Download Reat Numbers and Their Properties

C h a p t e r Pl
Prerequisites
ElReatNumbersand Their Properties
In arithmetic we learn facts about the real numbersand how to perfonn operations
with them. Sincealgebrais an extensionof arithmetic,we begin our study of algebra
with a discussionof the real numbersand their properties.
The Rea[ Numbers
A set is a collectionof objectsor elements.The setcontainingthe numbers1,2, and
3 is written as {1,2,3}. To indicatea continuingpattern,we use three dots as in
{ 1, 2 , 3 , . . . } . T h e s e to f r e a ln u m b e r si s a c o l l e c t i o no f m a n yt y p e so f n u m b e r sT. o
better understandthe real numberswe recall some of the basic subsetsof the real
nurnbers:
n
i
l: 1 1 , 2 , 3 , .).
I
{ 0 ,r , 2 , 3 , l
- 2 , 1 , 0 ,t , 2 , 3 ., . . j
^
ffibiliil j
i1 :.::"lltr.1$#i
(tr/)
Wholenumbers
i
(-r)
Integers
l,
j
lunit
+#
32-l
0123
* FigureP.1
Numberscan be picturedas pointson a line, the number line. To draw a numline,
ber
draw a line and label any convenientpoint with the number 0. Now choose
a convenientlength, one unit, and use it to locate evenly spacedpoints as shown in
Fig. P.l. The positive integersare locatedto the right of zero and the negativeintegers to the left of zero. The numbers correspondingto thc points on the line are
calledthe coordinatesofthe points.
The integersand their ratios form the set of rational numbers, Q.The rational
numbers also correspondto points on the number line. For example,the rational
number l/2 is found halfwaybetween0 and I on the numberline. In set notation,
the setof rationalnumbersis writtenas
la
it
a and bareintegerswith 6 + 0 .
)
This notation is read "The set of all numbersof the formalb such that a and b are
integerswith b not equal to zero." ln our set notationwe used lettersto representintegers.A letter that is usedto representa numberis called a variable.
Thereare infinitely many rationalnumberslocatedbetweeneachpair of consecutive integers,yet thereare infinitely many points on the number line that do not correspondto rational numbers.The numbersthat correspondto thosepoints are called
irrational numbers.In decirnalnotation,the rational numbersare the numbersthat
are repeatingor terminatingdecirnals,and the irrational numbersare the nonrepeating nonterminatingdecimals.For example,the number 0.595959 . . . is a rational
number becausethe pair 59 repeatsindefinitely.By contrast,notice that in the number 5.0I 0010001. . . , eachgroup of zeroscontainsone more zero than the previous
group. Becauseno group of digits repeats,5.010010001
. . . is an irrational number.
E
I
* FigureP.2
Numbers such as fi
o, n are also irrational. We can visualize \6 as the
length of the diagonalof a squarewhose sidesare one unit in length. SeeFig. P.2.In
any circle, the ratio of the circumferencec to the diameter d is r 1rr : cld).
P.1I rffi
RealNumbersand TheirProperties
See Fig. P.3.It is difficult to seethat numbers like f 2 and z are irrational because
their decimal representationsare not apparent. However, the irrationality of zr was
proven in 1767 by Johann Heinrich Lambert, and it can be shown that the square
root ofany positive integer that is not a perfect squareis irrational.
Since a calculator operateswith a fixed number of decimal places,it gives us a
ffi
'Figure P.3
rational approximation for an irrational number such as Y 2 or zr. SeeFig. P.4. tr
The set of rational numbers, Q, togetherwith the set of irrational numbers,I is
called the set of real numbers,,R. The following are examplesof real numbers:
-3,
3.14t592654
{2)
1.414213$62
-0.025,
0,
I
7,
J
0.595959...,
ft,
n,
5.010010001
Thesenumbersaregraphedon a numberline in Fig.P.5.
I
<
, FigureP.4
I
-3
|
-2
5
\2
.l .. l.
|
I
-l /o
\r
2
-0.025 0.s9s9...
r Figure P.5
tr
l.
3
4
s.01m1...
'.
>
s
Since there is a one-to-one correspondencebetween the points of the number line
and the real numbers, we often refer to a real number as a point. Figure P.6 shows
how the various subsetsofthe real numbers are related to one another.
