a) Solution for compound inequality x > 1 and x > 4 is the INTERSECTION

   of partial solution sets  {x > 1}  and  {x > 4}, which (intersection) is the set  {x > 4}.



b) Solution for compound inequality x > 1 or x > 4 is the UNION

   of partial solution sets  {x > 1}  and  {x > 4}, which (union) is the set  {x 1 4}.


The difference between the first system of inequalities and the second system of inequalities is in the used service word.



In the first system, the service word is "and", and it prescribes to use the intersection of partial solutions.


In the second system, the service word is "or", and it prescribes to use the union of partial solutions.