BMayer@ChabotCollege.edu MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§4.2 Compound§4.2 Compound InEqualities InEqualities
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §4.1 → Solving Linear InEqualities
Any QUESTIONS About HomeWork• §4.1 → HW-11
4.1 MTH 55
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt3
Bruce Mayer, PE Chabot College Mathematics
Compound InEqualitiesCompound InEqualities
Two inequalities joined by the word “and” or the word “or” are called compound inequalities
Examples
3 9 0 5x and x
7 1 8 8x or x
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt4
Bruce Mayer, PE Chabot College Mathematics
Intersection of Sets Intersection of Sets
The intersection of two sets A and B is the set of all elements that are common to both A and B. We denote the intersection of sets A and B as
A B
.A B
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt5
Bruce Mayer, PE Chabot College Mathematics
Example Example Intersection Intersection
Find the InterSection of Two Sets
, , , , , , , , , , .a b c d e f g a e i o u
SOLUTION: Look for common elements
The letters a and e are common to both sets, so the intersection is {a, e}.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt6
Bruce Mayer, PE Chabot College Mathematics
Conjunctions of SentencesConjunctions of Sentences
When two or more sentences are joined by the word and to make a compound sentence, the new sentence is called a conjunction of the sentences.
This is a conjunction of inequalities:
−1 < x and x < 3.
A number is a soln of a conjunction if it is a soln of both of the separate parts. For example, 0 is a solution because it is a solution of −1 < x as well as x < 3
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt7
Bruce Mayer, PE Chabot College Mathematics
Intersections & ConjunctionsIntersections & Conjunctions
Note that the soln set of a conjunction is the intersection of the solution sets of the individual sentences.
| 3 x x
| 1 x x
| 1 3 x x and x
-1
-1 3
| 1 } { | 3 x x x x 3
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt8
Bruce Mayer, PE Chabot College Mathematics
Example Example “anded” InEquality “anded” InEquality
Given the compound inequality
x > −5 and x < 2 Graph the solution set and write the
compound inequality without the “and,” if possible.
Then write the solution in set-builder notation and in interval notation.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt9
Bruce Mayer, PE Chabot College Mathematics
Example Example “anded” InEquality “anded” InEquality
SOLUTION → Graph x > −5 & x < 2
(
)
( )
x > 5
x < 2
x > 5 and
x < 2
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example “anded” InEquality “anded” InEquality
SOLUTION → Write x > −5 & x < 2
x > −5 and x < 2 Without “and”: −5 < x < 2 Set-builder notation: {x| −5 < x < 2} Interval notation: (−5, 2)
• Warning: Be careful not to confuse the interval notation with an ordered pair.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt11
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve “&” InEqual Solve “&” InEqual
Given InEqual → 2 1 3 3 12,x and x
Graph the solution set. Then write the solution set in set-builder notation and in interval notation.
SOLUTION: Solve each inequality in the compound inequality
2 1 3x 2 4 x
2 x
3 12x 4x
and
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt12
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve “&” InEqual Solve “&” InEqual
SOLUTION: Write for
Without “and”: −2 ≤ x < 4
Set-builder notation: {x| −2 ≤ x < 4}
Interval notation: [−2, 4)
2 1 3 3 12,x and x
[ )
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt13
Bruce Mayer, PE Chabot College Mathematics
““andand” Abbreviated” Abbreviated
Note that for a < b• a < x and x < b can be abbreviated a < x < b
and, equivalently,• b > x and x > a can be abbreviated b > x > a
So 3 < 2x +1 < 7 can be solved as3 < 2x +1 and 2x + 1 < 7
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt14
Bruce Mayer, PE Chabot College Mathematics
Mathematical use of “Mathematical use of “andand””
The word “and” corresponds to “intersection” and to the symbol ∩
Any solution of a conjunction must make each part of the conjunction true.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt15
Bruce Mayer, PE Chabot College Mathematics
No Conjunctive SolutionNo Conjunctive Solution
Sometimes there is NO way to solve BOTH parts of a conjunction at once.
A B
A B
In this situation, A and B are said to be disjoint
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt16
Bruce Mayer, PE Chabot College Mathematics
Example Example DisJoint Sets DisJoint Sets
Solve and Graph: 5 10 4 3.x and x
SOLUTION:
5 1.x and x 5 10 4 3x and x
Since NO number is greater than 5 and simultaneously less than 1, the solution set is the empty set Ø• The Graph:
0
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt17
Bruce Mayer, PE Chabot College Mathematics
Union of SetsUnion of Sets
The union of two sets A and B is the collection of elements belonging to A or B. We denote the union of sets, A or B, by
A B
A B
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt18
Bruce Mayer, PE Chabot College Mathematics
Example Example Union of Sets Union of Sets
Find the Union for Sets
, , , , , , , , , .a b c d e a e i o u
SOLUTION: Look for OverLapping (Redundant) Elements
Thus the Union of Sets
, , , , , , , .a b c d e i o u
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt19
Bruce Mayer, PE Chabot College Mathematics
DisJunction of SentencesDisJunction of Sentences
When two or more sentences are joined by the word or to make a compound sentence, the new sentence is called a disjunction of the sentences
Example x < 2 or x > 8 A number is a solution of a disjunction if
it is a solution of at least one of the separate parts. For example, x = 12 is a solution since 12 > 8.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt20
Bruce Mayer, PE Chabot College Mathematics
Disjunction of SetsDisjunction of Sets
Note that the solution set of a disjunction is the union of the solution sets of the individual sentences.
