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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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Important IdeaPrinciples for solvinginequalities:
2. Multiply or divide bothsides of the inequality bythe same positive number.
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Important IdeaPrinciples for solvinginequalities:
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
Solve. Write your answerusing interval notation.
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Example
4 3 5 18x
Solve. Write your answerusing interval notation.
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
Find the exact solutions:
22 3 4 0x x e
Confirm with your calculator
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Try This
2 3 2 0x x e
Find the exact solutions:
3 17 3 17,2 2
-
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Example
Find the exact solutions:
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?