Macos – The volume Macintosh HD was found corrupt and needs to be repaired

My Macbook Air (first gen) is very slow to start up probably because of the volume Macintosh HD is corrupt.

It takes infinity time to startup in normal mode and it takes about 1 hour to start up in safe mode.

In safe mode, I tried to run the "Disk Utility" and to execute "Repair Disk Permission" and "Repair Disk" without success.

Here is the logout (1) of "Disk Utility".
I tried to repair disk doing boot from external DVD but it takes infinity time to load programs from dvd (I did wait for about 3 hours without any result).
I tried also to mount a USB pen in safe mode to buck-up data without success.

Any idea?

Thanks,
Antonio

(1)

Verify permissions for “Macintosh HD”
Warning: SUID file "System/Library/CoreServices/RemoteManagement/ARDAgent.app/Contents/MacOS/ARDAgent" has been modified and will not be repaired.

Permissions verification complete

Repairing permissions for “Macintosh HD”
Warning: SUID file "System/Library/CoreServices/RemoteManagement/ARDAgent.app/Contents/MacOS/ARDAgent" has been modified and will not be repaired.

Permissions repair complete

Verifying volume “Macintosh HD”
Performing live verification.
Checking Journaled HFS Plus volume.
Checking extents overflow file.
Checking catalog file.
Checking multi-linked files.
Checking catalog hierarchy.
Checking extended attributes file.
Checking volume bitmap.
Checking volume information.
Invalid volume file count
(It should be 675058 instead of 675059)
Invalid volume directory count
(It should be 189754 instead of 189753)
The volume Macintosh HD was found corrupt and needs to be repaired.
Error: This disk needs to be repaired. Start up your computer with another disk (such as your Mac OS X installation disc), and then use Disk Utility to repair this disk.