Linear Inequalities & Graphs at the 700-800 Level

Hard-tier inequality items are small optimization problems: several constraints, with the answer sitting at whichever constraint binds first.

Binding constraints. When two resources limit output (boards and bolts, rods and fabric), compute the limit each imposes and take the smaller. The test deliberately makes the ignored constraint produce the more attractive number, then prices it into revenue asks.

Substitute-and-combine. Given a point like (6, b) or (k, 9) in a system, substitute the known coordinate into BOTH inequalities, get a bound from each, and keep the tighter one. Strict inequalities exclude their boundaries: b < 4 makes the greatest integer 3, and the excluded boundary value 4 is always an option. When one bound is strict and one is not, strictness alone can decide the answer.

Requirements plus budgets. Fund any minimum requirement first, then divide the leftover by the unit cost. Rounding direction is the recurring kill shot: budgets and capacities round DOWN, requirements ("at least") round UP, and both wrong directions appear in every option set. Stated policy caps are not automatically attainable — the clock or budget may bind before the cap does.

For which-pair-is-a-solution items, test options in the harder inequality first and watch for the option engineered to land exactly on a boundary. In Desmos, shading both inequalities and reading the overlap (or dropping the line x = 6 across it) settles coordinate questions in seconds — dashed boundaries excluded, solid ones included.