Chapter 2: Solving systems
of linear equations and
inequalities
Section 2-6: Solving systems of
linear inequalities
Objectives
Graph systems of inequalities
Find the maximum and minimum value of
a function defined for a polygonal convex
set.
Real-life situation modeled by a
system of inequalities
UPS adds extra charges for oversized parcels or those requiring
special handling. An oversize package is one in which the sum of
the length and the girth exceeds 84 inches. For a rectangular
package, its girth is the sum of twice the width and twice the height.
A package requiring special handling is one in which the length is
greater than 60 inches. What size packages qualify for both oversize
and special handling charges?
Solution:
First write two inequalities that represent each type of charge. Let L
represent the length of the package and G represent the girth.
Oversize: L+G > 84
Special Handling: L > 60
What size packages qualify for both oversize and special
handling charge?
L+G > 84
L > 60
Since both of these inequalities do not include the points on the
boundary line, they are dashed.
Find the boundary lines for both inequalities.
L+G=84
L=60
Next test a point not on the boundary line L+G= 84 and see if it makes
the inequality L+G > 84 true. Test say (100,100). True so shade
above the boundary line. Next test a point not on the boundary line
L=60 and see if it makes the inequality L > 60 true. Test (80,20).
Works so shade the right side of the line.
Where the blue and yellow shaded areas overlap (green) is the
solution. So for example (90,20) is a length greater than 90 inches
and a girth of 20 inches which represents an oversized package that
requires special handling.
The Graph of the problem
Not every system of inequalities
has a solution
When graphs have no points in common,
there is no solution. The inequalities are
y > x+3 and y < x-1
Polygonal Convex Set
A system of more than two linear
inequalities can have a solution that is a
bounded set of points. A bounded set of all
points on or inside a convex polygon
graphed on a coordinate plane is called a
polygonal convex set.
Example
Solve the system of inequalities by graphing. Name the coordinates of
the polygonal convex set.
Solution: Do each one separately.
x≥0
Find the boundary line, determine if it is dashed or
y≥0
solid, graph it. Then test a point not on the line with
x + y ≤ 5 the inequality.
The shaded region on the graph shows points that
satisfy all three inequalities. The region is a triangle whose vertices
are (0,0), (5,0), (0,5)
Vertex Theorem
Since it is impossible to evaluate all the
points in a polygonal convex set, the
vertex theorem allows us to only evaluate
the function at the vertices.
The maximum or minimum value of f (x, y) =
ax + by + c on a polygonal convex set
occurs at a vertex of the polygonal
boundary.
Example
Find the maximum and minimum values of f (x, y) = y-2x+5 for the
polygonal convex set determined by the system of inequalities.
First graph each inequality and find the coordinates
x ≥1 y ≥ 2
of the vertices of the resulting polygon.
y ≤8 x+y≥5
2x + y ≤ 14
The vertices are
(3,8),(1,8), (1,4),(3,2)(6,2)
Now evaluate the function at each of these vertices.
f (x, y)=y-2x+5
f(3,8)=8-2(3)+5=7
f (1,4)=4-2(1)+5=7
f (6,2)=2-2(6)+5=-5
f(1,8)=8-2(1)+5=11
f (3,2)=2-2(3)+5=1
Maximum is 11 at (1,8) and the minimum is -5 at (6,2)
HW # 12
Section 2-6
Pp 110-111
#9,11,14,17,19,30,33