Real numbers (R)
Irrational numbers(1)
Numbersthat arenot rational
2,-{3, a 0.3030030003...
r Figure P.6
To indicate that a number is a member of a set, we write a e A, which is read
"a is a member of set1." We write a G. A for "a is not a member of setl. " SetI is a
subsetof set B U g B) meansthat every member of setl is also a member of set.B,
and I is not a subsetof B (A G B) means that there is at least one member of A thal
is not a member of B.
foa*/e
I
ctassifyingnumbersandsetsof numbers
Determine whether each statementis true or false and explain. SeeFig. P.6.
a.0€.R
b.rreQ
c.ReQ
d. IgQ
".fieg
4
ChapterP r r=
Prerequisites
Solution
a. True, because0 is a member of the set of whole numbers, a subsetof the set of
real numbers.
b. False,becausez'is irrational.
c. False,becauseevery irrational number is a member of R but not e.
d. True, becausethe irrational numbers and the rational numbers have no numbers
in common.
e. False, because the square root of any integer that is not a perfect square is
irrational.
T F V T / a l . T r u e o r f a l s ea?. 0 e I
b.Ir- R
".fiet
Propertiesof the ReatNumbers
I n a r i t h m e t i c w e c a n o b s e r v e t h*a t43: 4
+ 3,6 + 9:9
+ 6,etc.Wegetthe
same sum when we add two real numbers in either order. This property of addition
of real numbers is the commutative property. Using variables,the commutative
propertyofadditionisstatedasa I b: b + aforany realnumbersaandb.There
is also a commutativeproperty of multiplication, which is written as a . b : b . a
or ab : ba. There are many properties concerning the operations of addition and
multiplication on the real numbers that are useful in algebra.
Propertiesof the
RealNumbers
For anyrealnumbersa, b, andc:
q * b andab arerealnumbers
a*b:b*aandab:ba
(a+b) *cand
a+(b *c):
a(bc): (ab)c
Closureproperty
Commutativeproperties
Associativeproperties
a(b+ c): ab * ac
0 * a: aandl. a: q
(Zerois the additive identity, and I
is the multiplicative identity.)
0.a:0
For eachrealnumbera, thereis a
uniquerealnumber-c suchthat
a + (-a) : 0. (-a is theadditive
inverseofa.)
Distributive property
For each nonzero real number a,
I
J
j
i
msre
there is
ls a uruque
unique real
real numDer
number l/4
I /a
s u c h t h a t a , l af = l . Q l a i s t h e
multiplicative inverseor reciprocal
of a.)
i --"-""*,,*,..
_**_*
Identity properties
Multiplication property of zero
Additive inverseproperty
Multiplicative inverseproperty
j
I
I
I
-".,-'I
The closure property indicatesthat the sum and product ofany pair ofreal numbers is a real number.The commutative properties indicate that we can add or multiply in either order and get the same result. Since we can add or multiply only a pair
of numbers,the associativeproperties indicate two different ways to obtain the result
P.1 r r m
RealNumbersand TheirProperties
when adding or multiplying three numbers. The operations within parenthesesare
performed first. Becauseof the commutative property, the distributive property can
be used also in the form (b + c)a : ab I ac.
Note that the properties stated here involve only addition and multiplication,
consideredthe basic operationsofthe real numbers. Subtraction and division are de- b : a * (-b) and
fined in terms of addition and multiplication. By definition a
a + b : a . t l b f o r b * 0 . N o t e t h a t a b i s c a l l e d t h e d i f f e r e n c e oafa n d b a n d
a + b is calledthe quotient of a andb'
foatrTle B usingthe properties
Complete each statementusing the property named.
a. a7 :
b. 2x -t 4 : -,
:
c. 8(-)
I
d. ;(3x)
commutative
distributive
l, multiplicative inverse
associative
J
Sotution
a. aJ :7a
..'(}):'
b.2x*4:2(x+2)
o.lt:"r:(+,)'
77V TIAS Completethe statementx'3 :
property.
using the commutative
T
Additive Inverses
-(2-3)
?,
I
,mFigureP.7
-7 (negative seven).If
The negative sign is used to indicate negative numbers as in
-b
it is read as "additive inverse" or "opthe negative sign precedesa variable as in
-b is negative
posite'because-b could be positiveor negative.If b is positivethen
-b
is positive.
