Chabot Mathematics
§4.2 Compound
InEqualities
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Review § 4.1
MTH 55
 Any QUESTIONS About
• §4.1 → Solving Linear InEqualities
 Any QUESTIONS About HomeWork
• §4.1 → HW-11
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Compound InEqualities
 Two inequalities joined by the word
“and” or the word “or” are called
compound inequalities
 Examples
3x  9  0 and x  5
7 x  1  8 or x  8
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
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.
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  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}.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Conjunctions 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
−1 < x and x < 3.
of inequalities:
 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
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Intersections & Conjunctions
 Note that the soln set of a conjunction
is the intersection of the solution sets of
the individual sentences.
 x | 1  x 
-1
x | x  3 
3
x | 1  x} {x|x  3 
x | 1  x and x  3 
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-1
3 Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  “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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  “anded” InEquality
 SOLUTION → Graph x > −5 & x < 2
x > 5
(
)
x<2
x > 5
and
x<2
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(
)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  “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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Solve “&” InEqual
 Given InEqual → 2x  1  3 and 3x  12,
 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 x  1  3
3x  12
and
2 x  4
x4
x  2
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Solve “&” InEqual
 SOLUTION: Write for 2x  1  3 and 3x  12,
)
[
 Without “and”: −2 ≤ x < 4
 Set-builder notation: {x| −2 ≤ x < 4}
 Interval notation: [−2, 4)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
“and” 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 as
3 < 2x +1 and 2x + 1 < 7
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Mathematical use of “and”
 The word “and” corresponds to
“intersection” and to the symbol ∩
 Any solution of a conjunction must
make each part of the conjunction
true.
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
No 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
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  DisJoint Sets
 Solve and Graph: 5  x  10 and x  4  3.
 SOLUTION: 5  x  10 and x  4  3
x  5 and x  1.
 Since NO number is greater than 5 and
simultaneously less than 1, the solution
set is the empty set Ø
• The Graph:
0
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Union 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
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
DisJunction 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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Disjunction of Sets
 Note that the solution set of a
disjunction is the union of the solution
sets of the individual sentences.
x | x  8 
8
x | x  2 
x | x  2} {x|x  8 
x | x  2 or x  8 
2
2
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8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Disjunction InEqual
 Given Inequality → 2 x  1  3 or 3x  3.
 Graph the solution set. Then write the
solution set in set-builder notation and in
interval notation
 SOLUTION: First Solve for x
2x  1  3
2x  2
x 1
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or
3x  3
x  1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Disjunction InEqual
 SOLUTION Graph → 2 x  1  3 or 3x  3.
x 1
x  1
[
)
)
[
x  1 or x  1
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Disjunction InEqual
 SOLN Write → 2 x  1  3 or 3x  3.
 Solution set: x < −1 or x ≥ 1
 Set-builder notation: {x|x < −1 or x ≥ 1}
 Interval notation: (−, −1 )U[1, )
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Disjunction InEqual
 Solve and Graph → 1 x  7  x or 4 x  3  x
 SOLUTION:
1 x 
 7  x or
6  2x
or
x  3
or




 
Solution set is (3,)



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
or
4x  3  x
3x  3
x  1

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  Disjunction InEqual
 Solve and Graph → 2 x  1  3 or 3x  3.
 SOLUTION:
2 x  1  3 or 3x  3
2 x  2 or 3x  3
x  1 or x  1.
[
)
−1
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0
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  [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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  [10°C, 20°C] → °F
 Familiarize: The formula for converting
Celsius temperature C to Fahrenheit
temperature F is
9
F  C  32.
5
 Use this Formula to determine the
temperature we expect to find in London
during the visit there
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Example  [10°C, 20°C] → °F
10 ≤ C ≤ 20.
9
9
9
10   C  20 
5
5
5
 State: the
9
temperature
18  C  36
range of 10º
5
to 20º Celsius
9
corresponds 18  32  5 C  32  36  32
9

to a range of
50   C  32   68
50º to 68º
5

Fahrenheit
50  F  68
 Carry Out
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
Solving Inequalities Summarized

“and” type Compound Inequalities
1. Solve each inequality in the compound
inequality
2. The solution set will be the intersection
of the individual solution sets.

“or” type Compound Inequalities
1. Solve each inequality in the compound
inequality.
2. The solution set will be the union of the
individual solution sets
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
WhiteBoard Work
 Problems From §4.2 Exercise Set
• Toy Prob (ppt), 22, 32, 58, 78

Electrical
Engineering
Symbols for
and & or
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
P4.2-Toys
More than
10%
 Which Toys
Fit Criteria
More than
40%
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• More than
40% of Boys
OR
• More than
10% of Girls
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
P4.2-Toys
 Toys That fit
the or Criteria
• DollHouses
• Domestic
Items
• Dolls
• S-T Toys
• Sports
Equipment
• Toy Cars &
Trucks
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt
All Done for Today
Spatial
Temporal
Toy
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BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_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
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt