The mo. Septic tank inspector said that I need to run my gray water [washer ]into the septic tank doesn't this hurt the septic system?

The mo. septic tank inspector said that i need to run my gray water [washer ]into the septic tank
Your inspector gave you advice that could clog your system. Kitchen, especially if you have a garbage disposal) and washer discharge should be routed through a grease trap, (see image) before going into the septic tank and out to the drain field.
The reason for this is to trap and contain the lint and fiber from the washer combined with the grease and garbage from the kitchen before they get out in the drainfield and clog up the holes in the drainfield tiles so they can no longer dissipate the liquid from the septic tank.
When that happens you're faced with the expense of repairing the drainfield. In my area septic systems are never inhstalled with out a grease trap. It only makes good sense. Good luck, tom

In my opinion, modern septic systems are designed to handle laundry waste. Use biodegradeable detergent and limit chlorine use and space out laundry loads. If your laundry is discharging to a ditch, that is a violation.