W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

1
Chabot Mathematics
4.4 2-VarInEqualities
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
2
Review

• Any QUESTIONS About
• 4.3b ? Absolute Value InEqualities
• Any QUESTIONS About HomeWork
• 4.3b ? HW-13

3
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.

4
Example ? Graph y x
y

• 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

6












y x
5
4
3
2
1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
-3
-4
-5
5
Example ? Graph y x

• 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
6












5
4
3
y x
2
1
x
-5 -4 -3 -2 -1 1 2 3
4 5
-1
-2
-3
-4
-5
6
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 itself
  whichis drawnsolid

y
6












5
4
3
y x
2
For any point here, y gt x.
1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
-3
-4
For any point here, y x.
-5
7
Example ? Graph y lt 3 - 8x

• SOLUTION
• Since the inequality sign is lt , points on the
  line y 3 8x do not represent solutions of the
  inequality, so the line is dashed.

y 3 8x
y
6












5
4
3
2
(3, 1)
1
x
-5 -4 -3 -2 -1 1 2 3
4 5
-1
-2
-3
-4

• Using (3, 1) as a test point, we see that it is
  NOT a solution

-5

• Thus points in the other ½-plane are solns

8
Graphing Linear InEqualities

1. Replace the inequality sign with an equals sign
   and graph this line as the boundary. If the
   inequality symbol is lt or gt, 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

9
Graphing Linear InEqualities

• Shade Above or Below the Line
• If the inequality is of the form y lt mx b or y
  mx b shade below the line.
• If the inequality is of the form y gt mx b or y
  mx b shade above the line.
• 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

10
Example ? Graph

• Draw Graph and test (3,3) (xtest, ytest)

• Check Location of Test Value
• ytest gt (1/6)xtest - 1 ?
• 3 gt (1/6)(3) - 1 ?
• 3 gt 2 - 1 ?
• Since 3 gt 1 the pt (3,3) IS a Soln, so shade on
  that side

y
6












(3,3)
5
4
3
2
1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
y (1/6)x 1
-3
-4
-5
11
Example ? Graph x -3

• Draw Graph

y

• Test (4,-2) (1, 3)

6












5

• Since both 4 1 aregreater than -3, thenpoints
  to the right ofthe line are solutions

4
(1,3)
3
2
1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
-3
(4,-2)
-4
-5
12
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.

13
Example ? x y gt 3 x - y 3

• SOLUTION
• First graphx y gt 3 in red.

y
y gt -x 3
6












5
4
3
2
1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
-3
-4
-5
14
Example ? x y gt 3 x - y 3

• SOLUTION
• Next graphx - y 3in blue

y
y x - 3
6
5
4
3
2
1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
-3
-4
-5
15
Example ? x y gt 3 x - y 3
Solution set to the system

• SOLUTION
• Now find the intersection of the regions

y
6












5
4
3
2

• The Solution is the OverLappingRegion
• CLOSED dot indicates that theIntersection is
  Part of the Soln

1
-5 -4 -3 -2 -1 1 2 3
4 5
x
-1
-2
-3
-4
-5
16
Example ? Graph -1 lt y lt 5

• SOLUTION
• Break into TwoInequalities andGraph
• -1 lt y
• y lt 5

y
6












5
4
3
Solution set
2
1
x
-5 -4 -3 -2 -1 1 2 3
4 5
-1
-2
-3

• The Solution is the OverLappingRegion

-4
-5
17
Intersection of Two Inequalities

• Graph 3x 4y 12 and y gt 2
• Graph Each InEquality Separately

18
Intersection of Two Inequalities

• Graph 3x4y12 and ygt2
• Shade Region(s) common to BOTH

19
Union of Two Inequalities

• Graph 3x 4y 12 or y gt 2
• Again Graph Each InEquality Separately

20
Union of Two Inequalities

• Graph 3x4y12 or ygt2
• Shade Region(s) covered by EITHER soln

21
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

22
Example ? Graph of System

• Graph System

y
6
(3, 5)
Blue












5
4
Green
3
2
(1, 1 )
Red
1
-5 -4 -3 -2 -1 1 2 3
4 5
x

• Draw Graph

-1
-2
(3, 3)
-3

• 3 LinesIntersecting at 3 locations

-4
-5
23
Example ? Graph of System

• Graph System

y
6
Blue












5
4
Green
3
2
Red
1
-5 -4 -3 -2 -1 1 2 3
4 5
x

• The Solution isthe EnclosedRegion a PolyGon
• A TriAngle in this case
• Check that, say, (2, 2) works in all threeof the
  InEqualities

-1
-2
-3
-4
-5
24
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.

25
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 -x y 2
• Solving find Vertex pt (-4, -2)
• The system of equations from inequalities (1) and
  (3)
• y 2 0 x y 0

26
Example ? 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)

(-1,-1)
(2,-2)
(-4,-2)
27
Example ? Graph of System

• Graph the following system. Find the coordinates
  of any vertices formed.

• Graph by Lines

• The CoOrd of the vertices are (0, 3), (0, 4),
  (3, 4) and (3, 1)

28
Types of Eqns InEquals

• Graph

29
Types of Eqns InEquals

• Graph

30
Types of Eqns InEquals

• Graph

31
Types of Eqns InEquals

• Graph

32
Types of Eqns InEquals

• Graph

33
Types of Eqns InEquals

• Graph

34
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
• Write an inequality that describes the amount of
  revenue the stand must make to be profitable.
• Graph the inequality.
• Find two combinations of large and small popcorns
  that must be sold to be profitable

35
Example ? PopCorn Revenue

• Translate by Tabulation

Category Price Number Sold Revenue
Large 4.00 x 4x
Small 3.00 y 3y

• 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

36
Example ? PopCorn Revenue

1. Graph 4x 3y 400

37
Example ? PopCorn Revenue

• 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.

38
WhiteBoard Work

• Problems From 4.4 Exercise Set
• 46 (ppt), 62

• PopCornBag BucketSizes

39
P4.4-46 Graph System

• Graph2x y 6

• Test (0,0)
• 2(0)0 6?
• 0 6 ?
• ShadeBELOWLine

40
P4.4-46 Graph System

• Graphx y 2

• Test (0,0)
• 00 2?
• 0 2 ?
• ShadeABOVELine

41
P4.4-46 Graph System

• Graph1 x 2

• Test (0,0)
• 1 0 2 ?
• Test (1.5,0)
• 1 1.5 2 ?
• ShadeBETWEENLines

42
P4.4-46 Graph System

• Graphy 3

• Test (0,0)
• 0 3 ?
• ShadeBELOWLine

43
P4.4-46 Graph System

• Now CheckFor OverLapRegion

• Found Onea five sidedPolyGon

44
P4.4-46 Graph System

• Thus Solution

45
All Done for Today
HealthyHeartWorkOut
46
Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

47
Graph y x

• Make T-table

48
(No Transcript)