Clarification about demand elasticity and monopoly

I am having a difficulty understanding certain things about elasticity and would like to clarify them.

What I understand. Elasticity is the percent change in quantity due to a percent change in price, that is, how does output change due to a change in price. A good is inelastic, if there is only 1(or few) producers of this firm, since, there are no perfect substitutes for this good. Which implies that if the price where to increase quantity would not decrease as consumers have no other options. On the other hand, a good is elastic, when there are various other firms selling this same good. Thus if there is an increase in price of a good, consumers will chose to purchase from other firms who offer the same product.

Now, this is where I get really confused.
The sentence that "monopolists always operate on the elastic portion of their demand curve." The reason this confuses me, is that I thought if you are a monopoly then your good is inelastic as mentioned above.

The only explanation that I can provide to myself is that there are two levels of elasticity going on here. That is 1. elasticity between monopoly and perfect competition and 2. depending on which one there is another elasticity inside. To further explain what I mean, is if for instance we are dealing with a monopolist, we know that his demand will be inelastic. But not within this inelastic demand there is an elastic and inelastic portion. Am I completely incorrect? Please help me understand this concept.

If the assumption that there is one elasticity within the other is incorrect, this is my ultimate question.

If monopolists have inelastic demand (since they are the only producer of this good) how can they operate on an elastic portion?