CHAPTER P Prerequisites P. Rel Numers P.2 Crtesin Coordinte System P.3 Liner Equtions nd Inequlities P.4 Lines in the Plne P.5 Solving Equtions Grphiclly, Numericlly, nd Algericlly P.6 Comple Numers P.7 Solving Inequlities Algericlly nd Grphiclly Lrge distnces re mesured in light yers, the distnce light trvels in one yer. Astronomers use the speed of light, pproimtely 86,000 miles per second, to pproimte distnces etween plnets. See pge 35 for emples.
2 CHAPTER P Prerequisites Biliogrphy For students: Gret Jos for Mth Mjors, Stephen Lmert, Ruth J. DeCotis. Mthemticl Assocition of Americ, 998. For techers: Alger in Technologicl World, Addend Series, Grdes 9 2. Ntionl Council of Techers of Mthemtics, 995. Why Numers Count Quntittive Litercy for Tommorrow s Americ, Lynn Arthur Steen (Ed.). Ntionl Council of Techers of Mthemtics, 997. Chpter P Overview Historiclly, lger ws used to represent prolems with symols (lgeric models) nd solve them y reducing the solution to lgeric mnipultion of symols. This technique is still importnt tody. Grphing clcultors re used tody to pproch prolems y representing them with grphs (grphicl models) nd solve them with numericl nd grphicl techniques of the technology. We egin with sic properties of rel numers nd introduce solute vlue, distnce formuls, midpoint formuls, nd equtions of circles. Slope of line is used to write stndrd equtions for lines, nd pplictions involving liner equtions re discussed. Equtions nd inequlities re solved using oth lgeric nd grphicl techniques. P. Rel Numers Wht you ll lern out Representing Rel Numers Order nd Intervl Nottion Bsic Properties of Alger Integer Eponents Scientific Nottion... nd why These topics re fundmentl in the study of mthemtics nd science. Ojective Students will e le to convert etween decimls nd frctions, write inequlities, pply the sic properties of lger, nd work with eponents nd scientific nottion. Motivte Ask students how rel numers cn e clssified. Hve students discuss wys to disply very lrge or very smll numers without using lot of zeros. Representing Rel Numers A rel numer is ny numer tht cn e written s deciml. Rel numers re represented y symols such s -8, 0,.75, 2.333 Á, 0.36, 8/5, 23, 23 6, e, nd p. The set of rel numers contins severl importnt susets: The nturl (or counting) numers: 5, 2, 3, Á 6 The whole numers: 50,, 2, 3, Á 6 The integers: 5 Á, -3, -2, -, 0,, 2, 3, Á 6 The rces 5 6 re used to enclose the elements, or ojects, of the set. The rtionl numers re nother importnt suset of the rel numers. A rtionl numer is ny numer tht cn e written s rtio / of two integers, where Z 0. We cn use set-uilder nottion to descrie the rtionl numers: e `, re integers, nd Z 0 f The verticl r tht follows / is red such tht. The deciml form of rtionl numer either termintes like 7/4 =.75, or is infinitely repeting like 4/ = 0.363636 Á= 0.36. The r over the 36 indictes the lock of digits tht repets. A rel numer is irrtionl if it is not rtionl. The deciml form of n irrtionl numer is infinitely nonrepeting. For emple, 23 =.7320508 Á nd p = 3.459265Á. Rel numers re pproimted with clcultors y giving few of its digits. Sometimes we cn find the deciml form of rtionl numers with clcultors, ut not very often.
