2. A company produces two types of widgets. The profit for selling Widget A is $5 per unit while the profit for selling Widget B is $7 per unit. If the company wants to make a profit of at least $8000, and they can produce no more than 1200 widgets combined, what is the minimum number of Widget A units they must sell?

Let x be the number of Widget A units and y be the number of Widget B units. We have the following inequalities:

5x + 7y ≥ 8000 (profit constraint)

x + y ≤ 1200 (production constraint)

To minimize the number of Widget A units, we want to maximize the number of Widget B units. We can rewrite the production constraint as:

x ≥ 1200 - y

Substituting this into the profit constraint, we get:

5(1200 - y) + 7y ≥ 8000

Simplifying and solving for y, we have:

6000 - 5y + 7y ≥ 8000

2y ≥ 2000

y ≥ 1000

Since we want to minimize x, we can take the maximum possible value for y:

y = 1000

Now, we find the value of x:

x ≥ 1200 - y = 1200 - 1000 = 200

So, the minimum number of Widget A units the company must sell is 200.