A company sells two types of products, X and Y. The profit per unit of product X is $10, while the profit per unit of product Y is $15. If the company wants to make a total profit of at least $5000 in one month, what is the minimum number of units they must sell?
Let's use x to represent the number of units of product X sold and y to represent the number of units of product Y sold. We can then express the company's desired profit as an inequality:

10x + 15y ≥ 5000

To minimize the number of units sold, we want to maximize the contribution of each product to the total profit. Since product Y has a higher profit per unit ($15) than product X ($10), we should prioritize selling more units of product Y.

Let's first consider the case where only product Y is sold (i.e., x=0):

15y ≥ 5000

Dividing both sides by 15 gives us:

y ≥ 333.33

Since the company cannot sell a fraction of a unit, they must sell at least 334 units of product Y if no units of product X are sold.

Now let's consider the case where only product X is sold (i.e., y=0):

10x ≥ 5000

Dividing both sides by 10 gives us:

x ≥ 500

In this case, the company must sell at least 500 units of product X if no units of product Y are sold.

Comparing the two cases, selling 334 units of product Y requires fewer total units to be sold than selling 500 units of product X. Therefore, the minimum number of units the company must sell to achieve a profit of at least $5000 is 334 units (all of product Y).