and if b is negativethen
-(-5)
: 5
Using two "opposite" signshas a cancellationeffect.For example,
and - (- (-3)) : -3. Note that the additive inverseof a number can be obtained
-I '3 : -3.
b y m u l t i p l y i n g t h e n u m b e r b y- 1 . F o r e x a m p l e ,
6u1"u1utorsusually use the negative sign (-) to indicate opposite or negative
ffi
the
subtraction sign (-) for subtraction as shown in Fig. P.7. tr
and
f o r a n y r e a l n u m b e r sa a n d b , b u t i s
We know that a+b:b+a
a a n d b ? I n g e n e r a l ,a - b i s n o t e q u a l t o
n
u
m
b
e
r
s
a
f
o
r
a
n
y
r
e
a
l
b
b
:
a
7 : -4. So subtractionis not commutab - a.Forexample, 7 - 3 : 4 and 3
- b and b - 4 are oppotive. Since a - b I b - a : 0, we can conclude that a
sites or additive inversesof each other. We summarize these properties of opposites
as follows.
of Opposites
Properties
ChapterP rrr
Prerequisites
fua.nFla !
using propertiesof opposites
Usethe propertiesof oppositesto completeeachequation.
c. -1(x - h) :
b. -1(-2) :
a. -(-zr) :
I
.
Sotution
a. -(-rr) : v
b. -1(-2) : -(-)) :2
c. -1(r - h): -(x - h): h - x
-(1 - w) : -usingthepropertiesof
TfV 1hl. Completetheequation
opposites.
;
r
Retations
Symbolssuchas (, ), :, <, ffid > arecalledrelationsbecausetheyindicatehow
numbersare related.We can visualize theserelations by_usinga number l'ine' For
example,\/i islocatedto the right of 0 in Fig. P.5,to ffi > 0. Since V2 it to ttte
left of zr in Fig. P.5,\/, ( zr. In fact,if a andb areanytwo real numbers,we say
that a is lessthan b (written a < b) providedthat a is to the left of D on the number
line. We saythat c is greaterthan,b(written a > b) if a is to the right of b on the
numberline. We saya : b if a andb correspondto the samepoint on the number
line. The fact that thereare only threepossibilitiesfor orderinga pair of real numbersis calledthe trichotomy property.
J
i
TrichotomyProperty
1
;
The trichotomy property is very natural to use. For example,if we know that
r = t is false,thenwe canconclude(usingthe trichotomyproperty) rlnt eirherr ) t
or r < I is true. If we know ttat w + 6 > z is false, then we can concludethat
w t 6 s z is true. The following four propertiesof equality are also very naturalto
use,andwe often usethem without eventhinking aboutthem.
Propertiesof Equality
AbsotuteValue
The absolutevalue of a (in symbols, la l) canbe thought of asthe distancefrom a
to 0 on a numberline. Sinceboth 3 and -3 are threeunits from 0 on a numberline
asshowninFig.P.8onthenextpage,
l:l : land l-f l : 3'
I
;
)
.
i
P.1r rro
RealNumbersand TheirProoerties
3 units
3 units
-3-2-10i23
r FigureP.8
A symbolicdefinitionof absolutevalueis written asfollows.
Definition:AbsoluteValue
For any real numbera,
| |
'o' :
f a ifa=0
t-o ifa<0.
(
,q.calculator typically uses abs for absolutevalue as shown in Fig. P.9. tr
The symbolic definition of absolute value indicates that for a > 0 we use the
equation lol : o (the absolutevalue ofa isjust a). For c ( 0 we use the equation
-o (the absolutevalue of a is the oppositeof a, a positivenumber).
lol :
ffi
( -3)
. FigureP.9
foatt/a
I
Usingthe definition of absolutevatue
Use the symbolic definition of absolutevalue to simplify eachexpression.
a . l s . o lu . l o l c . l - 3 1
Sotution
a. Since5.6 > 0, we usetheequationlol : o to get 15.61: 5.6.
b . S i n c e 0> 0 , w e u s e t h e e q u a lt"i o
l : n o t o g e tl 0 l : 0 .
