Chapter1–AnIntroductiontoAlgebra1.1 AnIntroductiontoAlgebraSymbolsParenthesis/ParenthesesBracket/BracketsBrace/BracesAlgebraicExpressionsvs.AlgebraicEquationsOperationVariableConstantEx:Whatoperationsdoestheequation4y–14=5/6contain?Whatisthevariable?Whataretheconstants?

,- Grouping Symbols

( c )

| [ ] usual

derision - { } zztversionNo = =

+ - • ÷t

Multiplication

Unknown quantity , somethingx

, y ,

¥#Xthat changes l

,0

,@

, Sj5

,it ,e

,i

,0 ( not 01

411-14=55Operations : subtraction,

division

implicit multiplication.

Variable : yConstants :

14,

-14,

56,4

,E

1.2FractionsPrimenumber:Compositenumber:Primenumbersbetween1and100:Ex:Express1240asaproductofprimesDivisionTricks:Divisibleby2: Ex:Divisibleby3: Ex:Divisibleby5: Ex:Divisibleby6:

,

Divisible

by land itself

2,3 ,5,7 ,

1513,17,

1923,

. . .

4=2.2,

6=2.3,99,1-0,1314, 15,16 ,

18,292523. .TT?jl,}

* ''

Look them up:450fFa¥ke

10.12411 11 1240=31.522295.2 2.62 Prime Factorization

* :en

,ends in 0,2 , 4,68

17953115792=

Add the digits15 : 1+5=6 3goesinb6⇒3 goes

into 15

1026: 1+2+6=9 ⇒ 3 goesinto 1026

Ends in 5or0

11111115 100,000

: 220

IF the number is

divisible by 2 add 3.

Ex:Divisibleby9: Ex:Divisibleby10: Ex:PizzasandSquirrels

IS 36 divisible by 6 ?2 : 36 is even ✓

3 : 3+6=9 3 goes into 9,

so 3 goes into 36 ✓Same as the 3 rule

i+¥f⇒9goes into 117 /←Yt9£18✓

Ends in 0

100 ✓ 100,000,000 ✓

523,4563890 ✓

# pizzas

times= zt Each squirrel gets

Etgtf €half a pizza

÷¥\@Each squirrel gets34 half apit

A sad story:

¥ 0 pizzas ,5 hungry squirrels

Each squirrel get 0 pizza

¥0

A

confusing

story :

OI ←pitz⇒squirrels

¥ is

vndekna*¥%~k¥=0 ( No problem)

÷¥t*¥¥¥me

Ex:Simplify

1) !"!#

2) "$%&'(Ex:Performtheoperationsandsimplify,ifpossible

1) ") ∗'!

2) ") ÷'!

3) ") +'!

4) ") −'!

5) 2 &! ÷ 1(#

1.3TheRealNumbersNaturalNumbers–WholeNumbers–Integers–

a ¥t=§or ¥¥=÷

←=5¥t¥s⇒- 6-

7

=¥e¥OR = 28*4=51=±s÷=¥÷

.settee '=¥Coman .natr=6 45.5¥z÷ty=Et=¥±E

→Braces }

IN-1,33, .:}Set 9

N ={0,5¥99elements

21 = { . . .

-3,

-2, -50,1 , 2,3 ,:}

= { o ,±l,±2,±3 ...}

1

y¥

.it#F=5sEtsite⇒ = 49.2

¥ 35,

= 7.71. 2⇒N#k

= ¥-€Denominator

= top TRotton Improper=tFor word problems , ¥ sounds

funny .

14÷5s#±i4⇒2¥.

