Step 1: Find the ranges for x and y .

- Given \(x^2<81\) , taking the square root gives the range for x :


\(-9<x<9\)


- Given \(y^2<25\) , taking the square root gives the range for y :


\(-5<y<5\)



Step 2: Find the maximum possible upper bound for the expression x-2 y . To make x-2 y as large as possible, you need to maximize x and minimize y :
- The upper bound for x is \(9(x<9)\) .
- The lower bound for y is \(-5(y>-5) \). Multiplying this inequality by -2 reverses the inequality sign:


\(-2 y<10\)



Now, add the two upper bounds together:


\((x)+(-2 y)<9+10
\)


\(x-2 y<19\)



Step 3: Determine the largest prime number within this range.
The value of x-2 y must be strictly less than 19 . Since the question does not state that x and y must be integers, x-2 y can equal any real number up to (but not including) 19.

The largest prime number strictly less than 19 is 17.
(Example check: If x=8 and y=-4.5 , both satisfy the initial conditions, and 8- 2(-4.5)=17 .)

Correct Answer: D) 17