Answer
infinitely many
Work Step by Step
The first inequality represents the region INSIDE OR ON THE CIRCLE with radius $\sqrt{24}$, centered at $(-4,3)$
The second inequality represents the region OUTSIDE OR ON THE CIRCLE with radius $\sqrt{24}$, centered at $(-4,3)$
The intersection of the two solution sets
is the circle itself,
and it has an infinite number of points.
So, there will be infinitely many solutions to the system.
