To determine the possible range for the distance \( d \) between Lincoln, NE, and the third city, given the distances between Lincoln and Boulder (500 miles) and Boulder and the third city (200 miles), we can use the triangle inequality theorem. The theorem states that for any triangle with sides \( a \), \( b \), and \( c \):
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
In this context, let's consider:
- \( a = 500 \): the distance between Lincoln and Boulder.
- \( b = 200 \): the distance between Boulder and the third city.
- \( c = d \): the distance between Lincoln and the third city, which we need to find.
Applying the triangle inequality theorem:
1. \( 500 + 200 > d \)
Simplifying this inequality:
[tex]\[ 700 > d \][/tex]
2. \( 500 + d > 200 \)
Simplifying this inequality:
[tex]\[ d > 200 - 500 \][/tex]
[tex]\[ d > -300 \][/tex]
However, since \( d \) represents a distance and must be positive, this inequality does not provide a restrictive lower bound. Thus, this inequality can be considered always true.
3. \( d + 200 > 500 \)
Simplifying this inequality:
[tex]\[ d > 500 - 200 \][/tex]
[tex]\[ d > 300 \][/tex]
Combining the results from the valid inequalities, we get:
[tex]\[ 300 < d < 700 \][/tex]
Thus, the possible distance \( d \) between Lincoln, NE, and the third city must satisfy:
[tex]\[ \boxed{300 < d < 700} \][/tex]
