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[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§4.4 2-Var§4.4 2-VarInEqualitiesInEqualities
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §4.3b → Absolute Value InEqualities
Any QUESTIONS About HomeWork• §4.3b → HW-13
4.3 MTH 55
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt3
Bruce Mayer, PE Chabot College Mathematics
Graphing InEqualitiesGraphing InEqualities
The graph of a linear equation is a straight line. The graph of a linear inequalityinequality is a half-planehalf-plane, with a boundaryboundary that is a straight line.
To find the equation of the boundary line, we simply replace the inequality sign with an equals sign.
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt4
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph yy ≥ ≥ xx
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
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
y = x
-4
-5
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt5
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph yy ≥ ≥ xx
• Note that in the graph each ordered pair on the half-plane above y = x contains ay-coordinate that is greater than thex-coordinate. It turns out that any point on the same side as (–2, 2) is also a solution. Thus, if one point in a half- plane is a solution, then all points in that half-plane are solutions.
y
x -5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
y = x
-4
-5
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt6
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph yy ≥ ≥ xx
• Finish drawing the solution set by shading the half-plane above y = x.
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
y = x
-4
-5
For any point here, y > x.
For any point here, y = x.
• The complete solution set consists of the shaded half-plane as well as the boundary itself whichis drawnsolid
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt7
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph yy < 3 < 3 − 8− 8xx SOLUTION Since the inequality
sign is < , points on the line y = 3 – 8x do not represent solutions of the inequality, so the line is dashed.
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
y = 3 – 8x
-4
-5
(3, 1)
Using (3, 1) as a test point, we see that it is NOT a solution:
Thus points in the other ½-plane are solns trueNOT 24313831
??
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt8
Bruce Mayer, PE Chabot College Mathematics
Graphing Linear InEqualitiesGraphing 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
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt9
Bruce Mayer, PE Chabot College Mathematics
Graphing Linear InEqualitiesGraphing 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
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph
Draw Graph and test (3,3) = (xtest, ytest)
11.
6y x
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
y = (1/6)x – 1-4
-5
(3,3)
Check Location of Test Value• ytest > (1/6)·xtest − 1 ¿?
• 3 > (1/6)(3) − 1 ¿?
• 3 > 2 − 1 Since 3 > 1 the pt
(3,3) IS a Soln, so shade on that side
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt11
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph xx ≥ −3≥ −3
Draw Graph
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
Test (4,−2) & (1, 3)
(4,−2)
(1,3) Since both 4 & 1 are
greater than −3, thenpoints to the right ofthe line are solutions
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt12
Bruce Mayer, PE Chabot College Mathematics
Systems of Linear EquationsSystems 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.
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt13
Bruce Mayer, PE Chabot College Mathematics
Example Example xx + + yy > 3 > 3 && xx − − yy ≤ 3 ≤ 3
SOLUTION First graph
x + y > 3 in red.
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
y > −x + 3
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt14
Bruce Mayer, PE Chabot College Mathematics
Example Example xx + + yy > 3 > 3 && xx − − yy ≤ 3 ≤ 3
SOLUTION Next graph
x − y ≤ 3in blue
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
y ≥ x − 3
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt15
Bruce Mayer, PE Chabot College Mathematics
Example Example xx + + yy > 3 > 3 && xx − − yy ≤ 3 ≤ 3
SOLUTION Now find the
intersection of the regions
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
The Solution is the OverLappingRegion• CLOSED dot
indicates that theIntersection is Part of the Soln
Solution set to the system
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt16
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph −1 <−1 < yy < 5 < 5
SOLUTION Break into Two
Inequalities andGraph• −1 < y
• y < 5
The Solution is the OverLappingRegion
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
Solution set
5y
1y
5
and
1
y
y
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt17
Bruce Mayer, PE Chabot College Mathematics
Intersection of Two InequalitiesIntersection of Two Inequalities
Graph 3x + 4y ≥ 12 and y > 2 Graph Each InEquality Separately
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt18
Bruce Mayer, PE Chabot College Mathematics
Intersection of Two InequalitiesIntersection of Two Inequalities
Graph 3x+4y≥12 and y>2
Shade Region(s) common to BOTH
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt19
Bruce Mayer, PE Chabot College Mathematics
Union of Two InequalitiesUnion of Two Inequalities
Graph 3x + 4y ≥ 12 or y > 2 Again Graph Each InEquality Separately
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt20
Bruce Mayer, PE Chabot College Mathematics
Union of Two InequalitiesUnion of Two Inequalities
Graph 3x+4y≥12 or y>2
Shade Region(s) covered by EITHER soln
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt21
Bruce Mayer, PE Chabot College Mathematics
Graphing a System of InEqualsGraphing 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
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt22
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph of System Graph of System
Graph System2,
3,
,
x y
x
y x
Red
Blue
Green
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
(3, 5)
(3, –3)
(–1, 1 )
Draw Graph• 3 Lines
Intersecting at 3 locations
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt23
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph of System Graph of System
Graph System2,
3,
,
x y
x
y x
Red
Blue
Green
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-3
2
-2
3
-1
1
6
54
-4
-5
The Solution isthe EnclosedRegion; a PolyGon• A TriAngle in this case
– Check that, say, (2, 2) works in all threeof the InEqualities
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt24
Bruce Mayer, PE Chabot College Mathematics
Example Example Find Vertices Find Vertices
Graph the following system of inequalities and find the coordinates of any vertices formed:
2 0
2
0
y
x y
x y
Graph the related equations using solid lines.
