Does the ideal gas law apply to a moving vehicle? If for example you have a car travelling along a road. There is obviously a high pressure region in front of the car, where the air is forced around the vehicle. Likewise, I understand that cars will leave a region of low pressure air directly behind them.
Air is described as being incompressible below mach 0.3. Therefore, this would suggest that the density of the air would be the same at the front of the car as it is behind it.
However, this would contradict the ideal gas law which suggests that for there to be a change in pressure in the air, there must also be a change in density.
 A: You can model a gas as incompressible. But, that does not mean that it is incompressible.
The difference between compressible and incompressible arises in how you treat a kinetic energy term in the Bernoulli equation. With incompressible fluids, the term can be determined directly because conservation of mass flow also means conservation of volumetric flow. With compressible fluids, the kinetic energy term must include considerations for changes in the pressure and temperature of the gas.
Consider this reference. You will see the term $mv((p_i/p_e) - 1)$. A statement of being below Mach 0.3 is equivalent to stating that you have a low velocity $v$. The kinetic energy term is negligible relative to the pressure drop term.
Alternatively, you can also assume that a gas is incompressible when the ratio of the initial and end pressure is near to unity. This will also remove the difference in kinetic energies front to back.
A: "Incompressible" means different things in different contexts. 


*

*In thermodynamics, an "incompressible substance" is one whose properties are completely insensitive to pressure. This is a good model for solids and liquids but not for gases.

*In fluid dynamics, "incompressible flow" means that the velocities present are not high enough to cause significant density changes. 
The statement "air is incompressible below Mach 0.3" refers to incompressibility in the fluid dynamical sense; it means that, in air, velocities below Mach 0.3 are not capable of causing significant density changes. The statement does not imply that air is an incompressible substance; unless the temperature is very very low or the pressure is very very high, air is best modelled as an ideal gas and not an incompressible substance.