Mixed Number

on'¥÷¥÷¥

2£ ÷IF

ZE :

2.2¥= E

B : ego=¥

hE÷¥=,¥.sets ¥ =¥

RationalNumbers–

• TerminatingDecimalor

• RepeatingDecimalIrrationalNumbers–

• Non-terminating,non-repeatingdecimalRealNumbers–VennDiagram:

Set- builder notation

⇐dions-Q-fba1gbEyZ-Oo.333.oj@bandnele.net,¥ejab⇒°r=0.131313 ...

such that=O.T3:

⇐T=pi= 3.14 ...

e

112

All the numberswe know

€vra¥jndR@€€±¥¥

Ex:Classifythefollowingnumbers:{0.1,-2/7,45,-2,13/4,12/4,-67/8,pi, 2, 9,0}Graphing,OrderandtheRealNumberLineTheRealNumberLine–Ex:Graphtheseonanumberline:{0,5,-6,0.3,-1/2,-5½,0.333333….}Ex:Whatisanopposite?Giveexamples.

i. • • o @ • & •a •

± $

/N : 45,9T , 'Y4W : 45k , 144,0

21:45 ,R ,

'

44,0,

-2

Q : 45k ,Yt

, 'd ,

-2,0-5-2-13Irrational :p ;

,p

7,5,-683

.

- 5g- ¥

0.33¥.fi#..x4s. 6 -5-4-3-2

- I

-2 and 2

0 and 0

The opposite of 2 is -2

The opposite of -2 is 2

The opposite of × is - ×

InequalitiesEx:Use<or>tomakeeachstatementtrue:

1) 45

2) -45

3) -4-5

4) 4-5

5) -1-1.1

6) 0-2AbsoluteValueTheabsolutevalueofanumberisitsEx:|3|=|-3|=|0|=-|-3|=

i

⇐W'÷is

4EK§6fjjjgbi*<

"

less than"

<

>'

gireaterthan"

) tester

> to>

7) 0¥ " greater than

8) 0>-0 or equal

: 3

distancefrom zero ¥193 3

33

=3

1.4AddingRealNumbers,PropertiesofAdditionModelingAdditionontheNumberLineEx:4+5Step1:Startat0Step2:Move4unitstotherightStep3:Move5moreunitstotherightEx:Addbygraphing

1) 2+3

2) (-2)+3

3) 2+(-3)

4) (-2)+(-3)Whataretherulesofaddition?LikeSign–UnlikeSign-

ex

is✓ 5 ⇒ 2+3=5

In+3

C-4+3=10¥a

T"

z+c→I=÷0¥÷t20f4+fD= -5

okra- G) C- HE )

Add Absolute value,

use the

original sign( + )+G )

Subtract Absolute value,

thatuse the bigger one 's sign

PropertiesofAdditionTheCommutativePropertyofAddition:Ex:TheAssociativePropertyofAddition:Ex:AdditionPropertyof0(IdentityProperty)Ex:AdditionPropertyofOpposites(InverseProperty)Ex:1.5SubtractingRealNumbersTheMinusSymbol 5–18isreadas“Fiveminuseighteen” -5 isreadas“Negative5” -(-5) isreadas“Theoppositeofnegative5

a + b= bta

3+4 = 4 +: at b) + c = at ( b + c)

(3 + 4) + 5=3+(4+5)

a + 0 = a

3+0 = 3

a + C- a) = O

3 + C-3) = 0

C- 3) + 3 =

:,

Ex:Simplify1)-(-3)2)-(-(-3))3)-(0)4)-|-3|5)|-(-3)|ModelingSubtractionontheNumberLine5–4Step1:Startat0Step2:Move5unitstotherightStep3:Move4unitstotheleft(Subtractiontellsustochangethedirection.)Butwait–thenwhat’s5+(-4)=?Fact:Subtractionisaddingtheopposite.Ex:Performtheoperations.1)5–(-4)2)-5–(-4)

=3

. =-32.5 ) - fttttttc . 4) )I ))))) -54

I 23456789 C- odd )

=%±

- 131=3

015

5-4=1K¥1, Note : Same as

5+641=1

3-6=3+66 )10+4=10-7

5 9

= 5- C- 4) to

=5I '

( +11)=q ¥← y5-

= - 5- C- 4)5- C- 4)

= - SFG 'T) -

= -5+4=-1

3)-5–44)-24–(-28)–48+441.6MultiplyingandDividingRealNumbersNegativenumbersandgothkids(babybats)Ex:Multiply1)(-9)(-3)2)(-1/2)(-1/3)(-1/4)(-1/5)(-1/6)Fact:1timesanyrealnumberis:Fact:0timesanyrealnumberis:Fact:-1timesanyrealnumberis:Fact:Theproductofanonzerorealnumberanditsreciprocalis:Ex:Findthereciprocalofeachnumberandthenmultiply1)22)-2

- 9 -5

=- 5- (4) #= -5¥ f 4) 4µ= . 9 - 4

in .

