I am trying to differentiate between pressure altitude, the effects of temperature on pressure, and density altitude. This is for my CFI checkride which is coming up in a few weeks.

Pressure altitude is the altitude above the standard plane which would have a pressure of 29.92 inHg. Increases in temperature raise the pressure. As a result, a higher pressure would indicate a LOWER altitude on the altimeter (since higher pressure is found near the surface). Conversely, a decrease in temperature would lower pressure, resulting in a higher pressure indicated on the altimeter. This is general knowledge that a higher temperature gives higher pressure reading and on page 6-2 of the PHAK.

However, we start talking about Density altitude, which is pressure altitude corrected for non standard temperatures. In the PHAK, it says when the temperature is higher, the altitude will be higher than pressure altitude. However, in the paragraph above didn't I just prove the if the temperature is higher, the altitude will be LOWER?