| 2 x x
| 8 x x
| 2 8 x x or x
8
| 2} { | 8 x x x x
2 8
2
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt21
Bruce Mayer, PE Chabot College Mathematics
Example Example Disjunction Disjunction InEqualInEqual Given Inequality → 2 1 3 3 3.x or x
Graph the solution set. Then write the solution set in set-builder notation and in interval notation
SOLUTION: First Solve for x
2 1 3x 2 2 x
1 x
3 3x 1x or
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt22
Bruce Mayer, PE Chabot College Mathematics
Example Example Disjunction Disjunction InEqualInEqual SOLUTION Graph → 2 1 3 3 3.x or x
[
)
[)
1 x
1x
11 xx or
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt23
Bruce Mayer, PE Chabot College Mathematics
Example Example Disjunction Disjunction InEqualInEqual SOLN Write → 2 1 3 3 3.x or x
Solution set: x < −1 or x ≥ 1
Set-builder notation: {x|x < −1 or x ≥ 1}
Interval notation: (−, −1 )U[1, )
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt24
Bruce Mayer, PE Chabot College Mathematics
Example Example Disjunction Disjunction InEqualInEqual Solve and Graph →
SOLUTION:
1 x 7 x or 4x 3 x
6 2x or 3x 3
x 3 or x 1
Solution set is ( 3,)
or
1 x 7 x or 4x 3 x
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt25
Bruce Mayer, PE Chabot College Mathematics
Mathematical use of “or”Mathematical use of “or”
The word “or” corresponds to “union” and to the symbol ( or sometimes “U”) for a number to be a solution of a disjunction, it must be in at least one of the solution sets of the individual sentences.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt26
Bruce Mayer, PE Chabot College Mathematics
Example Example Disjunction Disjunction InEqualInEqual Solve and Graph →
SOLUTION:
2 1 3 3 3.x or x
2 1 3 3 3x or x
2 2 3 3x or x
1 1.x or x
0 1−1
[)
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt27
Bruce Mayer, PE Chabot College Mathematics
Example Example [10°C, 20°C] → °F [10°C, 20°C] → °F
The weather in London is predicted to range between 10º and 20º Celsius during the three-week period you will be working there.
To decide what kind of clothes to bring, you want to convert the temperature range to Fahrenheit temperatures.
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt28
Bruce Mayer, PE Chabot College Mathematics
Example Example [10°C, 20°C] → °F [10°C, 20°C] → °F
Familiarize: The formula for converting Celsius temperature C to Fahrenheit temperature F is
F 9
5C 32.
Use this Formula to determine the temperature we expect to find in London during the visit there
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt29
Bruce Mayer, PE Chabot College Mathematics
Example Example [10°C, 20°C] → °F [10°C, 20°C] → °F
Carry Out 10 ≤ C ≤ 20.9
510 9
5C
9
520
18 9
5C 36
18 32 9
5C 32 36 32
State: the temperature range of 10º to 20º Celsius corresponds to a range of 50º to 68º Fahrenheit 6850
68325
950
F
C
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt30
Bruce Mayer, PE Chabot College Mathematics
Solving Inequalities SummarizedSolving Inequalities Summarized “andand” type Compound Inequalities
1. Solve each inequality in the compound inequality
2. The solution set will be the intersection of the individual solution sets.
“oror” type Compound Inequalities1. Solve each inequality in the compound
inequality.2. The solution set will be the union of the
individual solution sets
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt31
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §4.2 Exercise Set• Toy Prob (ppt), 22, 32, 58, 78
Electrical Engineering Symbols for and & or
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt32
Bruce Mayer, PE Chabot College Mathematics
P4.2-ToysP4.2-Toys
Which Toys Fit Criteria• More than
40% of Boys
OR
• More than 10% of Girls
More than10%
More than40%
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt33
Bruce Mayer, PE Chabot College Mathematics
P4.2-ToysP4.2-Toys Toys That fit
the or Criteria• DollHouses
• Domestic Items
• Dolls
• S-T Toys
• Sports Equipment
• Toy Cars & Trucks
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt34
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
SpatialTemporal
Toy
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