A store sells two brands of mobile phones, Brand A and Brand...

A store sells two brands of mobile phones, Brand A and Brand B. Past sales records indicate that the store must have at least twice as many Brand $\mathbf{B}$ phones in stock as Brand A phones. The store must have at least two Brand A phones in stock. The maximum inventory the store can carry is 20 phones. (a) Use $x$ to denote the number of Brand A phones in stock. Use $y$ to denote the number of Brand $B$ phones in stock. Write a system of linear inequalities that describes the possible numbers of each brand of phone in stock. (b) Graph the region that represents all possible numbers of Brand A and Brand B phones the store could have in stock.

A store sells two brands of mobile phones, Brand A and Brand B. Past sales records indicate that the store must have at least twice as many Brand $\mathbf{B}$ phones in stock as Brand A phones. The store must have at least two Brand A phones in stock. The maximum inventory the store can carry is 20 phones.
(a) Use $x$ to denote the number of Brand A phones in stock. Use $y$ to denote the number of Brand $B$ phones in stock. Write a system of linear inequalities that describes the possible numbers of each brand of phone in stock.
(b) Graph the region that represents all possible numbers of Brand A and Brand B phones the store could have in stock.

Key Concepts

Decision Variables

Decision variables are symbols used to represent the quantities of interest in a model. They allow the formulation of real-world situations into mathematical terms by assigning variables to the unknowns, making it possible to express constraints and objectives algebraically.

Linear Inequalities

Linear inequalities are mathematical expressions that involve linear functions with inequality signs. They are used to establish constraints that specify ranges or limits within which the solution must lie, instead of providing a single isolated solution as in the case of linear equations.

Systems of Linear Inequalities

A system of linear inequalities consists of multiple linear inequalities that are considered simultaneously. The solution to such a system is the set of all points that satisfy every inequality in the system, representing all combinations of decision variables that meet the given restrictions.

Feasible Region

The feasible region is the set of all points that satisfy the entire system of linear inequalities. In the context of real-world problems, it represents all viable solutions or choices that adhere to imposed constraints, and it is typically visualized as an intersection of half-planes.

Graphical Representation

Graphical representation involves plotting the boundaries represented by the equality forms of the inequalities on a coordinate plane and then shading the region that satisfies the inequality conditions. This visual illustration helps to understand the relationships between the variables and to easily identify the feasible region.

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