SECTION P. Rel Numers 3 /6 55/27 /7 N.0625 2.037037037.0588235294 FIGURE P. Clcultor deciml representtions of /6, 55/27, nd /7 with the clcultor set in floting deciml mode. (Emple ) EXAMPLE Emining Deciml Forms of Rtionl Numers Determine the deciml form of /6, 55/27, nd /7. SOLUTION Figure P. suggests tht the deciml form of /6 termintes nd tht of 55/27 repets in locks of 037. 6 = 0.0625 nd 55 27 = 2.037 We cnnot predict the ect deciml form of /7 from Figure P.; however, we cn sy tht /7 L 0.0588235294. The symol L is red is pproimtely equl to. We cn use long division (see Eercise 66) to show tht Now try Eercise 3. 7 = 0.05882352947647. The rel numers nd the points of line cn e mtched one-to-one to form rel numer line. We strt with horizontl line nd mtch the rel numer zero with point O, the origin. Positive numers re ssigned to the right of the origin, nd negtive numers to the left, s shown in Figure P.2. 3 O π 5 4 3 2 0 Negtive rel numers 2 3 4 5 Positive rel numers FIGURE P.2 The rel numer line. Every rel numer corresponds to one nd only one point on the rel numer line, nd every point on the rel numer line corresponds to one nd only one rel numer. Between every pir of rel numers on the numer line there re infinitely mny more rel numers. The numer ssocited with point is the coordinte of the point. As long s the contet is cler, we will follow the stndrd convention of using the rel numer for oth the nme of the point nd its coordinte. Order nd Intervl Nottion The set of rel numers is ordered. This mens tht we cn use inequlities to compre ny two rel numers tht re not equl nd sy tht one is less thn or greter thn the other. Unordered Systems Not ll numer systems re ordered. For emple, the comple numer system, to e introduced in Section P.6, hs no nturl ordering. Order of Rel Numers Let nd e ny two rel numers. Symol Definition Red 7 - is positive is greter thn 6 - is negtive is less thn Ú - is positive or zero is greter thn or equl to - is negtive or zero is less thn or equl to The symols 7, 6, Ú, nd re inequlity symols.
4 CHAPTER P Prerequisites Opposites nd Numer Line 6 0 Q - 7 0 If 6 0, then is to the left of 0 on the rel numer line, nd its opposite, -, is to the right of 0. Thus, - 7 0. Geometriclly, 7 mens tht is to the right of (equivlently is to the left of ) on the rel numer line. For emple, since 6 7 3, 6 is to the right of 3 on the rel numer line. Note lso tht 7 0 mens tht - 0, or simply, is positive nd 6 0 mens tht is negtive. We re le to compre ny two rel numers ecuse of the following importnt property of the rel numers. Trichotomy Property Let nd e ny two rel numers. Ectly one of the following is true: 6, =, or 7. 3 2 0 2 3 4 5 () 3 2 0 2 3 4 5 () Inequlities cn e used to descrie intervls of rel numers, s illustrted in Emple 2. EXAMPLE 2 Interpreting Inequlities Descrie nd grph the intervl of rel numers for the inequlity. () 6 3 () - 6 4 SOLUTION () The inequlity 6 3 descries ll rel numers less thn 3 (Figure P.3). () The doule inequlity - 6 4 represents ll rel numers etween - nd 4, ecluding - nd including 4 (Figure P.3). Now try Eercise 5. 0.5 5 4 3 2 0 2 3 (c) 3 2 0 2 3 4 5 (d) FIGURE P.3 In grphs of inequlities, prentheses correspond to 6 nd 7 nd rckets to nd Ú. (Emples 2 nd 3) EXAMPLE 3 Writing Inequlities Write n intervl of rel numers using n inequlity nd drw its grph. () The rel numers etween -4 nd -0.5 () The rel numers greter thn or equl to zero SOLUTION () -4 6 6-0.5 (Figure P.3c) () Ú 0 (Figure P.3d) Now try Eercise 5. As shown in Emple 2, inequlities define intervls on the rel numer line. We often use 32, 54 to descrie the ounded intervl determined y 2 5. This intervl is closed ecuse it contins its endpoints 2 nd 5. There re four types of ounded intervls. Bounded Intervls of Rel Numers Let nd e rel numers with 6. Intervl Intervl Inequlity Nottion Type Nottion Grph 3, 4 Closed, 2 Open 3, 2 Hlf-open 6 6 6, 4 Hlf-open 6 The numers nd re the endpoints of ech intervl.
