A small company produces both doll houses and sets of doll furniture. The doll houses take 3
hours of labor to produce, and the furniture sets take 8 hours. The labor available is limited to
400 hours per week, and the total production capacity is 100 items per week. Existing orders
require that at least 20 doll houses and 10 sets of furniture be produced per week. Write a system
of inequalities representing this situation, where x is the number of doll houses and y is the
number of furniture sets.
Solution
Doll houses = x Furniture = y Labor: It takes 3 hours to make one x, so the total labor for making
all of them is 3x hours. It takes 8 hours to make one y, for a total labor of 8y hours. Those added
together must be less than or equal to 400 hours. 3x + 8y <= 400 Production Capacity: You can
only make 100 items total a week, so you add the x and y and they must be less than or equal to
100. x + y <= 100 Requirements: You must have at least 20 of x made each week: x >= 20 And
you must have at least 10 of y made each week: y >= 10 NOTE: >= means "greater than or
equal to" <= means "less than or equal to" So, you need to graph these four inequalities: x >=
20 y >= 10 3x + 8y <= 400 x + y <= 100