The Omens discussed in this part of the article are unrealistic to acquire in quantity in solo-self-found gameplay. Consider these strategies trade league exclusive.
The true power of Fracturing Orbs is their ability to further constrain what an Orb of Annulment can choose. Especially in conjunction with other tools that constrain Annuls in different ways.
Omens are having their drop rates double-and-a-halved in 0.2.0, although this Omen will likely remain very rare and very valuable. It also cannot drop below monster level 80 (Tier 16 maps or Corrupted and/or Irradiated maps).

By default, an Orb of Annulment can remove any modifier on an item, whether that modifier is wanted or not. If an item has 2 prefixes and 3 suffixes, an Orb of Annulment is 20% to delete any specific modifier on it.
If the item is fractured, however, this becomes 25% to delete any specific non-locked mod, and 0% to delete the locked mod.
The Isolation Principle
What if the locked mod is one of the two prefixes, and you use an Omen of Sinistral Annulment?
In this case, the Omen 'bans' your Annul from deleting the suffixes, and the fracture 'bans' it from deleting the prefix you do not want deleted. Four out of five mods are banned, so the Annul always hits the fifth.
This is the isolation principle - you take an RNG orb like an Annul, and 'isolate' one result by banning all the RNG choices except one of them. It's very commonly used in crafting (in both Path of Exile 1 and 2) and players generally covet the rarer tools that are used to isolate mods.
The Omen of Sinistral Annulment was already able to deterministically remove one prefix from an item in a one prefix, three suffix state, allowing you to exalt a new modifier in its place (and if that modifier is bad, the same process can remove it again).
But with the addition of fracturing, you can improve this to also deterministically remove an exact prefix even in a two prefix, three suffix state, and to grant a 50-50 chance to remove an exact prefix in a three prefix, three suffix state.
In the hands of a very wealthy player (perhaps one experienced in the art of currency flipping), this approach can be used repeatedly until the perfect second prefix and maybe even third prefix is added (via Exalted Orb).
You should always consider the alternative option of an Omen of Whittling, however. These may be better at isolating a specific unwanted modifier than the left/right annul omens in your specific use case.

Omens of Whittling are also extremely rare and extremely valuable, although they can drop in all Tier 11+ maps. They don't really interact with Fracturing Orbs at all, they are simply an alternative to consider.