To determine the possible range for the distance [tex]\( d \)[/tex] between Lincoln, NE, and the third city, we'll 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 the length of the third side.
Given:
- The distance between Lincoln, NE, and Boulder, CO is 500 miles.
- The distance between Boulder, CO, and the third city is 200 miles.
We can denote these distances as:
- [tex]\( AB = 500 \)[/tex] miles, where [tex]\( AB \)[/tex] represents Lincoln to Boulder.
- [tex]\( BC = 200 \)[/tex] miles, where [tex]\( BC \)[/tex] represents Boulder to the third city.
Let's denote the distance between Lincoln, NE, and the third city as [tex]\( d \)[/tex], or [tex]\( AC \)[/tex].
Now, according to the triangle inequality theorem, for a triangle with sides [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex], the following must be true:
1. [tex]\( AB + BC > AC \)[/tex]
2. [tex]\( AB + AC > BC \)[/tex]
3. [tex]\( BC + AC > AB \)[/tex]
Substituting the given distances into these inequalities:
1. [tex]\( 500 + 200 > d \)[/tex]
2. [tex]\( 500 + d > 200 \)[/tex]
3. [tex]\( 200 + d > 500 \)[/tex]
Simplifying each one:
1. [tex]\( 700 > d \)[/tex] or [tex]\( d < 700 \)[/tex]
2. [tex]\( 500 + d > 200 \)[/tex] which simplifies to [tex]\( d > -300 \)[/tex] (which is always true for positive [tex]\( d \)[/tex])
3. [tex]\( 200 + d > 500 \)[/tex] which simplifies to [tex]\( d > 300 \)[/tex]
Thus, combining the valid inequalities:
[tex]\[ 300 < d < 700 \][/tex]
Therefore, the possible distance [tex]\( d \)[/tex] between Lincoln, NE, and the third city lies within the range:
[tex]\[ 300 < d < 700 \][/tex]
So, the values representing the possible distance [tex]\( d \)[/tex] are [tex]\( 300 \)[/tex] miles and [tex]\( 700 \)[/tex] miles.
