[email protected] MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

[email protected] • MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§4.2 Compound§4.2 Compound InEqualities InEqualities



Review §Review §

Any QUESTIONS About• §4.1 → Solving Linear InEqualities

Any QUESTIONS About HomeWork• §4.1 → HW-11

4.1 MTH 55



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



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



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}.



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



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



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.




SOLUTION → Graph x > −5 & x < 2

(

)

( )

x > 5

x < 2

x > 5 and

x < 2




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.



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



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

[ )



““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



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.



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



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



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



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



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.



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



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



Example Example Disjunction Disjunction InEqualInEqual SOLUTION Graph → 2 1 3 3 3.x or x

[

)

[)

1 x

1x

11 xx or



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, )



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



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.



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

[)



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.




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




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



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



WhiteBoard WorkWhiteBoard Work

Problems From §4.2 Exercise Set• Toy Prob (ppt), 22, 32, 58, 78

Electrical Engineering Symbols for and & or



P4.2-ToysP4.2-Toys

Which Toys Fit Criteria• More than

40% of Boys

OR

• More than 10% of Girls

More than10%

More than40%



P4.2-ToysP4.2-Toys Toys That fit

the or Criteria• DollHouses

• Domestic Items

• Dolls

• S-T Toys

• Sports Equipment

• Toy Cars & Trucks



All Done for TodayAll Done for Today

SpatialTemporal

Toy



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

[email protected] MTH55_Lec-16_sec_4-2_Compound_Inequalities.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

Documents

chabot mathematics

compound inequality

sets solution

solution sets

intersection of sets

ab slide

conjunction of inequalities

setbuilder notation