If |4𝑥−8|=8 and  |5𝑦+5|=25, what is the greatest possible value of xy?

First, let's solve the absolute value equations separately.

For |4x - 8| = 8, we have two possible solutions:

1. 4x - 8 = 8, which simplifies to 4x = 16, and then x = 4.
2. -(4x - 8) = 8, which simplifies to -4x + 8 = 8, and then -4x = 0, and x = 0.

So, the possible values for x are 0 and 4.

For |5y + 5| = 25, we also have two possible solutions:

1. 5y + 5 = 25, which simplifies to 5y = 20, and then y = 4.
2. -(5y + 5) = 25, which simplifies to -5y - 5 = 25, and then -5y = 30, and y = -6.

So, the possible values for y are 4 and -6.

Now, we need to find the greatest possible value of xy. We have four combinations to check:

1. x = 0, y = 4, so xy = 0.
2. x = 0, y = -6, so xy = 0.
3. x = 4, y = 4, so xy = 16.
4. x = 4, y = -6, so xy = -24.

Therefore, the greatest possible value of xy is 16.