3
<
c. Since
0 , w e u s e t h e e q u a tliooln : - " t o g e t l - 3 1 : - ( - 3 ) : 3 .
Usethedefinitionof absolute
valueto simplify| -9I.
TfV Thl.
The definition of absolute value guaranteesthat the absolutevalue of any number is nonnegative.The definition also implies that additive inverses(or opposites)
have the same absolute value. These properties of absolute value and two others are
statedas follows.
Propertiesof
AbsoluteValue
For anyreal numbersa andb:
1. lo | = 0 (Theabsolutevalueof anynumberis nonnegative.)
2. l-ol : lo I lAdaitiveinverses
havethe sameabsolute
value.)
3. lo . bl : lal . lbl (fUeabsolutevalueofaproductistheproductofthe
absolutevalues.)
ll
rl
4. l+l : {!,t
lDl
lDl
+ O (Theabsolute
valueof a quotientis thequotientof the
absolutevalues.)
Absolute value is used in finding the distancebetweenpoints on a number line.
Since 9 lies four units to the right of 5, the distancebetween 5 and 9 is 4. In symbols,
Chapter P I rH
Prerequisites
d(5,9):4.Wecanobta4
i nb y 9 - 5 : 4 o r l 5 - 9 l : 4 . I n g e n e r a l , l-a b l
givesthe distancebetweena andb for anyvaluesof a and b. For example,the distancebetween-2 and1 in Fis. P.l0 is threeunitsand
3 units
-2-1012
GFigureP.10
d ( - 2 . t \: l - 2 - 1 l : l - 3 1 : 3 .
DistanceBetweenTwo
Pointson the NumberLine
If a andb are arrytwo points on the number line, then the distancebetweena
,
a n d b i sl o - b l . I n s y m b o l s , d ( a ,=b )l a - b l , , ,
Note that d(a,O) : la - 0l : lal, which is consistentwith the definitionof absolute value of a as the distancebetweena and 0 on the number line.
fua.r*/e
p
oistancebetweentwo points
on a number line
8 units
-3-2-1
0
l
2
Find the distancebetween -3 and 5 on the number line.
3
4
5
mFigureP.'l1
Sotution
The points correspondingto -3 and 5 are shown on the number line in Fig. P.11.
The distancesbetweentheseooints is found as follows:
d ( - 3 , s ) :l - 3 - s l : l - s l : a
Notice thatd(-3,5)
: d(5, -3):
d ( s , - 3 ) : l s - ( - 3 ) l: l s l : s
Wtren you use a calculator to find the absolute value of a difference or a sum,
ffi
you must use parentheses
as shown in Fig. P.12.
. FigureP.12
I fA
I
I Mt.
Find the distancebetween- 5 and - 9 on the number line.
ExponentialExpressions
We use positive integral exponentsto indicate the number of times a number occurs
in a product. For example, 2 . 2 . 2 . 2 iswritten as 24. We read 2a as "the fourth
power of 2" or "2 to the fourth power."
Definition:Positive
IntegralExponents
For any positive integern
qn:A.A,A.'..,A,
z factors ofa
We call a the base, n the exponent or power, anda' an exponential expression.
We read a' as "a to the rth power." For al we usually omit the exponentand write a.
We refer to the exponents2 and 3 as squaresand cubes. For example, 32 is read "3
squared,"23 is read "2 cube{" ra is read 'X to the fourth," bs is read "b to the fifth,"
P.1 r r !
Real Numbers and Their Prooerties
I
and so on. To evaluate an expression such as -32 we square 3 first, then take the
opposite.So -32 : -9 and (-3)' : (-f)( -f) : q.
fua*Tle
ll
rvatuatingexponentialexpressions
Evaluate.
a. 43 b. (-D4
Solution
r . 4 3: 4 . 4 , 4 :
c. -24
16,4:
64
b. eD4 : G2)(-2)(-z)(-4: rc
-16
c.-24:-(2.2.2,2):
TrV Thl.
