You are almost always better off using Solr in addition to a database
Does the sentence above mean:
You are better NOT using Solr with a database
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"Addition" is the noun form of "add", as in 2+2=4, or in a non-mathematical sense, to combine two or more things in some way.
So "in addition to" means to add one thing to another, that is, to have or use both. When we say "A in addition to B" we normally mean that you already have or plan to get B, and someone is saying that you will or should also get A. In the example you give, the writer is saying that it is expected or assumed that you will use a database, and he is saying you should also use "Solar".
Likewise one could say, "You should wear socks in addition to shoes", meaning we know you plan to wear shoes, but you should also wear socks. Etc.
Solr is a "blazing fast open source enterprise search platform from the Apache Lucene project." This is a software tool.
An equivalent statement might be, "You are almost always better off using a pick in addition to a shovel when digging."