I cannot understand only a small bit of how this is solved. Please see below. I have written about what I cannot understand in red.

"A company produces two brands of trail mix, regular and deluxe, by mixing dried fruit, nuts and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts and 1,500 pounds of cereal to be used in producing the mixes. The company makes a profit of $0.40 on each pound of regular mix and $0.60 on each pound of deluxe mix. How many pounds of each ingredient should be used in each mix in order to maximize the company's profit?

Type of Mix | Ingredients

Regular | At least 20% nuts

| At most 40% cereal

Deluxe | At least 30% nuts

| At most 25% cereal

Note: State the mathematical model (objective function and all constraints); no solution is required.

Then I identified the variables

Variables:

X1 = Nuts in Regular mix

X2 = Nuts in Deluxe Mix

X3 = Cereal in Regular Mix

X4 = Cereal in Deluxe Mix

X5 = Fruit in Regular Mix

X6 = Fruit in Deluxe Mix

Constraints:

First we need to limit total supplies:

http://www.sosmath.com/CBB/latexrend...b85a16a0f9.png

http://www.sosmath.com/CBB/latexrend...d262ac0fc3.png

http://www.sosmath.com/CBB/latexrend...061b89c0f5.png

Than we limit content:

http://www.sosmath.com/CBB/latexrend...f9dc1bd5d0.png This one just means that at least 20% of total regular mix is the nuts in the regular. We can rewrite this one as:

http://www.sosmath.com/CBB/latexrend...2c6a5791c0.png How does 0.2(X1+X3+X5) become -4X1+X3+X5<=0?

Than we follow the same idea to guarantee the nut content of the deluxe and limit the cereal content of both:

http://www.sosmath.com/CBB/latexrend...5226a3d615.png

http://www.sosmath.com/CBB/latexrend...88a3a35687.png How are those three also rewritten into what they are right now?

http://www.sosmath.com/CBB/latexrend...7622c19218.png

Finally, the objective function is to maximize:

http://www.sosmath.com/CBB/latexrend...b8b5d45504.png "