A nut wholesaler sells two types of mixes of cashews and peanuts. He makes a low-grade mix containing 8 pounds of peanuts and 4 pounds of cashews and high-grade mixture containing 6 pounds of peanuts and 6 pounds of cashews. Let x and y denote the numbers of low-grade and high-grade packages that the wholesaler can make from 100 pounds of peanuts and 80 pounds of cashews. Represent the possible combinations of packs of the two mixes that can be made.

Answer 1

Okay, for x pkgs of low-grade nuts, he needs 8x lbs of peanuts and 4x lbs of cashews.
For y pkgs of high-grade nuts, he needs 6x lbs of peanuts and 6x lbs of cashews.
That means that we need
8x+6y <= 100
4x+6y <= 80
Assume we use all the available nuts. That changes our inequalities to equations, with solution
x=5 and y=10
Now you can just list the combinations where x <=5 and y >= 10
or x >= 10 and y <= 5
where the totals meet the requirements. For example,
x=1 means
8+6y <= 100 so y <= 16
4+6y <= 80 so y <= 12
So, with x=1, y<=12 because there are not enough cashews to use up all the available peanuts.
And so on.

Answer 2

Or you can just plot both lines, and any lattice points (with positive integer coordinates) in the shaded region will be solutions.

https://www.wolframalpha.com/input/?i=solve+8x%2B6y+%3C%3D+100,+4x%2B6y+%3C%3D+80+over+the+positive+integers

Answer 3

thank u oobleck