= -24+(25+648) +44

= :)C- 4 )= +12=12

- (-43--4×-4)=4

= 27

¥DGIEDEDAI ,÷o÷ .÷÷÷te==4.5 . 6

Itself I .a=a

=¥ opposite

-1 .a= - a

a. 'z=F . to =/

→ I 2.12=1

→÷= . I Edt '⇒=1

3)2/34)-1/55)0Fact:DivisioncomesfrommultiplicationWhatis6dividedby2?Why?Ex:Divideandcheckwithmultiplication

1)2)

2)232

3)(-27)÷ −9

→ Z ÷÷=l

→ ÷= -5 t⇒ts)=l

→ No reciprocal

6 ÷ 2=3 # 2.3=6

6÷o= ? # 0 ? =6

6 ⇒ = 2#3zi=6✓±±3B÷=26÷ C- 6) = - 1 # C-6) atD= 6

= 3 ←→ C-9) . 3=-27

.

Ex:Overan8-yearperiod,thevalueofa$150,000housefellatauniformrateto$110,000.Findtheamountofdepreciationperyear.1.7ExponentsandOrderofOperationsAnexponentisusedtoindicaterepeatedmultiplication.Ittellshowmanytimesthebaseisusedasafactor.Ex:23=32=45=!))=

Onacalculator:

F Tlose value steady

peryeart -

year← divided by

depreciation per year =depreciation 150,000-(10,000

ye==g=40,000-8=5000

dollars peryear

$50001year$5000 per year2tthe3rd( power )

2 cubed

= 2.22=8

3 Squared3 to the # ( power )3.3=9

4. 4. 4.4.4=10244 tthesthlpowef

§ . } . }=⇒t÷÷÷

45=4 0×45=04 050405€

NegativebasesEx:(-2)2(-2)3(-2)4(-1)5(-1)6−!))=

-12=-33=-24=OrderofOperations:1)GroupingSymbols2)Exponents3)Division/Multiplication,LefttoRight4)Subtraction/Addition,LefttoRight

4 # parent← base

= f 2) C-2) = 4

= C-2) fz )tz )= - 8

= f 2) f 2) (e) (2)

÷- DTDFHGKD

= - 1

% x⇒⇐ ,= - -8

-1.1=71 Nole :tDEfDH)=l-3.33=-27 a the quantity

"

g. z 2.2=-16 =L )Pleaiexcusemydeorowntsally

( ) exponents . t.tt( ) [ ] brackets { } braces

Ex:Evaluate1)3*4-22)3–2+13)6 ÷ 3 ∗ 24)3*235)-4[2+3(8–42)2]–26)45–5|1–8|

�1��2�

# - 2

÷�2�

=T±= 2

�1� �2�

.

= 2.2= 4

=3 .F= 24

GroupingSymbols⇒ Inside Out

-4 [2+318-4-42]-2= - 4[ 2+3 (8-165)-2

III;i÷eyj÷¥E*'s:- 4[ 194 ] . Z

T= -4:[ 194 ]

-2=45-51-7-1-776-2=-778--45-5F) or 45-5.7

. .

