I have the following code to show a red area defined by inequalities:
ClearAll["Global`*"];
p = Reduce[y <= 3/10 x + 18 && y > x^2/8, {x, y}]
r = RegionPlot[p, {x, -15, 18}, {y, -5, 25},
GridLines -> {Table[i, {i, -15, 18}], Table[j, {j, -5, 25}]},
PlotStyle -> Directive[{Opacity[0.5], Red}]];
bg = Graphics[{Opacity[0.2], Yellow, Rectangle[{-16, -6}, {19, 26}]}];
range = First /@
Differences /@ (PlotRange /. Options[r]); target = 1; Show[{r, bg},
AspectRatio -> (Last[range]/First[range]/target)]
which shows:
How can I count those integer grids and highlight them with colored dots?
Reduce
can't be efficient however a neat approach withFrobeniusSolve
can yield all interesting solutions as in one of the answers in the link. For benchmarks see this answer Finding the number of solutions to a diophantine equation. $\endgroup$