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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