The hole idea of RS232 is the voltage swing from ±5 to 12 V for noise immunity.
To make it more confusing your +5 to 10V is a zero and -5 to 12 V is a 1.

Most of the RS232 devices will recognize logic levels 0-5V from a TTL source but you do not have the built in logic inverter, so you must ask your PIC to do it for you by setting the Invert mode.

The resistor solution is simple but your noise immunity is no longer there. You will have to use shielded cables and short runs.

If you are dealing with a harsh environment you must use the RS232 chip and still limited to relatively short runs (I think 50’ maximum) or go RS485 for fairly longer distances without dealing with charge pumps. The charge pumps in the RS232 chips just generate the ±5 to 12 V and invert your TTL signals so you can use one single 5V supply.

Regards,

Nick