Chabot Mathematics
§4.4 2-Var
InEqualities
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Review § 4.3
MTH 55
 Any QUESTIONS About
• §4.3b → Absolute Value InEqualities
 Any QUESTIONS About HomeWork
• §4.3b → HW-13
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Graphing InEqualities
 The graph of a linear equation is a
straight line. The graph of a linear
inequality is a half-plane, with a
boundary that is a straight line.
 To find the equation of the
boundary line, we simply replace
the inequality sign with an
equals sign.
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Graph y ≥ x
 SOLUTION
 First graph the
boundary y = x.
Since the inequality
is greater than or
equal to, the line is
drawn solid and is
part of the graph of
the Solution
Chabot College Mathematics
4
y
6
y=x
5
4
3
2
1
-5 -4 -3 -2 -1
1
2 3 4 5
-1
-2
-3
-4
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  Graph y ≥ x
• Note that in the graph each ordered pair on the
half-plane above y = x contains a
y-coordinate that is greater than the
y
x-coordinate. It turns out
6
5
that any point on the
4
3
same side as (–2, 2) is
y=x
2
also a solution. Thus, if
1
one point in a half- plane -5 -4 -3 -2 -1 1 2 3 4 5
-1
is a solution, then all
-2
-3
points in that half-plane
-4
-5
are solutions.
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  Graph y ≥ x
• Finish drawing the solution set by shading the
half-plane above y = x.
• The complete solution
set consists of the
shaded half-plane
as well as the
boundary For any point
itself which here, y > x.
is drawn
solid
For any point
here, y = x.
Chabot College Mathematics
6
y
6
5
4
3
y=x
2
1
-5 -4 -3 -2 -1
-1
-2
1
2 3 4 5
-3
-4
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  Graph y < 3 − 8x
 SOLUTION
y = 3 – 8x
y
6
 Since the inequality
5
4
sign is < , points on the
3
2
line y = 3 – 8x do not
(3, 1)
1
represent solutions of
x
-5 -4 -3 -2 -1
1 2 3 4 5
-1
the inequality, so the
-2
line is dashed.
-3
-4
 Using (3, 1) as a test
-5
point, we see that it ?
?
is NOT a solution: 13  83  13  24  NOT true
 Thus points in the other ½-plane are solns
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Graphing Linear InEqualities
1. Replace the inequality sign with an
equals sign and graph this line as the
boundary. If the inequality symbol is
< or >, draw the line dashed. If the
inequality symbol is ≥ or ≤, draw the
line solid.
2. The graph of the inequality consists of
a half-plane on one side of the line
and, if the line is solid, the line is part
of the Solution as well
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Graphing Linear InEqualities
3. Shade Above or Below the Line
•
•
If the inequality is of the form y < mx + b
or y ≤ mx + b shade below the line.
If the inequality is of the form y > mx + b
or y ≥ mx + b shade above the line.
4. If y is not isolated, either solve for y and
graph as in step-3 or simply graph the
boundary and use a test point. If the test
point is a solution, shade the half-plane
containing the point. If it is not a solution,
shade the other half-plane
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Graph
1
y  x  1.
6
 Draw Graph and test (3,3) = (xtest, ytest)
y
 Check Location of
6
Test Value
5
(3,3)
• ytest > (1/6)·xtest − 1 ¿?
4
3
• 3 > (1/6)(3) − 1 ¿?
2
• 3>2−1
1
-5 -4 -3 -2 -1
y = (1/6)x – 1
 Since 3 > 1 the pt
(3,3) IS a Soln,
so shade on that side
Chabot College Mathematics
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1
2 3 4 5
-1
-2
-3
-4
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  Graph x ≥ −3
 Draw Graph
y
 Test (4,−2) & (1, 3)
6
 Since both 4 & 1 are
greater than −3, then
points to the right of
the line are solutions
5
4
3
(1,3)
2
1
-5 -4 -3 -2 -1
1
2 3 4 5
-1
-2
-3
-4
(4,−2)
-5
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Systems of Linear Equations
 To graph a system of equations, we
graph the individual equations and then
find the intersection of the individual
graphs. We do the same thing for a
system of inequalities, that is, we
graph each inequality and find the
intersection of the individual
Half-Plane graphs.
