A grocery store sells two types of carrots, organic and...

A grocery store sells two types of carrots, organic and non-organic. Past sales records indicate that the store must stock at least three times as many pounds of organic carrots as non-organic carrots. The store needs to have at least 5 pounds of non-organic carrots in stock. The maximum inventory of carrots the store can carry is 30 pounds. (a) Use $x$ to denote the number of pounds of organic carrots in stock. Use $y$ to denote the number of pounds of non-organic carrots in stock. Write a system of linear inequalitics that describes the possible numbers of pounds of each type of carrot the store could keep in stock. (b) Graph the region that represents all possible amounts of organic and non-organic carrots the store could have in stock.

A grocery store sells two types of carrots, organic and non-organic. Past sales records indicate that the store must stock at least three times as many pounds of organic carrots as non-organic carrots. The store needs to have at least 5 pounds of non-organic carrots in stock. The maximum inventory of carrots the store can carry is
30 pounds.
(a) Use $x$ to denote the number of pounds of organic carrots in stock. Use $y$ to denote the number of pounds of non-organic carrots in stock. Write a system of linear inequalitics that describes the possible numbers of pounds of each type of carrot the store could keep in stock.
(b) Graph the region that represents all possible amounts of organic and non-organic carrots the store could have in stock.

Key Concepts

Linear Inequalities

Linear inequalities represent relationships where one linear expression is greater than or less than another. In a two-variable context, these inequalities form borders on a graph that separate regions where the inequality holds true from those where it does not.

System of Linear Inequalities

A system of linear inequalities consists of multiple linear inequalities that are considered simultaneously. The solution to the system is the set of all points that satisfy each inequality in the system, reflecting the combined conditions imposed by all constraints.

Feasible Region

The feasible region is the set of all points in the coordinate plane that satisfy a system of inequalities. This region represents all possible solutions to the problem under the given constraints and is typically a polygonal area bounded by the lines representing the inequalities.

Graphical Representation

Graphing linear inequalities entails plotting the boundary lines of each inequality on a coordinate plane and shading the area where the inequality holds true. The overlapping shaded region from all inequalities gives the feasible region, which provides a visual interpretation of the solution set.

Translating Real-World Constraints

Translating real-world conditions into mathematical inequalities involves identifying variables that represent the quantities of interest and writing inequalities that express the constraints or requirements described in the problem. This process is fundamental in formulating models that can be analyzed mathematically.

Transcript

Please give Ace some feedback