Problem

For $5x~+~10y~\le~50$, $x$ counts $\$5$ small items and $y$ counts $\$10$ large items. Interpret boundary point $(4,~3)$ using both quantities, dollars used, and budget status.

Big Picture

What this problem is really about

A boundary point tells us both what the coordinates mean and which resource limit is met exactly. We’ll match each coordinate to its item type, multiply each count by its corresponding price, and add the two cost contributions. Comparing that total with the budget explains why equality places the point on an allowed boundary rather than outside the region.

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small items = 4; large items = 3; dollars used = $7; budget status = below limit

small items = 4; large items = 3; dollars used = $50; budget status = not allowed at equality

small items = 3; large items = 4; dollars used = $55; budget status = exceeds limit

small items = 4; large items = 3; dollars used = $50; budget status = exactly reaches limit

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For \(5x~+~10y~\le~50\), \(x\) counts \(\$5\) small items and \(y\) counts \(\$10\) large items. Interpret boundary point \((4,~3)\) using both quantities, dollars used, and budget status.

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