Evaluate.a. 52 b. -52
Arithmetic Expressions
Theresultof writing numbersin a meaningfulcombinationwith the ordinaryoperations of arithmeticis calledan arithmetic expressionor simplyan expression.The
value of an arithmetic expressionis the real numberobtainedwhen all operations
areperformed.Symbolssuchasparentheses,
brackets,braces,absolutevaluebars,
andfractionbarsarecalledgrouping symbols.Operationswithin groupingsymbols
areperformedfirst.
fua't/e
I
Evaluatingan arithmeticexpression
with groupingsymbots
Evaluateeachexpression.
a . ( - 7 . 3 ) + ( s . 8 ) v . -- #
z
?-o
-
\-J)
c . 3 - 1 5- ( 2 . e ) l
Sotution
first andremove
theparentheses:
withintheparentheses
a. Performtheoperations
( - 7 ' 3 ) + ( 5 ' 8 ) : - 2 t + 4 o: 1 9
thenumerator
and
b. Sincethefractionbaractsasa groupingsymbol,we evaluate
denominatorbeforedividine.
-6
3- g
-r-,):
3:-z
c. First evaluatewithin the innermostgroupingsymbols:
3 - 15 - (2' 9)l : g - 15 - 181 Innermostgroupingsvmbols
:
:3
:
g - 10
l-t3
13
l
Innermostgroupingsymbols
Evaluatetheabsolutevalue.
Subtract.
a. (-1 + 3)(s- 6) b. 2 - l: - ql
77y Titt. Evaluate.
r
10
ChapterP rrffi
Prerequisites
The Order of Operations
When some or all grouping symbols are omitted in an expressionwe evaluatethe
expressionusing the following order of operations.Any operationscontainedwithin
grouping symbols are performed first, using the order of operations.
Orderof Operations
l. Evaluateexponentialexpressions.
2. Performmultiplication and division in order from left to right.
3. Performadditionand subtractionin order from left to right.
II
I
j
I
fuaw/e
ll
usingthe orderof operations
to evaluate an expression
Evaluateeachexpression.
a.3 - 4.23 b. 5.8 + 4.2
c. 3 - 4+9 -2
d. 5 -2(3 - 4.D2
Solution
a. By the orderof operationsevaluate23,thenmultiply,andthensubtract:
3 - 4.2t : 3- 4. 8 : 3 - 32: -29
b. In an expressionwith only multiplicationand division,the operationsare performedfrom left to right:
5 . 8 + 4. 2 -- 40+ 4. 2 : 10.2 : 20
c. In an expression
with only additionandsubtraction,
theoperationsareperformed
from left to right:
3 - 4 + 9 - 2: -l + 9 - 2 - 8 - 2: 6
d. Performoperationswithin parentheses
first, usingthe orderof operations:
s - 2 ( 3 - 4 ' z 7 z: s - 2 ( - r 2 : s - 2 ' 2 5 : - 4 5
IFV TI'tt. Evaluate. a. 3 - 6 . 2
b. 4 - s . 23
Algebraic Expressions
When we write numbers and one or more variables in a meaningful combination
with the ordinary operations of arithmetic, the result is called an algebraic expression, or simply an expression.The value of an algebraic expression is the value of
the arithmetic expression that is obtained when the variables are replaced by real
numbers.
fuoloF/e !l
Evaluatingan algebraicexpression
F i n d t h e v a l u e o f 6-' 4 a c w h e n
l.b:
-2.andc:3.
P.1 r r n
RealNumbersand TheirProoerties
11
Solution
Replacethe variablesby the appropriatenumbers:
b2- 4ac: (-D2 - 4er)(3) : 16
a2 - b2if a:
77V ThZ. Evaluate
-2andb:
-3.
The domain of an algebraic expressionin one variable is the set of all real numbers that can be used for the variable. For example, the domain of I lx is the set of
nonzero real numbers, because division by 0 is undefined. Two algebraic expressions in one variable are equivalent if they have the same domain and if they have
the same value for each member of the domain. The expressions I f x and xf xz are
equivalent.
A term is the product of a number and one or more variables raised to powers.
Expressionssuch as 3x,2kab2, and rrr2 are terms. Numbers or expressionsthat are
multiplied are called factors. For example, 3 and x are factors of the term 3x. The
coefficient ofany variable part ofa term is the product ofthe remaining factors in
the term. For example,the coefficient of x in 3x is 3. The coefficient of ab3 in2kab3
is 2k and the coefficient of 63 is 2ka. If two terms contain the same variables with
the sameexponents,then they are called like terms. The distributive property allows
us to combine like terms. For example,3x -l 2x : (3 + 2)x : 5x.
