Ah... our favorite... a bit of **algebra**!

*Alex has 395 cards*

AL = 395

*Michael has 250 cards*

MI = 250

*They both give the same number of cards to their friend Austin*

AU = 2x and AL = 395-x and MI = 250-x

*After giving cards to Austin, the numbers of cards Alex has is less than two times the amount of cards Michael has*

AL < 2 x MI or 395 - x < 2 (250 - x)

=> 395 - x < 500 - 2x

=>** x < 105**

now check boundaries:

*At 105:** AU = 210, AL = 290, MI = 145, 2MI = 290 and 290 = 290 (excluded, equal)*

*At 104:** AU = 208, AL = 291, MI = 146, 2MI = 292 and 291 < 292*

*At 1:** AU = 2, AL = 394, MI = 249, 2MI = 498 and 394 < 498*

*At 0: **AU = 0, AL = 395, MI = 250, 2MI = 500 and 395 < 500 (exluded because they "gave cards")*

*At -10: **AU = -20, AL = 405, MI = 260, 2MI = 520 and 405 < 520 (excluded because can't "give" minus cards)*

I think it's reasonable, based on the phrasing of the question, to exclude giving 0 or -ve number of cards to Austin. Even though this meets inequality x < 105.

**Alex can give between 1 and 104 cards, inclusive, to Austin. Between Alex and Michael, Austin can get between 2 and 208 cards, inclusive.**