Let's define the amounts of money that Joel and Hazel have:
Let \( H \) be the amount Hazel has initially.
According to the problem, Joel has $80 less than Hazel, so:
\[ J = H - 80 \]
After Hazel spends $128 on a vacuum cleaner, her remaining amount will be:
\[ H - 128 \]
If you want to know how much money they each have after Hazel's purchase or any further calculations, please provide more details or questions!