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Course 3, Unit 2 - Inequalities and Linear Programming

This unit reviews and pulls together students' prior work with graphing of linear, quadratic, and inverse variation functions; solving inequalities graphically; solving quadratic equations algebraically; graphing linear equations in two variables; and solving systems of linear equations in two variables. Linear-programming techniques are used to solve problems in which the goal is to find optimum values of a linear objective function.

Key Ideas from Course 3, Unit 2

  • Write inequalities to express questions about functions of one or two variables: In all the lessons of this unit, students represent situations expressed in words in algebraic representations, that is, inequalities and equalities of one and two variables. An example of a one-variable inequality is -a2 + 10a - 7 ≤ 14. An example of a two-variable inequality is 2x + y > 4.

  • Solve quadratic inequalities in one variable: Solution sets are described symbolically, as a number line graph, and using interval notation. (See pages 108-117.)

  • Solving a linear inequality in two variables: Solutions sets are represented graphically as half-planes. (See pages 128-130.)

  • Solving a system of two linear inequalities: Solution sets are represented graphically as the region represented by both inequalities, which is the intersection of the two half-planes. (See pages 130-131.)

  • Solve linear programming problems involving two independent variables: Linear inequalities often represent constraints in a context such as the production-scheduling problem on page 132. The feasible points are the points that satisfy all of the constraints. The task in a linear-programming problem is to find a maximum or minimum (optimal) solution for an objective function for the context. For the production-scheduling problem that continues on page 137, the objective function is the algebraic rule that shows how to calculate profit for the day. (See pages 132-142.)

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