Inspiration

Inspiration is more complicated than you may realize

Before you breath in, pressure in the lungs is the same as barometric pressure. Alveolar pressure in this situation is zero, meaning that there is zero difference between lung pressure and barometric pressure. Also before breathing in, your chest wall is naturally expanded a little while your lungs are naturally contracted a little. This creates a little pocket of low pressure in the pleural space between the chest wall and the lungs. The lungs can't contract too much because they are at a higher pressure than the intrapleural space. If the lungs weren't always at a higher pressure than the pleural space, they WOULD collapse (pneumothorax)

So you decide to breath in and as your chest wall expands, pressure in the pleural space decreases even more, extenuating the pressure gradient between lung and pleural space. Now the pressure in the lungs is so much greater than pressure in the pleural space that the lungs expand. As the lungs expand, lung volume naturally increases, lowering the alveolar pressure to a point well below barometric pressure. As a result, air now flows from high to low pressure, i.e. from outside into your lungs.