### Video Transcript

A candy manufacturer has 30 kilograms of chocolate cookies and 60 kilograms of vanilla cookies. Sales will be made in two different combinations. The first combination will be one-quarter chocolate cookies and three-quarters vanilla cookies by weight, while the second combination will be half chocolate and half vanilla cookies by weight. There is a contract requiring that at least 20 kilograms of the second combination should be supplied to a specific bakery.

Which of the following systems of inequalities represents the number of kilograms of the first and second combinations that will be sold? Let ๐ฅ be the number of kilograms of the first combination and ๐ฆ the number of kilograms of the second combination. Is it (A) ๐ฅ plus two ๐ฆ is less than or equal to 120, three ๐ฅ plus two ๐ฆ is less than or equal to 240, ๐ฆ is greater than or equal to 20, ๐ฅ is greater than or equal to zero? Is it (B) ๐ฅ plus two ๐ฆ is less than or equal to 120, three ๐ฅ plus two ๐ฆ is less than or equal to 240, ๐ฆ is less than or equal to 20, ๐ฅ is greater than or equal to zero, ๐ฆ is greater than or equal to zero? Is it (C) ๐ฅ plus two ๐ฆ is greater than or equal to 120, three ๐ฅ plus two ๐ฆ is greater than or equal to 240, ๐ฆ is greater than or equal to 20, ๐ฅ is greater than or equal to zero? Then, we have option (D), ๐ฅ plus two ๐ฆ is less than or equal to 120, three ๐ฅ plus two ๐ฆ is less than or equal to 240, ๐ฆ is greater than or equal to 20. And (E) ๐ฅ plus two ๐ฆ is less than or equal to 120, three ๐ฅ plus two ๐ฆ is less than or equal to 240, and ๐ฅ is greater than or equal to 20.

Letโs go ahead and clear options (D) and (E) to create some space. We donโt need the systems of inequalities on screen because we can work them out ourselves. We want to find the systems of inequalities that satisfy the conditions for the number of kilograms of two different types of cookies. Now, usually in this sort of question, we would begin by defining our variables. But weโre told to let ๐ฅ be the number of kilograms of the first combination and to let ๐ฆ be the number of kilograms of the second. Now, of course, since weโre working with numbers of kilograms, we know these values must be nonnegative. That is, ๐ฅ is greater than or equal to zero and ๐ฆ is greater than or equal to zero.

But wait! Weโre actually told that thereโs a contract requiring at least 20 kilograms of the second combination should be supplied to this specific bakery. So, in fact, the inequality ๐ฆ is greater than or equal to zero is superfluous. We can instead specify that ๐ฆ is greater than or equal to 20. Now, letโs go back to our two combinations. Specifically, weโre told that the first combination will be one-quarter chocolate and the second will be half chocolate. Weโre also told that there are 30 kilograms of chocolate cookies. So the combination of this one-quarter chocolate and half chocolate can be no more than 30.

Now, since the first combination weighs ๐ฅ kilograms, the number of chocolate cookies in this combination is a quarter ๐ฅ or ๐ฅ over four. Similarly, the second combination weighs ๐ฆ kilograms. So, if weโre making it of half chocolate cookies, thatโs ๐ฆ over two. The total weight of chocolate cookies then is the sum of this, and this must be less than or equal to 30.

Letโs repeat this for the vanilla cookies. Weโre told that the first combination is three-quarters vanilla and the second is half vanilla, then that the total is no more than 60 kilograms. So this time we can represent the total weight of vanilla cookies as three ๐ฅ over four plus ๐ฆ over two. That, of course, must be less than or equal to 60.

We actually now have all of the relevant inequalities, but weโre going to create integer values throughout. To achieve this, we multiply both of these latter inequalities by four. That gives us ๐ฅ plus two ๐ฆ is less than or equal to 120 and three ๐ฅ plus two ๐ฆ is less than or equal to 240. And so the system of inequalities that represents the number of kilograms of our combinations are ๐ฅ plus two ๐ฆ is less than or equal to 120, three ๐ฅ plus two ๐ฆ is less than or equal to 240, ๐ฅ is greater than or equal to zero, and ๐ฆ is greater than or equal to 20. Comparing that to the original set of options, thatโs option (A).