A. After final action, or in case no objections are received, the board may order its inspector to enter upon the premises described and eradicate the noxious weeds, shrubs, vines or grasses thereon. The inspector and his assistants are authorized to enter upon private property for that purpose.
B. The costs of eradication shall constitute a valid claim against the antinoxious weed district and shall be paid as other claims against the district are paid.