Lecture 1.4 Inequaliyies

8/7/2019 Lecture 1.4 Inequaliyies

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1.4: Inequalities

Learning Goals:

Use interval notationSolve linear and compoundlinear inequalities

Find exact solutions ofquadratic and factorableinequalities


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Important Idea

In previous sections, wehave been solving equalities,

or equations. Now we aregoing to solve inequalities.The methods of solving

equalities and inequalitiesare similar but there areimportant differences.


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Definition

The statementc


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Definition

The statementc>dmeansthatc is to the right ofdonthe number line.

d

c


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Important Idea

The statementcc

mean the same thing.


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Definition

The statementb


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Definition

,c d c x d

A,c d c x d e

Interval Notation:Letx,c & dbe real numbers with c


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Example

Write the following usinginterval notation:

2 5x

2 5xe

3 8xe e


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Try ThisWrite the following usinginterval notation:

3 8x e

A3,8


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Try This

What do you think thismeans?

? 19, g

, 0g


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Important IdeaPrinciples for solvinginequalities:

1. Add or subtract thesame number on bothsides of the inequality.


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2. Multiply or divide bothsides of the inequality bythe same positive number.


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3. Multiply or divide bothsides of the inequality bythe same negative numberand reverse the direction

of the inequality.


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Example

2 3 5 2 11x xe

Solve. Write your answerusing interval notation.


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Try This

5 2 1 7x x e e

? A2,8



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Example

4 3 5 18x


Graph your answer on anumber line.


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Try This

Solve. Write your answerusing interval notation.Graph your answer on anumber line.

2, 2

3

2 4 3 6x


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Important Idea

The solutions of the form( ) ( )f x g x

consist ofintervals on thex axis wherethe graph offis below thegraph ofg.


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Example( )f x

( )g x

( ) ( )f x g x


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Important Idea

The graph of ( ) ( )y f x g x!

lies above thex axis when( ) ( )f x g x o " and below

thex axis when( ) ( )f x g x o


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Example

Solve:4 3 2

10 21 40 80 x x x x " Hint: Rewrite theinequality.


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Try This

Solve: 4 3 212 4 10 x x x x "

2.97x or4.21x "


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Important Idea

Solving an inequalitydepends only on knowingthe zeros of the functionand where the graph isabove or below thex-

axis. The zeros are wherethe function touches the

xaxis.


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Example

Find the exact solutions:

26 0x x e


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Example


22 3 4 0x x e

Confirm with your calculator


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Try This

2 3 2 0x x e


3 17 3 17,2 2

-


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Example


Confirm with your calculator

6( 5)( 2) ( 8) 0 x x x e


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Important Idea

Steps for solving inequalities:1. Write the inequality in oneof these forms:

( ) 0f x " ( ) 0f x u

( ) 0f x ( ) 0f x e


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Important Idea

Steps for solving inequalities:

2. Determine the zeros off,exactly if possible,approximately otherwise.


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Important Idea

Steps for solving inequalities:

3. Determine the intervals onthex axis where the graph isabove or below the

xaxis.


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Example

A store has determined thecostC of ordering andstoringx laser printers.

300,0002C xx

!

The delivery truck can bringat most 450 printers. Howmany should be ordered to

keep the cost below $1600?

Lecture 1.4 Inequaliyies

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