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Sec 1.6 Solve Linear Inequalities

Slide 2

Goals

Equations and Inequalities

Definition<= /font>

Inequalities

Inequalities

Graphing Inequaliti= es

Compound Inequaliti= es

Conjunctions/Exampl= e 2a

Conjunctions

Disjunctions/Exampl= e 2b

Slide 13

Inequalities

Inequalities

Transformations=

Transformations=

Example 3

Example 3

Example 4

Slide 21

Example 5

Example 6

Slide 24

Example 7

Example 8

Example 8

Example 8

Closure

Assignment<= /font>

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Sec 1.6 Solve Linear Inequalities

Slide 2

Goals
Goal 1: To solve si= mple inequalities
Goal 2: To solve co= mpound inequalities

Equations and Inequalities
Equations and inequ= alities are both statements comparing two expressions.
In this section, we= will learn the similarities and differences, and we will learn to solve inequalities, just as you already know how to solve equations.

Definition<= /font>
Inequalities such a= s x £ 1 and 2n + 3= > 9 are examples of linear inequalities in one variable.

Inequalities
A linear inequality= in one variable can be written in one of the following forms, where a and b are = real numbers and a is not equal to zero:
ax+b<0
ax+b>0
ax+b<0
ax+b>0

Inequalities
A solution of an in= equality in one variable is a value that, when substituted for the variable, results = in a true statement.
Give some solutions= to the inequality x<5.
The graph of an ine= quality in one variable consists of all points on the number line that represents solutions.

Graphing Inequaliti= es
The graph of x>3= is
The graph of x£-2 is<= /td>

Compound Inequaliti= es
Sometimes we want m= ore than one inequality to be true at once.
For example the room temperature is comfortable if it is more than 68° and it is less than 78°
A compound inequality is two simple inequalities joined by “and” or “or”.

Conjunctions/Exampl= e 2a
A compound inequali= ty joined by “and” is called a conjunction.  T= here is a special way that only a conjunction can be written.

Conjunctions

Disjunctions/Exampl= e 2b

Slide 13

Inequalities
Write two true ineq= ualities involving integers, one using < and one using >.
Add, subtract, mult= iply, & divide each side of the inequality by 2 and –2. Highlight the ones that a= re true.

Inequalities
What general conclu= sion can you make about the operations you can perform on a true inequality to produce another true inequality.

Transformations=
Just as equations h= ad transformations that we used to help solve them (Addition Property of Equality and Multiplication Property of Equality are two of them), so do inequalities.
Here are the transf= ormations that produce equivalent inequalities.

Transformations=
Add the same number= to both sides.
Subtract the same n= umber from both sides.
Multiply both sides= by the same positive number.
Divide both sides b= y the same positive number.
Multiply both sides= by the same negative number and reverse the inequality.
Divide both sides b= y the same negative number and reverse the inequality.

Example 3
FAIR  You have $50 to spend in the county fair.  You spend $20 for admission.  You want to play a ga= me that costs $1.50.  Describe the possible numbers of times you can play the game.
First create a verb= al model then add labels:

Example 3
FAIR  You have $50 to spend in the county fair.  You spend $20 for admission.  You want to play a ga= me that costs $1.50.  Describe the possible numbers of times you can play the game.
Now solve the algeb= raic model:

Example 4
Solve and graph.

Slide 21

Example 5
Solve and graph.

Example 6
Solve and graph.

Slide 24

Example 7
The weight w (in po= unds) of an Icelandic saithe is given by w =3D 10.4t – 2.2 where t is the age of the = fish in years.  Describe the ages of a= group of Icelandic saithe that weigh up to 29 pounds.

Example 8
You have added enou= gh antifreeze to your car's cooling system to lower the freezing point to     -35°C and raise the boiling point = to 125°C.  The coolant will remain a liquid as l= ong as the temperature C (in degrees Celsius) satisfies the inequality –35<C<125.  Write the inequ= ality in degrees Fahrenheit.
What is the relatio= nship between C and F, solved for C?

Example 8
For this problem be= gin with your inequality and your equation.
-35<C<125

Example 8
Solve.

Closure
How does solving li= near inequalities compare to solving linear equations?

Assignment<= /font>
Sec 1.6
1, 2, 3-5, 9-17, 20= , 21, 22-31 LC, 33, 34, 36, 41, 42, 46, 47, 49, 50, 53, 54, 56, 58

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<= /td> =
Sec 1.6 Solve Linear
Inequalities
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Goals
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Equations and Inequalities
<= /td> = <= /td> <= /td> <= /td>
n Eq= uations and inequalities are both
st= atements comparing two expressions.
n In= this section, we will learn the
si= milarities and differences, and we will
le= arn to solve inequalities, just as you
al= ready know how to solve equations.
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Definition
<= /td> = =
n In= equalities such as x 1 and 2n + 3 >
9 = are examples of linear inequalities
in= one variable.
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Inequalities
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Inequalities
<= /td> <= /td> = = <= /td> <= /td> <= /td> <= /td>
n A = solution of an inequality in one
va= riable is a value that, when
s= ubstituted for the variable, results in a
tr= ue statement.
n Gi= ve some solutions to the inequality
x&l= t;5.
n Th= e graph of an inequality in one
va= riable consists of all points on the
nu= mber line that represents solutions.
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Graphing Inequalities
<= /td> =
n Th= e graph of x>3 is
n Th= e graph of x-2 i= s
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Compound Inequalities
<= /td> = = <= /td> <= /td> <= /td>
n So= metimes we want more than one
in= equality to be true at once.
n Fo= r example the room temperature is
co= mfortable if it is more than 68 and it
is= less than 78
n A = compound inequality is two simple
in= equalities joined by “and” or “or”.
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Conjunctions/Example 2a
<= /td> = <= /td>
n A = compound inequality joined by “and” is
ca= lled a conjunction.  There is a specia= l way
th= at only a conjunction can be written.
<= /td> = = <= /td>
n No= te here that x is BETWEEN -2 and 1.
n Al= ways put the smaller # on the left and the
la= rger on the right and use < or symbols.
n Th= e graph would look like:
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Conjunctions
<= /td> <= /td> =
n Wr= itten as a conjunction of two inequalities
us= ing the word “and” this compound inequality
wou= ld be:
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Disjunctions/Example 2b
<= /td> = <= /td>
n A = compound inequality joined by “or” is called a
di= sjunction.  A disjunction must have two<= br>
se= parate inequalities and the word “or”.
<= /td>
n Th= e graph would look like:
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Content-Location: file:///C:/EC825EC1/Ch1Sec64A_files/v3_slide0058.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Chapter 1 Sec 6
Inequalities
<= /td> <= /td> = =
n Wr= ite two true inequalities involving integers, one
us= ing < and one using >.
n Ad= d, subtract, multiply, & divide each side of the
in= equality by 2 and –2. Highlight the ones that are
tru= e.
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image/gif R0lGODlhEgATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAP AA4AgAAAAP8AAAIhhGOpG7h/XGqvSQsxhsBSHkmZJyrbZn6TiqYhSKlZV3IFADs= ------=_NextPart_01C7FEB6.90057C10 Content-Location: file:///C:/EC825EC1/Ch1Sec64A_files/v3_slide0058_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAsAgAAAAP8AAAIShB2nG7kPo5St2gautrNT/D0FADs= ------=_NextPart_01C7FEB6.90057C10 Content-Location: file:///C:/EC825EC1/Ch1Sec64A_files/v3_slide0059.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Chapter 1 Sec 6
Inequalities
<= /td> = <= /td>
n Wh= at general conclusion can you make about
th= e operations you can perform on a true
in= equality to produce another true inequality.
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Transformations
<= /td> = <= /td> = <= /td>
n Ju= st as equations had transformations that
we= used to help solve them (Addition
Pr= operty of Equality and Multiplication
Pr= operty of Equality are two of them), so do
in= equalities.
n He= re are the transformations that produce
eq= uivalent inequalities.
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Transformations
<= /td> <= /td> = <= /td> <= /td>
n Add the same number to both sides.
n Subtract the same number from both sides.
n Multiply both sides by the same positive
num= ber.
n Divide both sides by the same positive
num= ber.
n Multiply both sides by the same negative
nu= mber and reverse the inequality.
n Divide both sides by the same negative
nu= mber and reverse the inequality.
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Example 3
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Example 3
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Example 6
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Chapter 1 Sec 6
Example 7
<= /td> =
n Th= e weight w (in pounds) of an Icelandic saithe
is= given by w =3D 10.4t – 2.2 where t is the age of
th= e fish in years.  Describe the ages of a = group
of= Icelandic saithe that weigh up to 29 pounds.
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Example 8
<= /td> <= /td> = = <= /td>
n Yo= u have added enough antifreeze to your car's
co= oling system to lower the freezing point to     -
35= °C and raise= the boiling point to 125°C.  The
co= olant will remain a liquid as long as the
te= mperature C (in degrees Celsius) satisfies the
in= equality –35<C<125.  Write the inequality in
de= grees Fahrenheit.
n Wh= at is the relationship between C and F, solved
for= C?
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Example 8
<= /td> <= /td> <= /td>
n Fo= r this problem begin with your inequality and
yo= ur equation.
n -3= 5<C<125
<= /td> =
n Su= bstitute in for C in the inequality so that the
in= equality is in terms of F.
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Example 8
<= /td>
n Sol= ve.
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Closure=
<= /td> =
n Ho= w does solving linear inequalities
co= mpare to solving linear equations?
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Assignment
<= /td> <= /td> <= /td> =
n Sec= 1.6
= n 1,= 2, 3-5, 9-17, 20, 21, 22-31 LC, 33, 34,
36= , 41, 42, 46, 47, 49, 50, 53, 54, 56, 58
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Sec 1.6 Solve Linear Inequalities
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 1 Sec 6
Goals
nGoal 1: To solve simple inequalities
nGoal 2: To solve compound <= /span>inequalities
------=_NextPart_01C7FEB6.90057C10 Content-Location: file:///C:/EC825EC1/Ch1Sec64A_files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 1 Sec 6