SECTION P. Rel Numers 5 Intervl Nottion t ˆ Becuse - q is not rel numer, we use - q, 22 insted of 3- q, 22 to descrie 6 2. Similrly, we use 3-, q2 insted of 3-, q4 to descrie Ú -. The intervl of rel numers determined y the inequlity 6 2 cn e descried y the unounded intervl - q, 22. This intervl is open ecuse it does not contin its endpoint 2. We use the intervl nottion - q, q2 to represent the entire set of rel numers. The symols - q (negtive infinity) nd q (positive infinity) llow us to use intervl nottion for unounded intervls nd re not rel numers. There re four types of unounded intervls. Unounded Intervls of Rel Numers Let nd e rel numers. Intervl Intervl Inequlity Nottion Type Nottion Grph 3, q2 Closed Ú, q2 Open 7 - q, 4 Closed - q, 2 Open 6 Ech of these intervls hs ectly one endpoint, nmely or. EXAMPLE 4 Converting Between Intervls nd Inequlities Convert intervl nottion to inequlity nottion or vice vers. Find the endpoints nd stte whether the intervl is ounded, its type, nd grph the intervl. () 3-6, 32 () - q, -2 (c) -2 3 SOLUTION () The intervl 3-6, 32 corresponds to -6 6 3 nd is ounded nd hlf-open (see Figure P.4). The endpoints re -6 nd 3. () The intervl - q, -2 corresponds to 6 - nd is unounded nd open (see Figure P.4). The only endpoint is -. (c) The inequlity -2 3 corresponds to the closed, ounded intervl 3-2, 34 (see Figure P.4c). The endpoints re -2 nd 3. Now try Eercise 29. () () (c) 6 5 4 3 2 0 2 3 4 FIGURE P.4 Grphs of the intervls of rel numers in Emple 4. Bsic Properties of Alger Alger involves the use of letters nd other symols to represent rel numers. A vrile is letter or symol for emple,, y, t, u2 tht represents n unspecified rel numer. A constnt is letter or symol for emple, -2, 0, 23, p2 tht represents specific rel numer. An lgeric epression is comintion of vriles nd constnts involving ddition, sutrction, multipliction, division, powers, nd roots.
6 CHAPTER P Prerequisites Sutrction vs. Negtive Numers On mny clcultors, there re two - keys, one for sutrction nd one for negtive numers or opposites. Be sure you know how to use oth keys correctly. Misuse cn led to incorrect results. We stte some of the properties of the rithmetic opertions of ddition, sutrction, multipliction, nd division, represented y the symols +, -, * (or # ) nd, (or /), respectively. Addition nd multipliction re the primry opertions. Sutrction nd division re defined in terms of ddition nd multipliction. Sutrction: - = + -2 Division: =, Z 0 In the ove definitions, - is the dditive inverse or opposite of, nd / is the multiplictive inverse or reciprocl of. Perhps surprisingly, dditive inverses re not lwys negtive numers. The dditive inverse of 5 is the negtive numer -5. However, the dditive inverse of -3 is the positive numer 3. The following properties hold for rel numers, vriles, nd lgeric epressions. Properties of Alger Let u, v, nd w e rel numers, vriles, or lgeric epressions.. Commuttive property Addition: u + v = v + u Multipliction: uv = vu 2. Associtive property Addition: u + v2 + w = u + v + w2 Multipliction: uv2w = uvw2 3. Identity property Addition: u + 0 = u Multipliction: u # = u 4. Inverse property Addition: u + -u2 = 0 Multipliction: u # u =, u Z 0 5. Distriutive property Multipliction over ddition: uv + w2 = uv + uw u + v2w = uw + vw Multipliction over sutrction: uv - w2 = uv - uw u - v2w = uw - vw The left-hnd sides of the equtions for the distriutive property show the fctored form of the lgeric epressions, nd the right-hnd sides show the epnded form. EXAMPLE 5 Using the Distriutive Property () Write the epnded form of + 22. () Write the fctored form of 3y - y. SOLUTION () () + 22 = + 2 3y - y = 3-2y Now try Eercise 37. Here re some properties of the dditive inverse together with emples tht help illustrte their menings. Properties of the Additive Inverse Let u nd v e rel numers, vriles, or lgeric epressions. Property Emple --u2 = u --32 = 3 -u2v = u-v2 = -uv2. 