Suppose the post office sold only five-cent stamps and seven-cent stamps. Some amounts of postage can be made with just those two kinds of stamps. For example, 1 five-cent and 2 seven-cent stamps make 19 cents in postage, and 2 five-cent stamps makes 10 cents in postage. Which amounts of postage is it impossible to make using only five-cent and seven-cent stamps

You can make any multiple of 5 with those stamps. Also, you can make any number x with stamps proven that that number is big enough. If you start with a number that is a multiple of 5, then you can increase that number by 2 each time you replace a 5 with a 7 (in each example, we are starting in a postage of value 5j made by 5-cent stamps):

If x = 5j + 1, for an integer j, (j ≥ 4), then you can replace 4 five cent stamps with 3 7 cent stamps, to obtain x.
If x = 5j + 2, you can replace 1 five cent stamp with one seven cent stamp to obtain x (j≥1)
If x = 5j+3, you can replace 5 five cent stamps with 4 seven cent stamps to obtain x (j ≥5).
If z = 5j+4, you can replace 2 five-cent stamps with 2 seven-cent stamos to obtain x (j≥2)

Note therefore, that if j is at least 5, then we can obtain a postage with value x. This means that any postage of value at least 25 can be made witg five-cent of seven-cent stamps. Lets see for smaller values

1, 2, 3 and 4 cent postage are impossible to make
5 cent package is possible, so is 7
6,7, 8 and 9 are impossible to make (any package with at least 2 stamp has a value at least 10)
10 is possible (5+5)
11 is not possible (5+5 = 10, 5+7 = 12)
12 is possible (5+7)
13 is impossible (5+7 = 12, but 7+7 = 14. Also any combination of 3 stamps has a value at least 15)
14 is possible (7+7)
15 is possible (5+5+5)
16 is not possible (5+5+5 = 15, 5+5+7 = 17)
17 = 5+5+7 is possible
18 is not possible (5+7+7 = 19, and any postage with 4 stamps has a value of 20 or higher)
19 = 5+7+7 is possible
20 = 5+5+5+5 is possible
21 = 7+7+7 is possible
22 = 5+5+5+7 is possible
23 is not possible (you cant reach 23 with 3 stamps, and you will get a value at least 25 with 5. If you use 4 stamps, you can obtain only even values)
24 = 5+5+7+7 is possible

So we conclude that the amounts of postage that are impossible to make are: 1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18 and 23 cents.