A certain satellite has a kinetic energy of 8.5 billion joules at
perigee (closest to earth) and 4.5 billion joules at apogee (farthest
from earth). As the satellite travels from apogee to perigee, how much
work does the gravitational force do on it?

Under ideal conditions the path of an orbiting satellite is a
composition of its inertia & its radial acceleration, with its total
energy (kinetic + potential) being constant. The work done by gravity
then is just the difference between the two extremes' kinetic
energies: 4.5 billion - 8.5 bilion = -4 billion. The work in this case
is negative.

Try looking at it this way:

Is gravity always pulling on the satellite, whether it is moving
towards the earth or away from the earth?

As the satellite gets closer to earth (the opposite of the situation
presented in the problem), what happens to its kinetic energy? What is
the equation relating velocity and kinetic energy? What caused the
increase in velocity / energy as it gets closer to earth? What force
is working on the satellite to pull it closer to earth? How is this
work reflected in the change in velocity / kinetic energy?

Then, what is the satellite working against (being opposed by) as it
moves from being closer to earth to being furthest from earth? How is
the work done reflected in the satellite's velocity / kinetic energy
in this case?