So let's take up the case where a=8. Then, as above, since a is not 9, we infer that the original reading was x98888. As we add miles one-by-one up to 2, we get the right palindromes; and when the third mile is added, we get the reading x98891. Since this is a palindrome, we now have x=1, and 198,888 on the odometer at the outset.
The only issue now is to determine that there are no other solutions lurking about; i.e., that a can't be 9. Indeed, suppose it were, so that the original reading looked like xy9bb9. Here's where we have to be careful about carry digits. If b were 9 as well, then--because the last 5 digits are not palindromic-- we know that y is not 9. Hence one more mile gives the reading x(y+1)0000. But the last five digits now form a palindrome, making y+1=0, an impossibility. So the only way for a to be 9 is for b to be no more than 8. Thus when we add one mile, the reading is xy9b(b+1)0, making y=0 and b=8. This makes the original reading x09889. But then, when we add the second mile, the reading is x09891, with the middle four digits not forming a palindrome. This last contradiction tells us there are no solutions other than 198,888.