To simplify an expression means to find a simpler-looking equivalent expression. The properties of the real numbers are used to simpliff expressions.
@ using propertiesto simplify an expression
foanf/e
Simpli$ eachexpression.
a. -4x - (6 - 7x)
b.
:,
1.
c. -6(x - 3) - 3(s- 7x)
Sotution
a. -4x-
-4x* t-(6+ (-7x))l
-4x * [-l(6 + (-ix))]
-4x*l(-6)+jxl
(6-7x):
Definition of subtraction
First propcrty of opposites
Distributiveproperty
C o m r r u t a t i v ca n d a s s o c i a t i v e
propcrtics
l-4x+7xl+ (-6)
3x*6
o.tI *
J
2
-x
J
z+
4
,t^
C o m b i n cl i k c t c r m s .
Write
l2
as
11
to obtain a
c o n r n r o lrl l c n o nirn a t o r ' .
I
4""
C o m b i n el i k c t e r m s .
c. -6(x - 3) - 3(5 - 7x) : - 6 x - r 1 8- 1 5+
: l5x*3
4rV Tht. Simplifu-2(* - 3) - 3(l - x).
2lx
Distributivepropefty
Colrbinc Iiketerms.
ChapterP r rr
12
Prerequisites
For Thought
True or False?Explain.
l. Zero is the only number that is both rational and
irrational. p
6. If a < w andw = z,then a I z.F
1
2. Between any two distinct rational numbers there is
anotherrational number. t
For any real numbers a, b, and c, a - (b - c) :
(a-b)-c.p
8. If a and b are any two real numbers, then the distance
3. Befween any two distinct real numbers there is an
irrational number. T
between a and b on the number line is a - b. p
9. Calculators give only rational answers.T
4. Every real number has a multiplicative inverse.F
10. For any real numbers a and b, the opposite of a * b is
5. If a is not less than and not equal to 3" then a is
greater than 3. r
l|f
Exercises
Match each given statementwith its symbolicform and determine
whether the statementis true or false. If the statement isfalse,
correct it. (Examplel)
1. The numbe, fi
ira real number. e, true
2. The number \6
Ir rational. a,fatse,t/j
'The
a-b.r
Completeeachstatementusingthepropertynamed.(Example
2)
15. 7 + x:
(5 . 4)y
associative
K. 5(y)
Ge
t7. 5(x + 3) :
number 0 is not an irrational number. h, true
18. -3(x - 4) :
" 4. The number -6 is not an integer.b, false,-6 € -/
19. 5x 1- 5 : -,
3.
- 7
-,commutativex
distributive
5.r+ 15
disfributive-3x + t2
distributive5(x + l)
5. The set ofintegers is a subsetofthe real numbers. g, true
20. -5x + 10 =
6. The set ofirrational numbers is a subsetofthe rationals.
d,,false,
I I Q
7. The set ofreal numbers is not a subsetofthe rational numbers.
c, true
21. -13 + (4 + x) : _,associative
8. The set of natural numbers is not a subsetof the whole
numbers. f, false,N C Il
u.fieg
d. IcQ
g.JCR
b. -6#.J
".
fien
C.RgQ
f. Ng14/
Determinewhich
elements
of theset{-3.5, - f Z, -1,0,1, ft,
3.14,n,4.3535.. . , 5.090090009
. . .) aremembers
of thefollowing sets. (Examplel)
11. Irrational numbers
t- \,5, \,6, n', 5.090090009
13. Whole numbers i0, 1]
(-t3 + 4) +:r
commutative
xy
23. 0.125(-)
24. -3 + (-)
: 1,multiplicativeinverse8
= 0, additiveinverse3
Usethepropertiesofoppositesto completeeachequation.
(Example 3)
h.0er
9. Real numbers All
22. yx =
-s(x - 2)
distributive
2s.-(-v5):
27. -l(x2
- y2)
y'-12
10. Rationalnumbers
{ - 3 . 5 -, 1 , 0 , 1 , 3 . r 4 , 4 . 3. .5. 3} s
12. Integers
{- 1,0, 1}
14. Naturalnumbers{1}
r,5 za.
