Why you should know: The ‘a’ vs ‘an’ conundrum is not about what letter actually begins the word, but instead about how the sound of the word starts.
For example, the ‘h’ in ‘hour’ is silent, so you would say ‘an hour’ and not ‘a hour’. A trickier example is Ukraine: because the ‘U’ is pronounced as ‘You’, and in this case the ‘y’ is a consonant, you would say “a Ukraine” and not “an Ukraine”.
Tip: when in doubt, sound it out(loud).
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Schwa is a vowel, so it would be the long e, not schwa on “the”.
A possible exception is when the following word begins with a long e, and people might actually break the rule to make it clear where one word ends and the other begins. Or rather they insert a glottal stop before the vowel sound - I believe this is called “hard attack” - and since a glottal stop is technically a consonant, that allows the rule-break.
That is, something like “the eel” could go either way, but there’d be a very obvious glottal stop before “eel” if the speaker chose the schwa version of “the”, and they would have made that choice for clarity, to avoid sounding like they’d said “theel”.