Turn the phrases "at least 12 boxes" and "at most 300 cookies" into inequalities using your variables. You should get one inequality for boxes and one for cookies.

To maximize the number of large boxes, would you want extra small boxes beyond the minimum required? Use the box inequality to express the number of small boxes in terms of the number of large boxes.

After substituting, you should get an inequality involving only the number of large boxes. Solve it, then pick the largest whole number that satisfies it.

In Desmos, type y = 216 + 12x. Here, $x$ represents the number of large boxes, and $y$ is the total number of cookies when you use the minimum number of small boxes (so that $s + l = 12$).

Type another line: y = 300. This is the maximum allowed number of cookies. Look at where the line $y = 216 + 12x$ meets or crosses $y = 300$, and note the $x$-value of that intersection.

From the graph or by tracing along y = 216 + 12x, find the largest integer $x$ for which $y$ is still at or below 300. That integer $x$ is the maximum number of large boxes that can be shipped.

Let

• $s$ = number of small boxes
• $l$ = number of large boxes

Translate the conditions into inequalities:

• At least 12 boxes total: $s + l \ge 12$
• At most 300 cookies: $18s + 30l \le 300$

We want to maximize $l$ (the number of large boxes).

For a given number of large boxes $l$, adding more small boxes only increases the number of cookies. That makes it harder to stay at or below 300 cookies.

So to allow as many large boxes as possible, we should use the fewest small boxes that still satisfy the box requirement.

The requirement is $s + l \ge 12$. To minimize cookies for a given $l$, take exactly 12 boxes:

So

Substitute $s = 12 - l$ into the cookie inequality $18s + 30l \le 300$:

Simplify step by step:

• Distribute 18: $18 \cdot 12 - 18l = 216 - 18l$
• Combine with $30l$:

So the inequality becomes

Solve $216 + 12l \le 300$:

So $l$ cannot be more than 7. Since $l$ must be a whole number of boxes and we want the maximum possible number of large boxes, the answer is 7 large boxes.