Course 3 Unit 2 - Inequalities and Linear Programming

Solving systems of linear inequalities was informally addressed in Course 1 Unit 3, Linear Functions. In Course 2 Unit 1, Functions, Equations, and Systems, and Unit 2, Matrix Methods, students solved linear systems of two equations in two variables by graphing, substitution, and linear combinations, and by using matrix methods. This unit reviews and extends students' understanding of and their ability to solve inequalities in one and two variables.

Unit Overview
This unit develops student ability to reason both algebraically and graphically to solve inequalities in one and two variables, introduces systems of inequalities in two variables, and develops a strategy for optimizing a linear function in two variables within a system of linear constraints on those variables. Topics included are inequalities in one and two variables, number line graphs, interval notation, systems of linear inequalities, and linear programming.

Objectives of the Unit
  • Write inequalities to express questions about functions of one or two variables
  • Solve quadratic inequalities in one variable, and describe the solution set symbolically, as a number line graph, and using interval notation
  • Solve and graph the solution set of a linear inequality in two variables
  • Solve and graph the solution set of a system of inequalities in two variables
  • Solve linear programming problems involving two independent variables

Sample Overview
The sample material provided below is Investigation 3, "Complex Inequalities," from Lesson 1, "Inequalities in One Variable." The goal of this investigation is to generalize students' graphic and algebraic understanding and technique for solving inequalities (developed in the first two investigations of this lesson) to somewhat more complex situations and algebraic expressions. Students use algebraic methods to locate intersection points of graphs and use the visual images provided by graphs to inform the solution of the related inequalities. Interval notation is introduced in this investigation.
Selected homework tasks from the Connections and Reflections sections are provided. See Implementing Core-Plus Mathematics page 13 for more detail on the design features of the On Your Own homework section.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
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How the Algebra and Functions Strand Continues
In Course 3, there are two more algebra and functions strand units, Unit 5, Polynomial and Rational Functions, and Unit 8, Inverse Functions. Unit 5 extends student ability to represent and draw inferences about polynomial and rational functions using symbolic expressions and manipulations. Topics included are the definition and properties of polynomial and rational expressions, operations on these types of expressions, completing the square, proof of the quadratic formula, solving quadratic equations (including complex number solutions), and the vertex form of quadratic functions.
Unit 8 develops student understanding of inverses of functions with a focus on logarithmic functions and their use in modeling and analyzing problem situations and data patterns. Topics studied in this unit include inverses of functions, logarithmic functions and their relation to exponential functions, properties of logarithms, equation solving with logarithms, and inverse trigonometric functions and their applications to solving trigonometric equations.
     Course 3 Unit 7, Recursion and Iteration, is technically a discrete mathematics unit, but working with sequences and series helps students strengthen their symbolic skills.
     Course 4: Preparation for Calculus extends student algebraic skills and understandings in equations and functions in algebra units but also in geometry units such as Unit 2, Vectors and Motion, and Unit 6, Surfaces and Cross Sections. (See the CPMP Courses 1-4 descriptions.)

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