4s.- ( t
a--
- o2):
|
Use the symbolic deJinition of absolute yalue to simplify each
expression. (Example4)
2 e . 1 7. 2 17. 2
3 0 .l o / 3 l o
3 1 l. - L l f t
3 2 .| - 3 1 4 1 3 1 4
P.l rrm
Find the distance on the number line betweeneach pair of
Exercises
13
83. (a - b)(a2 + ab + b2)
84. (a + b)(o2 - ab + b2) le
85. ao *
86. (a + c)b s
numbers.(Exumpla
5)
3 3 . 8 , 1 35
34. 1 ,9 9 e t J
3 6 . 2 2 ,- 9 3 l
J
1l
3e. -t,
o'' -r'
/.
35.
-6, -18 r2
-5
4'.
-3, -141r
41. 238
+tlq
42. 34st
\ 4/
trt,!4
Evaluate each expression.(Ettnry;lc7)
4 e .( 2 . s ) - ( 3. 6 ) 8
s 0 .( s - 3 ) ( 2- 6 )- 8
s r . 1 3- ( 4 . 5 ) l- s 1 2
s z .s - l q - Q . z ) l t
s 4 .( - 8 . 3 )- | - 3 . 7 1 4 s
s 3 .l - 4 . 3 l - l - 3 . 5 l - 3
-\
-
(-6\
-
:l
I -Ul
Jo,
4 - /-1)
- J - - -/ l z
t-r/
58.4 + 2(-6)2tr,
4l
59.3- 4+ 5-7
-4-7
60.4-3 +2-
5 + 64
61.3.6*2.4ztt
6 2 .- 2 . 9 + 3 . 5
63.26.L=).tt,
u o . : . 5 o ( 0 . 7+5z) z s
6s.(3.4 - 1)(l+ 2.4)tv1
6 6 .- 2 - 3 ( 5 - 2 . 8 )3 l
67.2-33-4.01-or
6 8 .1 - ( 3 - l 1 - 2 . 3 1 ) 3
69.72- 2(-3)(-6) l3
70.(-3)2- 4eDes) 3r
7 r . 3 - 4 ( s* 3 . D 2 |
7 2 .l - 3 ( 6. s - 4 . 8 ) 2- r r
'tt< - )P
Evaluate each expressionfa
7 5 . b 2 - 4 a cq t
rI.
,
D-C
o
3
(2-3.$2
74. -----;------;La
tt.-fi_f2
J
:
T+
-2, b : 3, and c : 4. (Exumpla9)
76. (b - 4ac)2tzz.s
/,
v
n. !--!
D'tc
90.;r*3-0.9x0.1-r+3
91. -3(2xy)
6r.r'
92.
I
93.;(6-4x)3-2.r
9a.
s
79, a2 - b2 -5
8 0 .a 2 + b 2 t z
8r. (a - b)(a + b) -s
82. (a + b)z t
95.
6x-2v
2r
lr
I
1(8wz)
1
o(8x
+r':
- 4) 2x
t- /9 6 . - 9 - 6 x
-J .
.r'
(x-9)
97. (3-ax)+
3.r'-6
99. -2(4 - x) - 3(3 - 3x)
ll.r- 17
Solve eachproblem.
l0l.
Use the order ofoperations to evaluate each expression.(Exumplt'il)
57.4-5.32
89. x - 0.15x0.85-r
44. -92 rJr
43. -72 -49
4 s (. - 4 ) 2r 6 4 6 .( - r 0 ) 2r ' r r4 ? .f - + ) ' - t t 6 44 s (. - + ) '
-)
Use the properties of the real numbers to simplifu each expression.
(Erumplcl0)
87. -5x t 3x -2.r
88. -5x - (-8;) 3.r
l3
4tl4
(t)
(Exuntplc
Evaluateeachexponentialexpression.