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  x + y > 3 & x − y ≤ 3
 SOLUTION
 First graph
x+y>3
in red.
y
6
y > −x + 3
5
4
3
2
1
-5 -4 -3 -2 -1
1
2 3 4 5
-1
-2
-3
-4
-5
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  x + y > 3 & x − y ≤ 3
 SOLUTION
 Next graph
x−y≤3
in blue
y≥x−3
y
6
5
4
3
2
1
-5 -4 -3 -2 -1
1
2 3 4 5
-1
-2
-3
-4
-5
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  x + y > 3 & x − y ≤ 3
 SOLUTION
 Now find the
intersection of
the regions
 The Solution is
the OverLapping
Region
Solution set to
the system
y
6
5
4
3
2
1
-5 -4 -3 -2 -1
• CLOSED dot
indicates that the
Intersection is Part of the Soln
Chabot College Mathematics
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1
2 3 4 5
-1
-2
-3
-4
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  Graph −1 < y < 5
 SOLUTION
 Break into Two
Inequalities and
Graph
• −1 < y
• y<5
 The Solution is
the OverLapping
Region
Chabot College Mathematics
16
y 5
Solution
set
y
6
1  y
5
4
3
and
y5
2
1
-5 -4 -3 -2 -1
1
-1
-2
2 3 4 5
y  1
-3
-4
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Intersection of Two Inequalities
 Graph 3x + 4y ≥ 12 and y > 2
 Graph Each InEquality Separately
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Intersection of Two Inequalities
 Graph 3x+4y≥12
and y>2
 Shade Region(s)
common to BOTH
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Union of Two Inequalities
 Graph 3x + 4y ≥ 12 or y > 2
 Again Graph Each InEquality Separately
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Union of Two Inequalities
 Graph 3x+4y≥12
or y>2
 Shade Region(s)
covered by
EITHER soln
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Graphing a System of InEquals
 A system of inequalities may have
a graph that consists of a polygon
and its interior.
 To construct the PolyGon we find
the CoOrdinates for the corners, or
vertices (singular vertex), of such a
graph
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Graph of System
 Graph System
x  y  2,
x  3,
y   x,
(3, 5)
5
4
3
Green
2
Red
• 3 Lines
Intersecting at
3 locations
22
6
Blue
 Draw Graph
Chabot College Mathematics
y
(–1, 1 )
1
-5 -4 -3 -2 -1
1
-1
-2
-3
-4
2 3 4 5
x
(3, –3)
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Graph of System
 Graph System
x  y  2,
x  3,
y   x,
y
6
Blue
5
4
3
Green
2
Red
1
 The Solution is
the Enclosed
Region; a PolyGon
-5 -4 -3 -2 -1
1
2 3 4 5
-1
-2
-3
-4
-5
• A TriAngle in this case
– Check that, say, (2, 2) works in all three
of the InEqualities
Bruce Mayer, PE
Chabot College Mathematics
23
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
x
Example  Find Vertices
 Graph the following
system of inequalities
and find the coordinates
of any vertices formed:
 Graph the related
equations using
solid lines.
 Shade the region
common to all
three solution sets.
Chabot College Mathematics
24
y20
x  y  2
x y0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Find Vertices
 To find the vertices, we solve
three systems of 2-equations.
 The system of equations from
inequalities (1) and (2)
• y+2=0
&
y20
x  y  2
x y0
−x + y = 2
 Solving find Vertex pt (−4, −2)
 The system of equations from
inequalities (1) and (3):
• y+2=0
Chabot College Mathematics
25
&
x+y=0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Find Vertices
y20
x  y  2
x y0
 The Vertex for The system of equations
from inequalities (1) & (3): (2, −2)
 The system of equations from
inequalities (2) and (3):
• −x + y = 2 & x + y = 0
 The Peak Vertex
Point is (−1, 1)
(−1,−1)
(2,−2)
(−4,−2)
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  Graph of System
 Graph the following system. Find the
coordinates of any vertices formed.
