# 0.2 ABSOLUTE VALUE AND DISTANCE ON THE REAL NUMBER LINE

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1 360040_0002.q 1/3/05 11:17 AM Pge HAPTER 0 A Preclculus Review 0.2 ABSOLUTE VALUE AND DISTANE ON THE REAL NUMBER LINE Fin the solute vlues of rel numers n unerstn the properties of solute vlue. Fin the istnce etween two numers on the rel numer line. Define intervls on the rel numer line. Fin the mipoint of n intervl n use intervls to moel n solve rel-life prolems. Asolute Vlue of Rel Numer TEHNOLOGY Asolute vlue epressions cn e evlute on grphing utility. When n epression such s 3 8 is evlute, prentheses shoul surroun the epression, s in s 3 8. Definition of Asolute Vlue The solute vlue of rel numer is,, if 0 if < 0. At first glnce, it my pper from this efinition tht the solute vlue of rel numer cn e negtive, ut this is not possile. For emple, let 3. Then, ecuse 3 < 0, you hve The following properties re useful for working with solute vlues. Properties of Asolute Vlue 1. Multipliction: 2. Division:, 3. Power: n n 4. Squre root: 2 0 Be sure you unerstn the fourth property in this list. A common error in lger is to imgine tht y squring numer n then tking the squre root, you come ck to the originl numer. But this is true only if the originl numer is nonnegtive. For instnce, if 2, then ut if 2, then The reson for this is tht (y efinition) the squre root symol the nonnegtive root. enotes only

2 360040_0002.q 1/3/05 11:17 AM Pge 0-9 SETION 0.2 Asolute Vlue n Distnce on the Rel Numer Line 0-9 Distnce on the Rel Numer Line onsier two istinct points on the rel numer line, s shown in Figure The irecte istnce from to is. 2. The irecte istnce from to is. 3. The istnce etween n is or. In Figure 0.9, note tht ecuse is to the right of, the irecte istnce from to (moving to the right) is positive. Moreover, ecuse is to the left of, the irecte istnce from to (moving to the left) is negtive. The istnce etween two points on the rel numer line cn never e negtive. Distnce Between Two Points on the Rel Numer Line The istnce etween points 1 n 2 on the rel numer line is given y Directe istnce from to : Directe istnce from to : Distnce etween n : FIGURE 0.9 or Note tht the orer of sutrction with n oes not mtter ecuse n EXAMPLE 1 Fining Distnce on the Rel Numer Line Determine the istnce etween 3 n 4 on the rel numer line. Wht is the irecte istnce from 3 to 4? Wht is the irecte istnce from 4 to 3? SOLUTION The istnce etween 3 n 4 is given y s shown in Figure Distnce = 7 or or FIGURE 0.10 The irecte istnce from 3 to 4 is The irecte istnce from 4 to 3 is TRY IT 1 Determine the istnce etween 2 n 6 on the rel numer line. Wht is the irecte istnce from 2 to 6? Wht is the irecte istnce from 6 to 2?

3 360040_0002.q 1/3/05 11:17 AM Pge HAPTER 0 A Preclculus Review Intervls Define y Asolute Vlue EXAMPLE 2 Defining n Intervl on the Rel Numer Line Fin the intervl on the rel numer line tht contins ll numers tht lie no more thn two units from 3. SOLUTION Let e ny point in this intervl. You nee to fin ll such tht the istnce etween n 3 is less thn or equl to 2. This implies tht 3 2. Requiring the solute vlue of 3 to e less thn or equl to 2 mens tht 3 must lie etween 2 n 2. So, you cn write units 2 units Solving this pir of inequlities, you hve Solution set FIGURE 0.11 So, the intervl is 1, 5, s shown in Figure TRY IT 2 Fin the intervl on the rel numer line tht contins ll numers tht lie no more thn four units from 6. Two Bsic Types of Inequlities Involving Asolute Vlue Let n e rel numers, where > 0. if n only if. Inequlity if n only if or. Interprettion Grph ALGEBRA REVIEW Be sure you see tht inequlities of the form hve solution sets consisting of two intervls. To escrie the two intervls without using solute vlues, you must use two seprte inequlities, connecte y n or to inicte union. All numers whose istnce from is less thn or equl to. All numers whose istnce from is greter thn or equl to. + +

