Represent constraints with equations, inequalities, and systems; interpret viable and non-viable solutions in context.
Test whether a point is viable for a system of constraints
Problem
At a supply station, \(x\) deluxe kits and \(y\) standard kits are nonnegative whole-number counts. Test \((4,~5)\) against \(C_{1}\): \(3x~+~2y~\le~25\); \(C_{2}\): \(x~+~y~\le~10\); \(C_{3}\): \(x~\ge~0\); and \(C_{4}\): \(y~\ge~0\). State the constraints it satisfies and violates, then determine whether the proposal is viable.
Big Picture
What this problem is really about
A point is viable for a system only if it passes every constraint, not just most of them. We’ll substitute the same two coordinates into each labeled inequality, evaluate every comparison, and also check the stated number domain. One failed test would reject the proposal; only a complete set of true statements makes it viable.
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