0 x3
0 y4
2x  3y  9
 Graph by Lines
 The CoOrd of the
vertices are: (0, 3),
(0, 4), (3, 4) and (3, 1)
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Types of Eqns & InEquals
Type
Example
Solution
Linear Equations
in one variable
2x – 8 = 3(x + 5)
A number in
One Variable
 Graph
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Types of Eqns & InEquals
Type
Example
Solution
Linear InEqualities
in one variable
–3x + 5 > 2
A set of numbers;
an interval
 Graph
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Types of Eqns & InEquals
Type
Linear Equations
in two variable
Example
Solution
2x + y = 7
A set of ordered
pairs; a line
 Graph
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Types of Eqns & InEquals
Type
Example
Solution
Linear InEqualities
in two variable
x+y≥
?4
A set of ordered
pairs; a half-Plane
 Graph
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Types of Eqns & InEquals
Type
System of
Equations in
two variables
Example
Solution
x+y=3
5x - y = -27
An ordered pair or
a (possibly empty)
set of
ordered pairs
 Graph
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Types of Eqns & InEquals
Type
Example
Solution
System of
two variables
6x – 2y ≤? 12
y – 3 ≤? 0
x + 7 ≥? 0
A set of ordered
inequalities in
pairs; a region of
a plane
 Graph
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  PopCorn Revenue
 A popcorn stand in an amusement park sells
two sizes of popcorn. The large size sells for
$4.00 and the smaller for $3.00 The park
management feels that the stand needs to
have a total revenue from popcorn sales of
at least $400 each day to be profitable
a) Write an inequality that describes the amount of
revenue the stand must make to be profitable.
b) Graph the inequality.
c) Find two combinations of large and small
popcorns that must be sold to be profitable
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  PopCorn Revenue
 Translate by Tabulation
Category
Large
Small
Price
4.00
3.00
Number Sold Revenue
x
4x
y
3y
a) The total revenue would be found by
the expression 4x + 3y. If that total
revenue must be at least $400, then
we can write the following inequality:
4x + 3y ≥ 400
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  PopCorn Revenue
b) Graph
4x + 3y
≥ 400
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Example  PopCorn Revenue
c) We assume that fractions of a
particular size are not sold, so we will
only consider whole number
combinations.
•
One combination is 100 large and 0 small
popcorns which is exactly $400.
•
A second combination is 130 large and
40 small, which gives a total revenue of
$640.
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
WhiteBoard Work
 Problems From §4.4 Exercise Set
• 46 (ppt), 62

PopCorn
Bag &
Bucket
Sizes
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
P4.4-46 Graph System
1 x  2
y3
y
5
 Graph
2x + y ≤ 6
2x  y  6 x  y  2
4
3
 Test (0,0)
2
• 2(0)+0 ≤ 6?
1
• 0≤6
 Shade
BELOW
Line
x
0
-3
-2
-1
39
1
2
3
-1
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
0
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
2x  y  6 x  y  2
P4.4-46 Graph System
y
5
 Graph
x+y≥2
1 x  2
y3
Y2
4
3
 Test (0,0)
2
• 0+0 ≥ 2?
1
• 0≥2
 Shade
ABOVE
Line
x
0
-3
-2
-1
40
1
2
3
-1
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
0
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
P4.4-46 Graph System
3
2
• 1≤0≤2
1
 Test (1.5,0)
x
• 1 ≤ 1.5 ≤ 2  -3
Chabot College Mathematics
41
y3
4
 Test (0,0)
 Shade
BETWEEN
Lines
1 x  2
y
5
 Graph
1≤x≤2
2x  y  6 x  y  2
0
-2
-1
0
1
2
3
-1
-2
M55_§JBerland_Graphs_0806.xls
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
P4.4-46 Graph System
 Graph
y≤3
1 x  2
Y5
3
2
• 0≤3
1
x
0
-3
-2
-1
0
1
2
3
-1
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
42
y3
4
 Test (0,0)
 Shade
BELOW
Line
y
5
2x  y  6 x  y  2
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
P4.4-46 Graph System
y
5
 Now Check
For OverLap
Region
2x  y  6 x  y  2
1 x  2
y3
Y2
Y5
4
3
2
 Found One;
a five sided
PolyGon
1
x
0
-3
-2
-1
0
1
2
3
-1
-2
M55_§JBerland_Graphs_0806.xls
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
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-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
P4.4-46 Graph System
 Thus
Solution
4
3
2x  y  6 x  y  2
1 x  2
y
5
y3
2
1
x
0
-3
-2
-1
0
1
2
3
-1
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
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-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
All Done for Today
Healthy
Heart
WorkOut
Chabot College Mathematics
45
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
46
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
47
5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt
4
5
6
y
5
4
3
2
1
x
0
-3
-2
-1
0
1
2
3
4
5
-1
-2
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
48
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt