A heating air conditioning company wants to order no more...

A heating air conditioning company wants to order no more than $$ 2,000 worth of electronic air cleaners and humidifiers from a wholesaler that charges $500 for air cleaners and $$ 200 for humidifiers. If the company wants more humidifiers than air cleaners, use the illustration to graph a system of inequalities that will show the possible combinations of the number of air cleaners $(x)$ and the number of humidifiers $(y)$ that can be ordered. Give two possible solutions. (GRAPH CANNOT COPY)

Key Concepts

Systems of Linear Inequalities

A system of linear inequalities involves several inequalities that together describe constraints on two or more variables. In many applications, such as budgeting or planning, each inequality represents a restriction, and the solution to the system is the set of all possible values of the variables that satisfy every constraint simultaneously.

Cost Constraint Formulation

This concept involves translating monetary limitations into a mathematical inequality. For instance, if items have fixed prices and a total budget is imposed, a cost constraint is formed by multiplying the number of items by their individual prices and ensuring the sum does not exceed the budget. This is a key step in modeling real-life problems using linear inequalities.

Quantity Relationship Constraint

In many practical problems, there can be requirements not only on the total cost but also on the relationship between the quantities of items purchased. This requires writing an inequality that directly expresses the stipulated relationship, such as one type of item being ordered in greater numbers than another.

Graphing Linear Inequalities

Graphing linear inequalities involves first drawing the boundary line, which is derived from the corresponding equality, and then shading the region of the coordinate plane that satisfies the inequality. This graphical representation helps visualize the feasible solution set where all inequalities in the system overlap.

Feasible Region

The feasible region is the intersection of all the individual regions defined by each inequality in the system. It represents all the combinations of variable values that meet every constraint. In optimization or planning problems, this region is key to identifying valid solutions and further analyses such as finding optimal points.

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