2. 3. 4. -2u = -u -423 = 4-32 = -4 # 32 = -2 -u2-v2 = uv -62-72 = 6 # 7 = 42-25 = -5 5. -u + v2 = -u2 + -v2-7 + 92 = -72 + -92 = -6
SECTION P. Rel Numers 7 Integer Eponents Eponentil nottion is used to shorten products of fctors tht repet. For emple, -32-32-32-3) = -32 4 nd 2 + 22 + 2 = 2 + 2 2. Eponentil Nottion Let e rel numer, vrile, or lgeric epression nd n positive integer. Then n = # # Á #, n fctors where n is the eponent, is the se, nd to the nth power. n is the nth power of, red s The two eponentil epressions in Emple 6 hve the sme vlue ut hve different ses. Be sure you understnd the difference. Understnding Nottion -32 2 = 9-3 2 = -9 Be creful! EXAMPLE 6 Identifying the Bse () In -32 5, the se is -3. () In -3 5, the se is 3. Now try Eercise 43. Here re the sic properties of eponents together with emples tht help illustrte their menings. Properties of Eponents Let u nd v e rel numers, vriles, or lgeric epressions nd m nd n e integers. All ses re ssumed to e nonzero. Property. u m u n = u m+n u m 2. u n = um-n 3. u 0 = Emple 5 3 # 5 4 = 5 3+4 = 5 7 9 4 = 9-4 = 5 8 0 = 4. u -n = u n y -3 = y 3 5. uv2 m = u m v m 2z2 = 2 5 z 5 = 32z 5 6. u m 2 n = u mn 2 2 3 = 2 # 3 = 6 7. 7 u m = 7 v = um v m 7 To simplify n epression involving powers mens to rewrite it so tht ech fctor ppers only once, ll eponents re positive, nd eponents nd constnts re comined s much s possile.
8 CHAPTER P Prerequisites Moving Fctors Be sure you understnd how eponent property 4 permits us to move fctors from the numertor to the denomintor nd vice vers: v -m u -n = un v m EXAMPLE 7 Simplifying Epressions Involving Powers () 2 3 25 2 5 2 = 0 2 2 3 5 2 = 0 3 8 () u 2 v -2 u - v 3 = u2 u v 2 v 3 = u3 v 5 (c) 2-3 Now try Eercise 47. 2 = 2 2-3 2-3 = -6 23 = -3 2 6 = 8 6 Scientific Nottion Any positive numer cn e written in scientific nottion, c * 0 m, where c 6 0 nd m is n integer. This nottion provides wy to work with very lrge nd very smll numers. For emple, the distnce etween the Erth nd the Sun is out 93,000,000 miles. In scientific nottion, 93,000,000 mi = 9.3 * 0 7 mi. The positive eponent 7 indictes tht moving the deciml point in 9.3 to the right 7 plces produces the deciml form of the numer. The mss of n oygen molecule is out 0.000 000 000 000 000 000 000 053 grm. In scientific nottion, 0.000 000 000 000 000 000 000 053 g = 5.3 * 0-23 g. The negtive eponent -23 indictes tht moving the deciml point in 5.3 to the left 23 plces produces the deciml form of the numer. EXAMPLE 8 Converting to nd from Scientific Nottion () () 2.375 * 0 8 = 237,500,000 0.000000349 = 3.49 * 0-7 Now try Eercises 57 nd 59. EXAMPLE 9 Using Scientific Nottion 360,00024,500,000,0002 Simplify, without using clcultor. 8,000 SOLUTION 360,00024,500,000,0002 8,000 = 3.6 * 05 24.5 * 0 9 2.8 * 0 4 = 3.624.52 * 0 5+9-4.8 = 9 * 0 0 = 90,000,000,000 Now try Eercise 63. Using Clcultor Figure P.5 shows two wys to perform the computtion. In the first, the numers re entered in deciml form. In the second, the numers re entered in scientific nottion. The clcultor uses 9E0 to stnd for 9 * 0 0.
