linear inequality in one variable

7/31/2019 Linear Inequality in One Variable

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ObjectivesObjectives

ExercisesExercises

CreditsCredits

PresentationPresentation

Prepared by: FLORDELITA G. MALEPrepared by: FLORDELITA G. MALE

ContentContent


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1. To determine the domainof the variable of an

inequality;2. To verify if a real

number is a solution of

an inequality;


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An equation states that thetwo algebraic expressionsare equal while an inequalitystates that two algebraic

expressions are not equal ina particular way.

PresentationPresentation


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x 7x 7 x 5x 5

x + 4 x + 4

x -3x -3

(x +5) 0(x +5) 0

5x + 4 2x5x + 4 2x-3-3

We also use the symbols and .

we have a b, read aswe have a b, read as

a is less than or equal to b ,a is less than or equal to b ,e mean thate mean that

< b or a = b.< b or a = b.


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If we have a b,If we have a b,read asread as

a is greater than ora is greater than orequal to bequal to b

we mean thatwe mean that

a > b or a = b.a > b or a = b.word or means that at least one of theword or means that at least one of the

o statements must be true.o statements must be true.


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Solution of InequalitySolution of Inequality

solution of linear inequality in one variablesolution of linear inequality in one variablea real number which makes the inequality tra real number which makes the inequality tr

hen substituted to the variable.hen substituted to the variable.

ExampleExample::Is 5 a solution of the inequality 2x > 8?Is 5 a solution of the inequality 2x > 8?

Since 2(5) > 8 , is true ,ince 2(5) > 8 , is true ,hen 5 is a solution of 2x>8.hen 5 is a solution of 2x>8.


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Is 4 a solution of the inequality 2x Is 4 a solution of the inequality 2x 8

2(4) 8 , is true , 8 8,2(4) 8 , is true , 8 8,

since the statement 2x since the statement 2x 8,8,

means that eithermeans that either

2x > 8 or 2x = 8.2x > 8 or 2x = 8.Hence, 4 is a solution of 2x 8.Hence, 4 is a solution of 2x 8.

5 and 4 the only solutions of the inequality 25 and 4 the only solutions of the inequality 2


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The set of all the solutions of anThe set of all the solutions of an

inequality is called ainequality is called a solution setsolution set

of the inequality.of the inequality.Our example 2x 8, hasOur example 2x 8, has infinitelyinfinitely

many solutionsmany solutions..

Here any number greater than orHere any number greater than orequal to 4 is a solution.equal to 4 is a solution.

To write all the solutions we needTo write all the solutions we need

other ways of writing the solution,other ways of writing the solution,


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Our solution to 2x 8, is any numberOur solution to 2x 8, is any number

greater than or equal to 4.greater than or equal to 4.

We write this in set builder notation as:We write this in set builder notation as:

{x/x 4}{x/x 4}It is read asIt is read as

the set of all x such that x isthe set of all x such that x isgreater than or equal to 4.greater than or equal to 4.


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Our solution to 2x 8, is any numberOur solution to 2x 8, is any number

greater than or equal to 4.greater than or equal to 4.

We write this in set interval notation as:We write this in set interval notation as:

[4,+)[4,+)

It is read asIt is read as 4 to positive infinity.4 to positive infinity.

The bracket indicates that 4 is includedThe bracket indicates that 4 is included

in the solution set.in the solution set.

+ + is not a number, so a parenthesis isis not a number, so a parenthesis is

always used after it.always used after it.


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Assume that is constant ,Assume that is constant ,

we write the sets as follows.we write the sets as follows.

S t i S t B ildSet in Set B ilder S t i I t lSet in Inter al TT pe


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Set in Set BuilderSet in Set BuilderNotationNotation

Set in IntervalSet in IntervalNotationNotation

TypeType

Ope

nOpen

half open orhalf-closed

half-open or

half-closed

OpenReal Numberseal Numbers


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1. {x/ x > 5}1. {x/ x > 5}

2. { x/x < 5}2. { x/x < 5}

3. {x/x -5}3. {x/x -5}

4. { x/x -5}4. { x/x -5}

A. [-5, + )A. [-5, + )

B. ( -,-5]B. ( -,-5]

C. ( -, 5)C. ( -, 5)

D. (5,+)D. (5,+)

ExercisesExercises

Good Job!!!Good Job!!! nextnext


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A. Determine by inspection the solution setA. Determine by inspection the solution set

of the inequality.of the inequality.

Write your answer in set-builder and inWrite your answer in set-builder and in

interval notation.interval notation.

