Final answer:
The nurse should first use the head-tilt, chin-lift method to open the airway.
Step-by-step explanation:
The nurse should first take the action of using the head-tilt, chin-lift method to open the airway in order to address the client's swollen tongue obstructing the airway. This technique involves gently tilting the head back and lifting the chin to help ensure proper airway alignment and prevent airway obstruction. By using this method, the nurse can immediately address the client's breathing difficulty caused by the swollen tongue.