For each problem, write the constraints and graph the...

For each problem, write the constraints and graph the solution set to the system of constraints. See Example 2 . Making boats. A company make kayaks and canoes. Each kayak requires $\$ 80$ in materials and 60 hours of labor. Each canoe requires $\$ 120$ in materials and 40 hours of labor. The company has at most $\$ 12,000$ available for materials and at most 4800 hours of labor. Let $x$ represent the possible number of kayaks and $y$ represent the possible number of canoes that can be built.

For each problem, write the constraints and graph the solution set to the system of constraints. See Example 2 .
Making boats. A company make kayaks and canoes. Each kayak requires $\$ 80$ in materials and 60 hours of labor. Each canoe requires $\$ 120$ in materials and 40 hours of labor. The company has at most $\$ 12,000$ available for materials and at most 4800 hours of labor. Let $x$ represent the possible number of kayaks and $y$ represent the possible number of canoes that can be built.

Key Concepts

Graphical Representation of Inequalities

Graphically representing inequalities involves drawing each inequality on a coordinate system as a boundary line and shading the areas that satisfy the inequality. The overlapping area, or feasible region, provides a visual understanding of which combinations of variable values are valid. This method facilitates the analysis of the problem and is especially useful in problems with two variables.

Feasible Region

The feasible region is the set of all points that satisfy the entire system of inequalities simultaneously. Graphically, this region is typically represented on a coordinate plane as the intersection of half-planes defined by each inequality. It represents all possible solutions that meet the given constraints and is crucial in determining optimal solutions in optimization problems.

Constraint Modeling

Constraint modeling is the process of translating real-world limitations, such as resource availability, time, or costs, into mathematical expressions. This practice forms the basis for creating systems of inequalities that define the boundaries within which feasible solutions can exist. It is essential for problem solving in fields like operations research and economics.

Systems of Linear Inequalities

This concept involves using multiple inequalities to model restrictions or conditions that a set of variables must satisfy. It is a fundamental tool in linear programming and optimization, where real-world constraints such as budgets, resources, or capacities are expressed algebraically. By representing limits with inequalities, the relationships among variables become clear and manageable.

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