= 45-35 = 45-35

= To =P

7)" $3' 6 '273!&&

-

= 4121+171,

I=

4-(2) + (7)

¥= said ,=¥⇒

1.8AlgebraicExpressionsEx:Identifythetermsandthecoefficientsofeachterminthefollowingexpression:

7x2–x+6DeclaringVariablesEx:Writeanexpressionthatrepresentstheareaofasquare(Reallygoodchartonpage70)Ex:Writeeachphraseasanalgebraicexpression1)13morethanx2)13lessthanx3)xlessthan134)13timesx5)Theratioof13tox6)Doublex7)Triplex8)8greaterthantwicex

7×2,

coefficient = 7- X

,"

= - 1

6 ," = 6

×Dx× Aan¥×gth. width

yllet* ofhonkygtdhe

= xr A=×2

××@3orb+x13 - ×

Bx or 13 ( × ) or 13 . ×

¥Zx

3X

€8or 8+2 ×

9)8lessthantwicexEx:Writeanexpressionthatrepresentseachsituation1)ppoundsofPeanutsweremixedwithcpoundsofcashewstomake100poundsofamixture.2)Howmanyfeetarethereinyyards?3)Ifoneeggisworthgcents,findthevalue(incents)ofonedozeneggs.4)Theexpression20,000–3sgivesthenumberofsquarefeetofsodthatareleftinafieldaftersstripshavebeenremoved.Supposeacityorders7,000stripsofsod.Evaluatetheexpressionandexplaintheresult.

2×-8

⇒= 100

Let f= # feet 3 feet.

- 1 yard6 feet = Zyds

f =3y 9 feet =3 yds

Let V= value of the eggs

V = 12g

Ends# Seato#⇐ ↳ ooo . ss

20,000 - 3 ( 7,000)= 20,000 - 21,000

The field was = - 1,000tooshy

1.9SimplifyingAlgebraicExpressionsUsingPropertiesofRealNumbersCommutativePropertyofAddition:CommutativePropertyofMultiplication:AssociativePropertyofAddition:AssociativePropertyofMultiplication:New–TheDistributivePropertyaka“VisitingtheFamily”Ex:4(5+3)=?OrderofOperations:Breakingupthemultiplicationintotwopieces:Didwegetthesameanswereachtime?TheDistributiveProperty: a(b+c)=ab+ac a(b–c)=ab-acEx:Multiplya)3(4x+5)b)3(4x-5)c)-3(4x-5)

atb-btaab.ba

( at b) + C .

- at ( btc )

( abk = a Cbc)

4 (5-+3)= 4 (8) =

4.8=325415+31428++42 }Yes !

÷U

a=3 . (4×+5)= 3.4×+35

= 12×+15

= 374×-5 )= 3.4×-3.5=12×-15

~~= f 3) . 4× . C- 3) . 5

= - 12×+15

d)10 9

! +)(

e)-0.5(2t–3+0.2w)CombiningLikeTermsLiketerms:Unliketerms:Ex:Add3x+4xusingtheDistributivePropertyEx:Simplifybycombiningliketerms1)9z–72)9z–7–z3)9z–7–z–19z4)43s3–44s3

=D . E + lots or F. ftp.3=,#5¥+'F¥|=.

# + ¥~j#×+6 - 5×+6

÷0.5 )2t - (-0.513+60.5) 0.2W

= - It + 1.5 - 0.1W= - t + 1.5 - 0.1W

Same variables

\# ×,

× ×Y,Y×x

'

,×

'

xy#xyNot the same variables

3×+4*(3+4) x

= 7×

Cannot be SimplifiedAlready Simplified

= 92=-7= 82--7

III.fan /9-1-19

= - 11

= - |s3= - s

3 / 43-44= - 1

5)43s4–44s3Ex:Simplify1)x+x2)x*x3)x+x+x4)2x+x+55)6x–y+2y–3x–126)3z–y2+2y–10z–4y+3

Can't Be

T 7 SimplifiedUnlike

terms

= ZX

: - 3×

-

= 3×+5

= 6×-3×-1/+2 y- 12

÷+ y - 12

- • W - m

=- 7z - y2 -2/+3

5)43s4–44s3Ex:Simplify1)x+x2)x*x3)x+x+x4)2x+x+55)6x–y+2y–3x–126)3z–y2+2y–10z–4y+3

/