Shade the region common to all three solution sets.
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt25
Bruce Mayer, PE Chabot College Mathematics
Example Example Find Vertices 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 & −x + y = 2
Solving find Vertex pt (−4, −2) The system of equations from
inequalities (1) and (3):• y + 2 = 0 & x + y = 0
2 0
2
0
y
x y
x y
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt26
Bruce Mayer, PE Chabot College Mathematics
Example Example Find Vertices Find Vertices
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)
2 0
2
0
y
x y
x y
(−4,−2)
(2,−2)
(−1,−1)
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt27
Bruce Mayer, PE Chabot College Mathematics
Example Example Graph of System Graph of System
Graph the following system. Find the coordinates of any vertices formed.
0
2 3
0 3
4
9x y
y
x
Graph by Lines The CoOrd of the
vertices are: (0, 3), (0, 4), (3, 4) and (3, 1)
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt28
Bruce Mayer, PE Chabot College Mathematics
Types of Eqns & InEqualsTypes of Eqns & InEquals
Graph
Type Example Solution
Linear Equations in one variable
2x – 8 = 3(x + 5) A number in One Variable
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt29
Bruce Mayer, PE Chabot College Mathematics
Types of Eqns & InEqualsTypes of Eqns & InEquals
Graph
Type Example Solution
Linear InEqualities in one variable
–3x + 5 > 2 A set of numbers;
an interval
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt30
Bruce Mayer, PE Chabot College Mathematics
Types of Eqns & InEqualsTypes of Eqns & InEquals
Graph
Type Example Solution
Linear Equations in two variable
2x + y = 7 A set of ordered
pairs; a line
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt31
Bruce Mayer, PE Chabot College Mathematics
Types of Eqns & InEqualsTypes of Eqns & InEquals
Graph
Type Example Solution
Linear InEqualities in two variable
x + y ? 4 A set of ordered
pairs; a half-Plane
≥
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt32
Bruce Mayer, PE Chabot College Mathematics
Types of Eqns & InEqualsTypes of Eqns & InEquals
Graph
Type Example Solution
System of Equations in two variables
x + y = 3 5x - y = -27
An ordered pair or a (possibly empty)
set of ordered pairs
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt33
Bruce Mayer, PE Chabot College Mathematics
Types of Eqns & InEqualsTypes of Eqns & InEquals
Graph
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
≤≤≥
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt34
Bruce Mayer, PE Chabot College Mathematics
Example Example PopCorn Revenue 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 profitablea) 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
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt35
Bruce Mayer, PE Chabot College Mathematics
Example Example PopCorn Revenue PopCorn Revenue
Translate by Tabulation
Category Price Number Sold Revenue
Large 4.00 x 4x
Small 3.00 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
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt36
Bruce Mayer, PE Chabot College Mathematics
Example Example PopCorn Revenue PopCorn Revenue
b) Graph 4x + 3y ≥ 400
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt37
Bruce Mayer, PE Chabot College Mathematics
Example Example PopCorn Revenue 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.
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt38
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §4.4 Exercise Set• 46 (ppt), 62
PopCornBag & BucketSizes
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt39
Bruce Mayer, PE Chabot College Mathematics
P4.4-46 Graph SystemP4.4-46 Graph System
Graph2x + y ≤ 6
321
262
yx
yxyx
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
Test (0,0)• 2(0)+0 ≤ 6?
• 0 ≤ 6 Shade
BELOWLine
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt40
Bruce Mayer, PE Chabot College Mathematics
P4.4-46 Graph SystemP4.4-46 Graph System
Graphx + y ≥ 2
321
262
yx
yxyx
Test (0,0)• 0+0 ≥ 2?
• 0 ≥ 2 Shade
ABOVELine
Y2
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
x
y
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt41
Bruce Mayer, PE Chabot College Mathematics
P4.4-46 Graph SystemP4.4-46 Graph System
Graph1 ≤ x ≤ 2
321
262
yx
yxyx
Test (0,0)• 1 ≤ 0 ≤ 2
Test (1.5,0)• 1 ≤ 1.5 ≤ 2
ShadeBETWEENLines
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt42
Bruce Mayer, PE Chabot College Mathematics
P4.4-46 Graph SystemP4.4-46 Graph System
Graphy ≤ 3
321
262
yx
yxyx
Test (0,0)• 0 ≤ 3
ShadeBELOWLine
x
y Y5
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt43
Bruce Mayer, PE Chabot College Mathematics
P4.4-46 Graph SystemP4.4-46 Graph System
Now CheckFor OverLapRegion
321
262
yx
yxyx
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
Y2
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
x
y
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
Y5
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
Found One;a five sidedPolyGon
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt44
Bruce Mayer, PE Chabot College Mathematics
P4.4-46 Graph SystemP4.4-46 Graph System
Thus Solution
321
262
yx
yxyx
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt45
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
HealthyHeart
WorkOut
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt46
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt47
Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
[email protected] • MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt48
Bruce Mayer, PE Chabot College Mathematics
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls