The distance between lincoln, ne, and boulder, co, is about 500 miles. the distance between boulder, co, and a third city is 200 miles. assuming the three cities make a triangle on the map, which values represent the possible distance, d, in miles, between lincoln, ne, and the third city?

The sum of the lengths of any two sides of a triangle is greater than the length of the third side (Triangle Inequality Theorem)

Let 

A> Lincoln, NE

B> Boulder, CO

C> third city

we know that

in the triangle ABC

AB=500 miles

BC=200 miles

AC=x

Applying the Triangle Inequality Theorem

1) 500+200 > x> 700 > x> x < 700 miles

2) 200+x > 500> x > 500-200> x > 300 miles

the solution for x is 

300 < x < 700

the interval is> (300,700)

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