To determine the possible distances, [tex]\( d \)[/tex], between Lincoln, NE, and the third city, given the distances between Lincoln and Boulder (500 miles) and between Boulder and the third city (200 miles), we need to use the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
In our scenario:
- Let [tex]\( a \)[/tex] represent the distance between Lincoln and Boulder, which is 500 miles.
- Let [tex]\( b \)[/tex] represent the distance between Boulder and the third city, which is 200 miles.
- Let [tex]\( c \)[/tex] represent the distance [tex]\( d \)[/tex] between Lincoln and the third city.
We need to find the range of values for [tex]\( c \)[/tex] (or [tex]\( d \)[/tex]).
### Applying the Triangle Inequality Theorem
1. [tex]\( a + b > c \)[/tex]
[tex]\[
500 + 200 > c
\][/tex]
[tex]\[
700 > c \quad \text{or} \quad c < 700
\][/tex]
2. [tex]\( a + c > b \)[/tex]
[tex]\[
500 + c > 200
\][/tex]
[tex]\[
c > 200 - 500
\][/tex]
Since [tex]\( 200 - 500 \)[/tex] is negative:
[tex]\[
c > -300
\][/tex]
Which is always true since distances cannot be negative. We skip this step in the effective range calculation.
3. [tex]\( b + c > a \)[/tex]
[tex]\[
200 + c > 500
\][/tex]
[tex]\[
c > 500 - 200
\][/tex]
[tex]\[
c > 300
\][/tex]
### Gathering the Findings
Combining these inequalities, we get:
[tex]\[
300 < d < 700
\][/tex]
Therefore, the possible distance [tex]\( d \)[/tex] between Lincoln, NE, and the third city is in the range:
[tex]\[
300 < d < 700
\][/tex]
So, the values representing the possible distance are:
[tex]\[
300 < d < 700
\][/tex]
