To determine the possible distance [tex]\( d \)[/tex] between Lincoln, NE, and the third city, given that the distance from Lincoln to Boulder is 500 miles and from Boulder to the third city is 200 miles, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the remaining side. Here is a step-by-step explanation to find the range for [tex]\( d \)[/tex]:
1. Identify given distances:
- Distance between Lincoln, NE, and Boulder, CO: 500 miles
- Distance between Boulder, CO, and the third city: 200 miles
2. Use the triangle inequality theorem:
- The sum of any two sides must be greater than the third side.
3. Derive inequalities:
- For the side [tex]\( d \)[/tex] (distance between Lincoln and the third city):
[tex]\[
\begin{align*}
d + 200 &> 500 \\
d + 500 &> 200 \\
500 + 200 &> d
\end{align*}
\][/tex]
4. Simplify the inequalities:
- From [tex]\( d + 200 > 500 \)[/tex]:
[tex]\[
d > 500 - 200 \implies d > 300
\][/tex]
- From [tex]\( d + 500 > 200 \)[/tex]:
[tex]\[
d > 200 - 500 \implies d > -300 \quad\text{(since distances cannot be negative, this inequality does not impose any additional constraints)}
\][/tex]
- From [tex]\( 500 + 200 > d \)[/tex]:
[tex]\[
d < 700
\][/tex]
5. Combine the inequalities:
- The distance [tex]\( d \)[/tex] must be greater than 300 and less than 700.
[tex]\[
300 < d < 700
\][/tex]
Therefore, the values that represent the possible distance [tex]\( d \)[/tex] in miles between Lincoln, NE, and the third city are [tex]\( 300 < d < 700 \)[/tex].
