Constraints and the feasible region

A constraint turns a resource limit into an inequality. The factory has only 10 kg of flour. Each kilogram of cookie mix needs 0.4 kg of flour and each kilogram of donut mix needs 0.5 kg. With $c$ kg of cookie and $d$ kg of donut, write total flour used as a linear expression in $c$ and $d$, then require it to stay at or below the 10 kg available.

To picture this, put cookie mix on the vertical axis and donut mix on the horizontal axis. Find where the flour limit meets each axis by setting the other mix to zero and solving for the intercept. Join those two points with a straight line; everything on or below the line uses at most 10 kg of flour.

Because you cannot make negative amounts, you also keep only the part where $c \ge 0$ and $d \ge 0$. The shaded area that survives is the set of all feasible combinations. Its corners have a name from the terminology alert: each corner is a vertex (plural vertices), and those corners turn out to be where the search for maximum revenue will focus.