < 6 .
We are more than 5 miles away, so d > 5 .
We are more than 4 miles away, so d > 4 .
The union of these statements is 5 < d < 6 ⟹ ( 5 , 6 ) .