SECTION P. Rel Numers 9 (360000)(4500000 000)/(8000) (3.6E5)(4.5E9)/(.8E4) N 9E0 9E0 FIGURE P.5 Be sure you understnd how your clcultor displys scientific nottion. (Emple 9) QUICK REVIEW P.. List the positive integers etween -3 nd 7. 2. List the integers etween -3 nd 7. 3. List ll negtive integers greter thn -4. 4. List ll positive integers less thn 5. In Eercises 5 nd 6, use clcultor to evlute the epression. Round the vlue to two deciml plces. 2-5.52-6 5. () 4-3.2 3 - -4.22 5 () 7.4-3.8 6. () 533-.2 2-4-0.52 3 4 () 5-2 + 2-4 In Eercises 7 nd 8, evlute the lgeric epression for the given vlues of the vriles. 7. 3-2 +, = -2,.5 8. 2 + + 2, = -3, = 2 In Eercises 9 nd 0, list the possile reminders. 9. When the positive integer n is divided y 7 0. When the positive integer n is divided y 3 SECTION P. EXERCISES Eercise numers with gry ckground indicte prolems tht the uthors hve designed to e solved without clcultor. In Eercises 4, find the deciml form for the rtionl numer. Stte whether it repets or termintes.. -37/8 2. 5/99 3. -3/6 4. 5/37 In Eercises 5 0, descrie nd grph the intervl of rel numers. 5. 2 6. -2 6 5 7. - q, 72 8. 3-3, 34 9. is negtive. 0. is greter thn or equl to 2 nd less thn or equl to 6. In Eercises 6, use n inequlity to descrie the intervl of rel numers.. 3-, 2 2. - q, 44 3. 4. 5. is etween - nd 2. 6. is greter thn or equl to 5. In Eercises 7 22, use intervl nottion to descrie the intervl of rel numers. 7. 7-3 8. -7 6 6-2 9. 20. 2. is greter thn -3 nd less thn or equl to 4. 22. is positive. In Eercises 23 28, use words to descrie the intervl of rel numers. 23. 4 6 9 24. Ú - 25. 3-3, q2 26. -5, 72 27. 28.
0 CHAPTER P Prerequisites In Eercises 29 32, convert to inequlity nottion. Find the endpoints nd stte whether the intervl is ounded or unounded nd its type. 29. -3, 44 30. -3, -2 3. - q, 52 32. 3-6, q2 In Eercises 33 36, use oth inequlity nd intervl nottion to descrie the set of numers. Stte the mening of ny vriles you use. 33. Writing to Lern Bill is t lest 29 yers old. 34. Writing to Lern No item t Srh s Vriety Store costs more thn $2.00. 35. Writing to Lern The price of gllon of gsoline vries from $.099 to $.399. 36. Writing to Lern Slry rises t the Stte University of Cliforni t Chico will verge etween 2% nd 6.5%. In Eercises 37 40, use the distriutive property to write the fctored form or the epnded form of the given epression. 37. 2 + 2 38. y - z 3 2c 39. 2 + d 2 40. 3 z + 3 w In Eercises 4 nd 42, find the dditive inverse of the numer. 4. 6 - p 42. -7 In Eercises 43 nd 44, identify the se of the eponentil epression. 43. -5 2 44. -22 7 45. Group Activity Discuss which lgeric property or properties re illustrted y the eqution. Try to rech consensus. () 32y = 3y2 () 2 = 2 (c) 2 + - 2 2 = 0 (d) + 32 2 + 0 = + 32 2 (e) + y2 = + y 46. Group Activity Discuss which lgeric property or properties re illustrted y the eqution. Try to rech consensus. () + 22 + 2 = () # + y2 = + y (c) 2 - y2 = 2-2y (d) 2 + y - z2 = 2 + y + -z22 = 2 + y2 + -z2 = 2 + y2 - z (e) 2 = = # = In Eercises 47 52, simplify the epression. Assume tht the vriles in the denomintors re nonzero. 4 y 3 3 2 2 2 y 4 47. 48. 2 y 5 3y 2 49. 50. 