Ch 2.6 – Solving Systems of
Linear Inequalities
&
Ch 2.7 – Linear Programming
Vocabulary (2.6)
• Polygonal Convex Set: when 2 or more
linear inequalities make a bounded set of
points.
Function Notation
• You are familiar with f(x) =ax + b,
where a and b are real numbers
• Functions may be defined with more than
one variable:
Ex) f(x, y) = ax + by + c
where a,b,c are real numbers
Vertex Theorem
The minimum or maximum of f(x, y) = ax + by + c on
a polygonal convex set occurs at a vertex of the
polygonal boundary.
Example
Vocabulary (2.7)
• Linear Programming: using polygonal
convex sets to find maximums or
minimums of real world problems
• Constraints: conditions of the real life
problem. The inequality systems that
creates the polygonal convex set.
Linear Programming Procedure
1. Define variables
2. Write constraints
3. Graph and find vertices of polygonal
convex set
4. Write an expression whose value is to be
maximized or minimized
5. Substitute values of the vertices in the
expression
6. Select the greatest or least value
The profit on each set of cassettes that is
manufactured by MusicMan, Inc., is $8. The
profit on a single cassette is $2. Machines A and
B are used to produce both types of cassettes.
Each set takes nine minutes on Machine A and
three minutes on Machine B. Each single takes
one minute on Machine A and one minute on
Machine B. If Machine A is run for 54 minutes
and Machine B is run for 42 minutes, determine
the combination of cassettes that can be
manufactured during the time period that most
effectively generates profit within the given
constraints.
Sometimes there might not be a
polygonal convex set
• Infeasible: when the constraints cannot be
satisfied simultaneously. (no solution)
• Unbounded: the region formed by the
constraints is not a closed region. (just a
minimum or just a maximum value)
More than one solution?
• Alternate Optimal Solutions: when two or
more vertices of a polygonal convex set
gives the same minimum or maximum