Attacking with two weapons

I know we had a question about attacking with two weapons, so I would like to clarify the ruling:

As per the DMG pg 70

Attacks With Two Weapons:
Characters normally using a single weapon may choose to use one in each
hand (possibly discarding the option of using a shield). The second
weapon must be either a dagger or hand ax. Employment of a second
weapon is always a penalty. The use of a second weapon causes the
character to attack with his or her primary weapon at -2 and the secondary
weapon at -4. If the user's dexterity is below 6, the reaction/attacking Adjustment
penalties shown in the PLAYERS HANDBOOK are added to EACH
weapon attack. If the user's dexterity is above 15, there is a downward adjustment
in the weapon penalties as shown, although this never gives a
positive (bonus) rating to such attacks, so that at 16 dexterity the secondary/
primary penalty is -3/-1, at 17 -2/0, and at 18 -l/O.
The secondary weapon does not act as a shield or parrying device in any
event.