得(え)ないUnable to, Cannot

得(え)る is a る-Verb that has the meaning of 'to gain' or 'to acquire'. When attached to the conjunctive form of verbs, it takes on the nuance of 'for (A) to be possible', or 'able to (A)'. In addition to this, when 得(え)る is combined with ない, that nuance switches to 'unable to (A)', or 'cannot (A)'.

Compared to できない, meaning that the potential to do something does not exist, 得(え)ない means something closer to 'there is no possibility that (A) can be done', or 'the possibility to do (A) has not been gained'.

Caution

得(え)る may be seen as either える or うる. The primary difference between the two being that うる is a relatively old form that is often seen only in literature, and almost never spoken. Despite this, うる is never used in the negative form with ない.

Fun-fact

得(え)る is far more formal/bookish than できる. However, there are several set phrases that are very common in daily speech that use this particular grammar point, such as あり得(え)る 'possible', and あり得(え)ない 'impossible'.