An exercise from English pronunciation in use (Elementary), J. Marks:
Seven of these numbers have /e/. Which are they?
Three, seven, eight, ten, eleven, twelve, thirteen, seventeen, eighteen, twenty, seventy, eighty, a hundred.
The correct answer: seven, ten, eleven, twelve, seventeen, twenty, seventy.
But what about: eight /eɪt/, eighteen /ˌeɪ'tiːn/ and eighty /'eɪti/?
As pointed out in the comments, the /e/ is a different phoneme from the diphthong /eɪ/.
A vowel can either be a monopthong or a diphthong. A monopthong is a vowel sound in which the speech organs remain in the same position throughout the production of the vowel sound (i.e. the vowel quality doesn't change). The /e/ in words like met, set, let etc., is a monopthong
A diphthong, on the other hand, involves a change in the position of the speech organs (i.e. the vowel quality changes). For example, the vowel in mate, sate, late is a diphthong /eɪ/.
In the OP, eight, eighty, eighteen have the diphthong /eɪ/ not the monopthong /e/, so they're excluded from the words having /e/.
/eɪt/, not/eit/(phonetically[eɪ̯t], not[eit]). 2. Diphthong/eɪ/is different from/e/and/ɪ/together, which can be denoted more explicitly phonetically by writing it as[eɪ̯](as I did) or with theɪin the superscript (like the2in²). For[eɪ̯eɪ]you'd know there's a diphthong there ([eɪ̯]), together with two other vowels:[e]and[ɪ]. 3. Nobody bothers with this in phonemic transcriptions in English as it probably doesn't occur. 4. If we trust CMUdict, there are no words (pronunciations) matchingEH\d? IH\d?.