sounds like a simple ripple edit?
mark the area you want to extract/ delete with in/outs points, then Shift + Delete (Or forward delete)
you could cause some unwanted edits in some layers this way, which you can then tidy up with the roll tool (hit r to summon)
the alternative is to mark the area with in/outs,
then summon up the RIPPLE tool (rr) and select the various edit points across the 6 tracks,
then enter a number on the number pad for how much you want to ripple by (the duration of the in/outs.
if you haven't used the ripple tool this could take a little getting used to, as you need to select the edit on the side you want to trim.
hold apple while you select other edit points on other tracks.
you can even do what they call Asymmetrical trimming, which is where different tracks extend/contract in different directions, depending on the side of the edit you select with the ripple tool.
powerful stuff!
nick