If -19/6<-4z+7, swap sides and change signs as appropriate: 4z-7<19/6. Also 13/6<4z-7.

16z-28=4(4z-7). So multiply the two inequalities by 4: 16z-28<19*4/6 and 13*4/6<16z-28.

So 16z-28<38/3 and 26/3<16z-28. Combine these, putting 16z-28 in the middle: 26/3<16z-28<38/3.

26/3=8+2/3 and 38/3=12+2/3. 16z-28 has to lie between the integers 9 and 12, because the nearest integer greater than 8+2/3 is 9 and the nearest integer smaller than 12+2/3 is 12. So the greater of 12 and 9 is 12. So the greatest possible integer value of 16z-28 is 12. Draw a number line and it will show you that 16z-28 lies between the integers 9 and 12, with 12 being the greatest.