Mike's Toy Truck Company manufactures two models of toy...

Mike's Toy Truck Company manufactures two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours ( $\mathrm{hr}$ ) for painting and 3 hr for detail work; each deluxe model requires 3 hr for painting and 4 hr for detail work. Two painters and three detail workers are employed by the company, and each works 40 hr per week. a) Using $x$ to denote the number of standard-model trucks and $y$ to denote the number of deluxe-model trucks,write a system of linear inequalities that describes the possible number of each model of truck that can be manufactured in a week. (b) Graph the system and label the corner points. (IMAGE CAN NOT COPY)

Mike's Toy Truck Company manufactures two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours ( $\mathrm{hr}$ ) for painting and 3 hr for detail work; each deluxe model requires 3 hr for painting and 4 hr for detail work. Two painters and three detail workers are employed by the company, and each works 40 hr per week.
a) Using $x$ to denote the number of standard-model trucks and $y$ to denote the number of deluxe-model trucks,write a system of linear inequalities that describes the possible number of each model of truck that can be manufactured in a week.
(b) Graph the system and label the corner points.
(IMAGE CAN NOT COPY)

Key Concepts

Feasible Region

The feasible region is the intersection of all the shaded areas derived from graphing each linear inequality. It represents the complete set of solutions that simultaneously satisfy all the constraints and is particularly important in optimization and linear programming problems.

Corner Points

Corner points, or vertices, are the points of intersection of the boundary lines of the inequalities. They are significant because they often represent optimal solutions in linear programming problems, as the optimum values typically occur at these vertices within the feasible region.

Graphing Linear Inequalities

Graphing linear inequalities entails plotting the boundary lines of each inequality on a coordinate system and then shading the regions that satisfy the inequalities. This visual method helps to identify the feasible set of solutions and understand the interactions between different constraints.

Constraints

Constraints are conditions or limitations defined by inequalities that restrict the possible values of decision variables. In practical applications, these constraints usually represent limits imposed by available resources, time, or capacity, ensuring that the solutions adhere to real-world restrictions.

Systems of Linear Inequalities

A system of linear inequalities consists of multiple linear inequality expressions that are considered at the same time. The solution to such a system is the set of all variable values that satisfy all the individual inequalities simultaneously, often representing various constraints in optimization problems.

Linear Inequalities

Linear inequalities involve linear expressions with inequality signs (such as ?, ?, <, and >) rather than an equality sign. They serve to define ranges of values that satisfy particular conditions or restrictions, making them fundamental in modeling constraints in various applications.

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