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
⇐[email protected],¥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
/