The annual Cheese Rolling happens in May at Gloucestershire, England. As the name suggests, large, 7-9 pound wheels of cheese are rolled down a steep hill, and people chase after them. The first person to the bottom wins cheese. Renaldo wants to go to the Cheese Rolling: He plans to leave from Atlanta and fly into London ( 4281 miles). On the return, he will fly back from London to New York City ( 3470 miles) to visit his aunt. Then Renaldo heads back to Atlanta. Atlanta, New York City, and London do not lie on the same line. Find the range of the total distance Renaldo could travel on his trip.

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According to the Triangle Inequality Theorem the sum of any two sides of a triangle must be greater than the measure of the third side. If $a, b$ are two known sides and $x$ is the unknown third side, we have (assume that $a\leq b$): $$\begin{align*} \begin{cases} a+b>x\ a+x>b\ b+x>a \end{cases}\ \begin{cases} a+b>x\ x>b-a\ x>a-b \end{cases}\ \end{align*}$$ As $a\leq b$ we can write: $$\begin{align*} \begin{cases} a+b>x\ x>b-a\ \end{cases}\ or\ b-a

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