Best use a Venn diagram, where circles are used to enclose the various sets. Start with a big circle representing 25 students. Inside the big circle are two interlocking circles, one for those needing a dental checkup and the other needing an eye checkup. Outside the interlocking circles, but within the big circle, write the number 5, representing the students needing neither checkup. Write 11 in the circle for dental checkups and 17 in the one for eye checkups. These numbers apply to the whole of each circle. Where the circles overlap is the area for students requiring both checkups, which we don't know yet.
The picture should help. In the big circle we have 5, so the interlocking circles between them must contain 25-5=20 students requiring one or both checkups. We have 11 students in one circle and 17 in the other, but we know there are only 20 students altogether in these two circles so the overlapping area must contain 11+17-20 students, that is 8 students, needing both checkups.