Modelling Constraints with Inequalities

Real situations come with limits, and limits have directions. 'At least 10 items' means you can hit 10 or go above it, so it becomes x + y ≥ 10. 'Under 60 dollars' means you can get close to 60 but never touch it, so it becomes 3x + 7y < 60. One points up from its boundary, the other points down toward zero, which is exactly why a spending ceiling and a minimum order look like they're pulling in opposite directions: they are.

Mechanically, every inequality is a line (or plane, with more variables) plus a side. Rewrite it so one variable sits alone, and you can read off how much of the other quantity is still allowed. For buying x items at 3 dollars and y items at 7 dollars, with at least 10 total and under 60 dollars spent, the system is x + y ≥ 10 and 3x + 7y < 60, both restricted to whole, non-negative values since you can't buy a negative or fractional item. Graphed together, the solution is the region where both shaded sides overlap — every point there is a buying plan that works.

Layer on a third limit, say a cap on how many of one item you can buy, and you add one more boundary and one more side to respect. The allowed region can only shrink or hold steady, never grow, because a valid plan now has to clear all three conditions at once. This is the same logic that connects to systems of linear equations: instead of one exact crossing point, you're narrowing down a whole patch of the plane, and each new rule trims that patch further.