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.)