5l'---
co 56
17))
I
-l+2.r
98. (9x-3)+
@-6x)
3.r+ I
100. 5(4 - 2x) - 2(x - s)
-12.r+30
TargetHeart Rate The expression0.60(220 - a - r) * r
is used to obtain the target heart rate for a cardiovascular
workout for a nonathletic male with age a and resting heart
rate r (StevensCreek Software, www.stevenscreek.com).
a. Simplify the expression.0.4(r' - 0.(r0a-r- 132
b. Find the target heart rate for a 20-year-old nonathletic
male with restingheart rate of 70 beatsper minute.
lzlltlrclrts/nrin
c. Simplify the expression0.60(205 - t - ,) * r, which
is used for an athleticmale. 0.40/'- 0.30a+ 123
102. TargetHeart Rate The expression0.60(226 - a - r) -l r
is used to obtain the target heart rate for a cardiovascular
workout for a nonathletic female with age a and resting heart
rate r (StevensCreek Software,www.stevenscreek.com).
a. Simplif' the expression.0.49 - 0.60o+ 135.6
b. The accompanying table shows the target heart rate for a
22-yearold nonathletic female for various resting heart
rates.Find the missing entries.148.4bcats/nin,
| 52.4bcats/nrin
' Tablefor Exercise102
Resting
Heart
Rate
Target
Heart
Rate
55
60
65
70
75
t44.4
t46.4
150.4
Chapter P I ffi,ffi Prerequisites
14
For Writing/Discussion
-]on a num103.Graph
thenumbers
+,-+,+,*\,o,i,and
ber line. Explain how you decided where to put the numbers.
Arrange these same numbers in order from smallest to
largest.Explain your method.Did you use a calculator?If
how it could be done without one.
so. exolain
l5l
I
5
I
S r t t : t l l c ' stlo l r t u e : t :
. \r.
lt
.r i [ ]
2],
n, ana
104. Use a calculatorto arrangethe numbers+, Vt0,
t57 .
you did
,n in order from smallestto largest.Explain what
to make your decisionson the order ofthese numbers.
Could thesenumbersbe arrangedwithout using a calculator? How do thesenumbersdiffer from those in the orevious
exercise?
S n r a l l c s1t ol a r g c s t
Ef
t57
-s0
ll
z.;.
V t0
turyry Qrbrda*llt7
?f t _
Paying Up A king agreed to pay his gardenerone dollar's worth
oftitanium per day for sevendaysofwork on the castlegrounds.
The king has a seven-dollarbar of titanium that is segmentedso
that it can be broken into sevenone-dollarpieces,but it is bad
luck to break a seven-dollarbar of titanium more than twice. How
can the king make two breaks in the bar and pay the gardenerexactly one dollar's worth oftitanium per day for sevendays?
prcces
Br.cali
inlo l-. 2_.and.l_dollar
l0
I
e,Figurefor ThinkingOutsidethe Box I
PopQutz
- 3- 2 . 4 1 . o
6. E v a l u a t e 5 1
l . Is 0 an irrational number?No
3 . S i m p l i f y- ( 1 - y ) . '
S i m p f i f yl - 2 1 . )
^t
Evaluate
a-
4. Find the distancebetween-3 and9. lz
5. E v a l u a t e 3 . 4 - 5 . 2 ' .
-2-h2
1
ila:2andb:-3.
I
l)
8. S i m p l i l3y ( x - 5 ) * 2 ( 5 - x ) .s r ' - 2 s
li.
Wh4tng@
F o r l n d i v i d u ao
lr
Group Explorations
NumberPuzzles
Puzzles concerning numbers are as old as numbers themselves.The best number
puzzles can be solved without a lot of mathematics, but that does not necessarily
make them easy.
a)
Think of a number,andaddf of this numberto itself.Fromthis sum,subtractj
ofits valueand saywhatyour answeris. Fromyour answersubtractfi ofyoui
answer.
Youwill now haveyour originalnumber(Rhindpapyrus,1849e.c.).
Exprainwhythisworks.(.
i,) lt' , :') ,l,lt.1,) ]( - l')l =.
b)
Think of a number between I and 10. Think of the product of your number and
9. Think of the sum of the digits in your answer.Think of that number minus 5.
Think of the letter in the alphabetthat correspondsto the number you are thinking about. Think of a statethat begins with the letter. Think of the secondletter
in the state.Think of a big animal that begins with that letter. Think of the color
of that animal. The color is gray. Explain. Biggrayclephant
ffi