What is the law of inertia? How does it apply to static equilibrium?

This law is something like this (at least, it's how I interpret it):

An object continues its motion or lack of motion at invariant velocities for as long as outside forces do not disrupt its motion in such a way so as to alter its static or dynamic equilibrium state.

This can be called the Law of Inertia because that is essentially the property it describes. There is no formula for this that I'm aware of, but there can be one created that partially describes it:

#sum F = 0#

It doesn't tell you the whole story, however, since it only describes the context in which inertia can be observed, not the inertial property of the object.

As an example of static equilibrium, let's say you have a box on a table. Of course, if the table is strong enough and perfectly flat, no friction is necessary to ensure that the box won't move anywhere. The box is said to be at rest. If someone nudges the box, the box can be moved, but as soon as they stop nudging it, the box stops moving and goes back to its static equilibrium state.

This demonstrates the box's property of inertia in that its original state is restored when your contact force no longer influences its state.

You could say that, with only the normal force of the table and the force due to gravity acting:

#sum F_y = F_g - F_N = 0#

#sum F_x = 0#

When you have your force working on it, let's say it's #F_i = "2 N"#. Then:

#sum F_x = F_i - F_s = "2 N" - mu_sF_N#

in some #hati# (horizontal) direction, where #F_s# is the resistive force due to static friction.

But again, it's only describing the forces on the object while the object is being moved, not exactly the inertial property of the object, which is the actual focus of the law.

(It can seem subtle, but writing a formula describing the forces acting on an object that has inertia is not the same as writing a formula that describes an object's inertial property.)