APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line


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1 APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente y coorinte system clle the rel numer line or is (see Figure D.1). The rel numer corresponing to point on the rel numer line is the coorinte of the point. As Figure D.1 shows, it is customry to ientify those points whose coorintes re integers The rel numer line Figure D Rtionl numers Figure D Irrtionl numers Figure D.3 e π The point on the rel numer line corresponing to zero is the origin n is enote y. The positive irection (to the right) is enote y n rrowhe n is the irection of incresing vlues of. Numers to the right of the origin re positive. Numers to the left of the origin re negtive. The term nonnegtive escries numer tht is either positive or zero. The term nonpositive escries numer tht is either negtive or zero. Ech point on the rel numer line correspons to one n only one rel numer, n ech rel numer correspons to one n only one point on the rel numer line. This type of reltionship is clle onetoonecorresponence. Ech of the four points in Figure D. correspons to rtionl numer one tht cn e written s the rtio of two integers. Note tht 4. 9 n Rtionl numers cn e represente either y terminting ecimls such s.4, 1 or y repeting ecimls such s Rel numers tht re not rtionl re irrtionl. Irrtionl numers cnnot e represente s terminting or repeting ecimls. In computtions, irrtionl numers re represente y eciml pproimtions. Here re three fmilir emples e (See Figure D.3.) D1
2 D APPENDIX D Preclculus Review 1 < if n only if lies to the left of. Figure D.4 Orer n Inequlities One importnt property of rel numers is tht they re orere. If n re rel numers, is less thn if is positive. This orer is enote y the inequlity <. This reltionship cn lso e escrie y sying tht is greter thn n writing >. When three rel numers,, n c re orere such tht < n < c, you sy tht is etween n c n < < c. Geometriclly, < if n only if lies to the left of on the rel numer line (see Figure D.4). For emple, 1 < ecuse 1 lies to the left of on the rel numer line. The following properties re use in working with inequlities. Similr properties re otine if < is replce y n > is replce y. (The symols n men less thn or equl to n greter thn or equl to, respectively.) Properties of Inequlities Let,, c,, n k e rel numers. 1. If < n < c, then < c. Trnsitive Property. If < n c <, then c <. A inequlities. 3. If <, then k < k. A constnt. 4. If < n k >, then k < k. Multiply y positive constnt.. If < n k <, then k > k. Multiply y negtive constnt. NOTE Note tht you reverse the inequlity when you multiply the inequlity y negtive numer. For emple, if < 3, then 4 > 1. This lso pplies to ivision y negtive numer. So, if > 4, then <. A set is collection of elements. Two common sets re the set of rel numers n the set of points on the rel numer line. Mny prolems in clculus involve susets of one of these two sets. In such cses, it is convenient to use set nottion of the form {: conition on }, which is re s follows. The set of ll such tht certin conition is true. { : conition on } For emple, you cn escrie the set of positive rel numers s : >. Set of positive rel numers Similrly, you cn escrie the set of nonnegtive rel numers s :. Set of nonnegtive rel numers The union of two sets A n B, enote y A B, is the set of elements tht re memers of A or B or oth. The intersection of two sets A n B, enote y A B, is the set of elements tht re memers of A n B. Two sets re isjoint if they hve no elements in common.
3 APPENDIX D.1 Rel Numers n the Rel Numer Line D3 The most commonly use susets re intervls on the rel numer line. For emple, the open intervl, : < < Open intervl is the set of ll rel numers greter thn n less thn, where n re the enpoints of the intervl. Note tht the enpoints re not inclue in n open intervl. Intervls tht inclue their enpoints re close n re enote y, :. Close intervl The nine sic types of intervls on the rel numer line re shown in the tle elow. The first four re oune intervls n the remining five re unoune intervls. Unoune intervls re lso clssifie s open or close. The intervls, n, re open, the intervls, n, re close, n the intervl, is consiere to e oth open n close. Intervls on the Rel Numer Line Intervl Nottion Set Nottion Grph Boune open intervl, : < < Boune close intervl, : Boune intervls (neither open nor close),, : < : < Unoune open intervls,, : < : > Unoune close intervls,, : : Entire rel line, : is rel numer NOTE The symols n refer to positive n negtive infinity, respectively. These symols o not enote rel numers. They simply enle you to escrie unoune conitions more concisely. For instnce, the intervl, is unoune to the right ecuse it inclues ll rel numers tht re greter thn or equl to.
4 D4 APPENDIX D Preclculus Review EXAMPLE 1 Liqui n Gseous Sttes of Wter Descrie the intervls on the rel numer line tht correspon to the temperture (in egrees Celsius) for wter in. liqui stte.. gseous stte. Solution. Wter is in liqui stte t tempertures greter thn C n less thn 1 C, s shown in Figure D.()., 1 : < < 1. Wter is in gseous stte (stem) t tempertures greter thn or equl to s shown in Figure D.(). 1, : 1 1 C, () Temperture rnge of wter (in egrees Celsius) () Temperture rnge of stem (in egrees Celsius) Figure D. A rel numer is solution of n inequlity if the inequlity is stisfie (is true) when is sustitute for. The set of ll solutions is the solution set of the inequlity. EXAMPLE Solving n Inequlity Solve < 7. Solution < 7 Write originl inequlity. < 7 A to ech sie. < 1 < 1 < 6 Divie ech sie y. The solution set is, 6. If, then ( ) If, then () NOTE In Emple, ll five inequlities liste s steps in the solution re clle equivlent ecuse they hve the sme solution set. If , then ( 7) Checking solutions of Figure D.6 9 < Once you hve solve n inequlity, check some vlues in your solution set to verify tht they stisfy the originl inequlity. You shoul lso check some vlues outsie your solution set to verify tht they o not stisfy the inequlity. For emple, Figure D.6 shows tht when or the inequlity < 7 is stisfie, ut when 7 the inequlity < 7 is not stisfie.
5 APPENDIX D.1 Rel Numers n the Rel Numer Line D EXAMPLE 3 Solving Doule Inequlity Solve 3 1. Solution [, 1] Solution set of 3 1 Figure D Write originl inequlity. 3 1 Sutrct from ech prt. 1 1 Divie ech prt y n reverse oth inequlities. 1 The solution set is, 1, s shown in Figure D.7. The inequlities in Emples n 3 re liner inequlities tht is, they involve firstegree polynomils. To solve inequlities involving polynomils of higher egree, use the fct tht polynomil cn chnge signs only t its rel zeros (the vlues tht mke the polynomil equl to zero). Between two consecutive rel zeros, polynomil must e either entirely positive or entirely negtive. This mens tht when the rel zeros of polynomil re put in orer, they ivie the rel numer line into test intervls in which the polynomil hs no sign chnges. So, if polynomil hs the fctore form r r 1 < r < r 3 <... 1 r... r n, < r n the test intervls re, r 1, r 1, r,..., r n 1, r n, n r n,. To etermine the sign of the polynomil in ech test intervl, you nee to test only one vlue from the intervl. EXAMPLE 4 Solving Qurtic Inequlity Solve < 6. Choose ( 3)( 3. ) Choose 4. ( 3( ) ) Solution < 6 6 < 3 < Write originl inequlity. Write in generl form. Fctor. 3 Co h ose ( 3) ( Testing n intervl Figure D.8 1. ) 3 4 The polynomil 6 hs n 3 s its zeros. So, you cn solve the inequlity y testing the sign of 6 in ech of the test intervls,,, 3, n 3,. To test n intervl, choose ny numer in the intervl n compute the sign of 6. After oing this, you will fin tht the polynomil is positive for ll rel numers in the first n thir intervls n negtive for ll rel numers in the secon intervl. The solution of the originl inequlity is therefore, 3, s shown in Figure D.8.
6 D6 APPENDIX D Preclculus Review Asolute Vlue n Distnce If is rel numer, the solute vlue of is,, The solute vlue of numer cnnot e negtive. For emple, let 4. Then, ecuse 4 <, you hve 4 if if < Rememer tht the symol oes not necessrily men tht is negtive. Opertions with Asolute Vlue Let n e rel numers n let n e positive integer. 1.., n n NOTE You re ske to prove these properties in Eercises 73, 7, 76, n 77. Properties of Inequlities n Asolute Vlue Let n e rel numers n let k e positive rel numer. 1.. k if n only if k k. 3. k if n only if k or k. 4. Tringle Inequlity: Properties n 3 re lso true if is replce y <. EXAMPLE Solving n Asolute Vlue Inequlity Solve 3. 1 units units 4 Solution set of 3 Figure D Solution Using the secon property of inequlities n solute vlue, you cn rewrite the originl inequlity s oule inequlity. 3 Write s oule inequlity A 3 to ech prt. 1 The solution set is 1,, s shown in Figure D.9.
7 APPENDIX D.1 Rel Numers n the Rel Numer Line D7 EXAMPLE 6 A TwoIntervl Solution Set (, ) (1, ) Solution set of Figure D.1 > 3 + Solution set of + Solution set of Figure D.11 Solve > 3. Solution Using the thir property of inequlities n solute vlue, you cn rewrite the originl inequlity s two liner inequlities. < 3 or > 3 < > 1 The solution set is the union of the isjoint intervls, n 1,, s shown in Figure D.1. Emples n 6 illustrte the generl results shown in Figure D.11. Note tht if >, the solution set for the inequlity is single intervl, wheres the solution set for the inequlity is the union of two isjoint intervls. The istnce etween two points n on the rel numer line is given y. The irecte istnce from to is n the irecte istnce from to is, s shown in Figure D.1. Distnce etween n Directe istnce from to Directe istnce from to Figure D.1 EXAMPLE 7 Distnce on the Rel Numer Line Distnce = Figure D The istnce etween 3 n 4 is (See Figure D.13.). The irecte istnce from 3 to 4 is c. The irecte istnce from 4 to 3 is or The mipoint of n intervl with enpoints n is the verge vlue of n. Tht is, Mipoint of intervl,. To show tht this is the mipoint, you nee only show tht is equiistnt from n.
8 D8 APPENDIX D Preclculus Review EXERCISES FOR APPENDIX D.1 In Eercises 1 1, etermine whether the rel numer is rtionl or irrtionl In Eercises 11 14, write the repeting eciml s rtio of two integers using the following proceure. If , then Sutrcting the first eqution from the secon prouces or Given <, etermine which of the following re true. () < (c) > (e) > () < () 1 < 1 (f) < 16. Complete the tle with the pproprite intervl nottion, set nottion, n grph on the rel numer line. Intervl Nottion, 4 1, 7 In Eercises 17, verlly escrie the suset of rel numers represente y the inequlity. Sketch the suset on the rel numer line, n stte whether the intervl is oune or unoune < < < 8 In Eercises 1 4, use inequlity n intervl nottion to escrie the set. 1. y is t lest 4.. q is nonnegtive. Set Nottion : 3 11 Grph 3. The interest rte r on lons is epecte to e greter thn 3% n no more thn 7%. 4. The temperture T is forecst to e ove 9 F toy. < In Eercises 44, solve the inequlity n grph the solution on the rel numer line < 3 < < 9. 3 > 3. > < > > 3 3. <, > < < 9 3 In Eercises 4 48, fin the irecte istnce from to, the irecte istnce from to, n the istnce etween n () 16, 7 () 16, () 9.34,.6 () 16 11, 7 In Eercises 49 4, use solute vlue nottion to efine the intervl or pir of intervls on the rel numer line = = = = = = = 3 = = = = =
9 APPENDIX D.1 Rel Numers n the Rel Numer Line D9 3. () All numers tht re t most 1 units from 1. () All numers tht re t lest 1 units from () y is t most two units from. () y is less thn units from c. In Eercises 8, fin the mipoint of the intervl () 7, 1 () 8.6, () 6.8, 9.3 () 4.6, Profit The revenue R from selling units of prouct is R 11.9 n the cost C of proucing units is C 9 7. = = = = To mke (positive) profit, R must e greter thn C. For wht vlues of will the prouct return profit? 6. Fleet Costs A utility compny hs fleet of vns. The nnul operting cost C (in ollrs) of ech vn is estimte to e C.3m 3 where m is mesure in miles. The compny wnts the nnul operting cost of ech vn to e less thn $1,. To o this, m must e less thn wht vlue? 61. Fir Coin To etermine whether coin is fir (hs n equl proility of lning tils up or hes up), you toss the coin 1 times n recor the numer of hes. The coin is eclre unfir if For wht vlues of will the coin e eclre unfir? 6. Dily Prouction The estimte ily oil prouction p t refinery is p,, < 1, where p is mesure in rrels. Determine the high n low prouction levels. In Eercises 63 n 64, etermine which of the two rel numers is greter. 63. () or () 11 or 73 () or () or 6. Approimtion Powers of 1 Light trvels t the spee of meters per secon. Which est estimtes the istnce in meters tht light trvels in yer? () (c) () () Writing The ccurcy of n pproimtion to numer is relte to how mny significnt igits there re in the pproimtion. Write efinition for significnt igits n illustrte the concept with emples. True or Flse? In Eercises 67 7, etermine whether the sttement is true or flse. If it is flse, eplin why or give n emple tht shows it is flse. 67. The reciprocl of nonzero integer is n integer. 68. The reciprocl of nonzero rtionl numer is rtionl numer. 69. Ech rel numer is either rtionl or irrtionl. 7. The solute vlue of ech rel numer is positive. 71. If <, then. 7. If n re ny two istinct rel numers, then < or >. In Eercises 73 8, prove the property Hint: 1 7., n n, n 1,, 3, k, if n only if k k, 8. if n only if k or k, k k >. 81. Fin n emple for which n n emple for which Then prove tht. for ll,. 8. Show tht the mimum of two numers n is given y the formul m, k >. >, Derive similr formul for min,.
0.1 THE REAL NUMBER LINE AND ORDER
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