2-3 4 2 2 y -3 y 2 2-4 5. y 6-4 2-2 52. 43 32 2 3 2 2 4 The dt in Tle P. give the ependitures in millions of dollrs for U.S. pulic schools for the 2005 2006 school yer. Tle P. U.S. Pulic Schools Ctegory Amount (in millions) Current ependitures 449,595 Cpitl outly 57,375 Interest on school det 4,347 Totl 528,735 Source: Ntionl Center for Eduction Sttistics, U.S. Deprtment of Eduction, s reported in The World Almnc nd Book of Fcts 2009. In Eercises 53 56, write the mount of ependitures in dollrs otined from the ctegory in scientific nottion. 53. Current ependitures 54. Cpitl outly 55. Interest on school det 56. Totl In Eercises 57 nd 58, write the numer in scientific nottion. 57. The men distnce from Jupiter to the Sun is out 483,900,000 miles. 58. The electric chrge, in couloms, of n electron is out -0.000 000 000 000 000 000 6. In Eercises 59 62, write the numer in deciml form. 59. 3.33 * 0-8 60. 6.73 * 0 6. The distnce tht light trvels in yer (one light yer) is out 5.87 * 0 2 mi. 62. The mss of neutron is out.6747 * 0-24 g. In Eercises 63 nd 64, use scientific nottion to simplify..3 * 0-7 22.4 * 0 8 2 63. without using clcultor.3 * 0 9 64. 3.7 * 0-7 24.3 * 0 6 2 2.5 * 0 7 Eplortions 65. Investigting Eponents For positive integers m nd n, we cn use the definition to show tht m n = m+n. () Emine the eqution m n = m+n for n = 0 nd eplin why it is resonle to define 0 = for Z 0. () Emine the eqution m n = m+n for n = -m nd eplin why it is resonle to define -m = / m for Z 0.
SECTION P. Rel Numers 66. Deciml Forms of Rtionl Numers Here is the third step when we divide y 7. (The first two steps re not shown, ecuse the quotient is 0 in oth cses.) 0.05 7.00 85 5 By convention we sy tht is the first reminder in the long division process, 0 is the second, nd 5 is the third reminder. () Continue this long division process until reminder is repeted, nd complete the following tle: Step Quotient Reminder 0 2 0 0 3 o 5 o 5 o () Eplin why the digits tht occur in the quotient etween the pir of repeting reminders determine the infinitely repeting portion of the deciml representtion. In this cse 7 = 0.05882352947647. (c) Eplin why this procedure will lwys determine the infinitely repeting portion of rtionl numer whose deciml representtion does not terminte. Stndrdized Test Questions 67. True or Flse The dditive inverse of rel numer must e negtive. Justify your nswer. 68. True or Flse The reciprocl of positive rel numer must e less thn. Justify your nswer. In Eercises 69 72, solve these prolems without using clcultor. 69. Multiple Choice Which of the following inequlities corresponds to the intervl 3-2, 2? (A) -2 (B) -2 (C) -2 6 6 (D) -2 6 (E) -2 6 70. Multiple Choice Wht is the vlue of -22 4? (A) 6 (B) 8 (C) 6 (D) -8 (E) -6 7. Multiple Choice Wht is the se of the eponentil epression -7 2? (A) -7 (B) 7 (C) -2 (D) 2 (E) 72. Multiple Choice Which of the following is the simplified form of, Z 0? 6 (A) (C) (E) -4 3 8 2 (B) (D) Etending the Ides The mgnitude of rel numer is its distnce from the origin. 73. List the whole numers whose mgnitudes re less thn 7. 74. List the nturl numers whose mgnitudes re less thn 7. 75. List the integers whose mgnitudes re less thn 7. 2 4