11. x -3 > 12. x -3 > 12

2. x + 5 < 202. x + 5 < 20

3. 2x < 63. 2x < 6

4. 3x -1 174. 3x -1 17

5. 5x + 2 275. 5x + 2 27

Answer all before you click for the answer.Answer all before you click for the answer.

Answer


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11. x - 3 > 12. x - 3 > 12

2. x + 5 < 202. x + 5 < 20

3. 2x < 63. 2x < 6

4. 3x -1 174. 3x -1 175. 5x + 2 275. 5x + 2 27

{x/x >15}{x/x >15} (15,+)(15,+)

{x/x < 15}{x/x < 15} (-,15)(-,15)

{x/x < 3}{x/x < 3}

{x/x 6}{x/x 6}{x/x 5}{x/x 5} (-,15](-,15]

[15, + )[15, + )

(-,3)(-,3)

In set-builder notationIn set-builder notationIn Interval NotationIn Interval Notation

If you get 6 or more correct answers, click next.If you get 6 or more correct answers, click next.If not click notes.If not click notes. nextnext NotesNotes


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B. Determine by inspection the solution setB. Determine by inspection the solution setIf the given number is a solution of the inequality.If the given number is a solution of the inequality.

11. x -3 > 12 ; x = 10. x -3 > 12 ; x = 10

2. x + 5 < 20 ; x = 1.52. x + 5 < 20 ; x = 1.53. 2x < 6 ; x = 53. 2x < 6 ; x = 5

4. 3x -1 17 ; x = 184. 3x -1 17 ; x = 185. 5x + 2 27; x =55. 5x + 2 27; x =5

Answer all before you click for the answer.Answer all before you click for the answer.

Answer


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B. Determine by inspection the solution setB. Determine by inspection the solution setIf the given number is a solution of the inequality.If the given number is a solution of the inequality.

11. 10 -3 > 12. 10 -3 > 12

2. 1.5 + 5 < 202. 1.5 + 5 < 203. 2(5) < 63. 2(5) < 6

4. 3(18) -1 174. 3(18) -1 175. 5(5)+ 2 275. 5(5)+ 2 27

Click for the answer.Click for the answer.

Not a solutionNot a solution

SolutionSolution

Not s solutionNot s solution

SolutionSolution

SolutionSolution

If your correct answer is less than 3, click notes.If your correct answer is less than 3, click notes.

Note

s

Note

s!!

N


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11. x - 3 > 12. x - 3 > 12

2. x + 5 < 202. x + 5 < 20

3. 2x < 63. 2x < 6

4. 3x -1 174. 3x -1 175. 5x + 2 275. 5x + 2 27

{x/x >15}{x/x >15} (15,+)(15,+)

{x/x < 15}{x/x < 15} (-,15)(-,15)

{x/x < 3}{x/x < 3}

{x/x 6}{x/x 6}{x/x 5}{x/x 5} (-,15](-,15]

[15, + )[15, + )

(-,3)(-,3)

In set-builder notationIn set-builder notationIn Interval NotationIn Interval Notation

nextnext

NotesNotes


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1. An equation states that

the two algebraic expressionsare equal while an inequalitystates that two algebraicexpressions are not equal in aparticular way.


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2. Any member in the domain2. Any member in the domainof a variableof a variable

for which the inequalityfor which the inequalityis true afteris true after

Substitution into theSubstitution into thevariable is calledvariable is called

thethe solutionsolutionof the inequalityof the inequality


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The domain of a variable inThe domain of a variable in

an inequality is the setan inequality is the setof all real numbers for whichof all real numbers for whichthe expressions involvedthe expressions involved

in the inequalityin the inequalityare defined.are defined.

DOMAINDOMAIN

Definition:Definition:

Credits


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Set-Builder

NotationInterval

notationType Open

The NotationsThe Notations


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Interval

notationnotation

Type

Set-Set-BuilderBuilder

NotationNotation


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Set-BuilderSet-BuilderNotationNotation

IntervalInterval

notationnotation

Type half-open or half-


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SetSet

IntervalIntervalnotationnotation

Real Numbers

OpenTypeBack


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Set Builder

NotationNotationIntervalIntervalnotationnotation

pe half-open or half-clos

C di


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Prepared by:Prepared by:Flordelita G. MaleFlordelita G. Male

San bartolome High School QC

CreditsCredits

San Bartolome High SchoolSan Bartolome High School

The Authors of XP Introductory AlgebraThe Authors of XP Introductory Algebra

linear inequality in one variable

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