4 360040_0002.q 1/3/05 11:17 AM Pge 0-11 SETION 0.2 Asolute Vlue n Distnce on the Rel Numer Line 0-11 Appliction EXAMPLE 3 Qulity ontrol A lrge mnufcturer hire qulity control firm to etermine the reliility of prouct. Using sttisticl methos, the firm etermine tht the mnufcturer coul epect 0.35% ± 0.17% of the units to e efective. If the mnufcturer offers money-ck gurntee on this prouct, how much shoul e ugete to cover the refuns on 100,000 units? (Assume tht the retil price is \$8.95.) SOLUTION Let r represent the percent of efective units (written in eciml form). You know tht r will iffer from y t most r Figure 0.12() Now, letting e the numer of efective units out of 100,000, it follows tht 100,000r n you hve Figure 0.12() Finlly, letting e the cost of refuns, you hve So, the totl cost of refuns for 100,000 units shoul fll within the intervl given y \$ r , ,000r , \$ Figure 0.12(c) r () Percent of efective units () Numer of efective units (c) ost of refuns FIGURE 0.12 TRY IT 3 Use the informtion in Emple 3 to etermine how much shoul e ugete to cover refuns on 250,000 units. In Emple 3, the mnufcturer shoul epect to spen etween \$1611 n \$4654 for refuns. Of course, the sfer uget figure for refuns woul e the higher of these estimtes. However, from sttisticl point of view, the most representtive estimte woul e the verge of these two etremes. Grphiclly, the verge of two numers is the mipoint of the intervl with the two numers s enpoints, s shown in Figure Mipoint = 2 = FIGURE 0.13 Mipoint of n Intervl The mipoint of the intervl with enpoints n is foun y tking the verge of the enpoints. Mipoint 2

5 360040_0002.q 1/3/05 11:17 AM Pge HAPTER 0 A Preclculus Review EXERISES 0.2 In Eercises 1 6, fin () the irecte istnce from to,() the irecte istnce from to, n (c) the istnce etween n , , , , , 61 5, In Eercises 7 18, use solute vlues to escrie the given intervl (or pir of intervls) on the rel numer line. 7. 2, , 3 9., 2 2, 10., 3 3, 11. 2, , 1 13., 0 4, 14., 20 24, 15. All numers less thn two units from All numers more thn si units from y is t most two units from. 18. y is less thn h units from c. In Eercises 19 34, solve the inequlity n sketch the grph of the solution on the rel numer line > 10 2 < 5 2 < 6 > < < 5 10 > < < 1, > 0 2 > 0 3 4, < 2, 5 2 >, In Eercises 35 40, fin the mipoint of the given intervl , , , , , 3 4 > 0 > 0 5 6, hemistry opper hs melting point M within 0.2 of Use solute vlues to write the rnge s n inequlity. 42. Stock Price A stock mrket nlyst preicts tht over the net yer the price p of stock will not chnge from its current price of \$ y more thn \$2. Use solute vlues to write this preiction s n inequlity. 43. Sttistics The heights h of two-thirs of the memers of popultion stisfy the inequlity h where h is mesure in inches. Determine the intervl on the rel numer line in which these heights lie. 44. Biology The Americn Kennel lu hs evelope guielines for juging the fetures of vrious rees of ogs. For collies, the guielines specify tht the weights for mles stisfy the inequlity w where w is mesure in pouns. Determine the intervl on the rel numer line in which these weights lie. 45. Prouction The estimte ily prouction t refinery is given y 200,000 25,000 where is mesure in rrels of oil. Determine the high n low prouction levels. 46. Mnufcturing The cceptle weights for 20-ounce cerel o re given y where is mesure in ounces. Determine the high n low weights for the cerel o. Buget Vrince In Eercises 47 50, () use solute vlue nottion to represent the two intervls in which epenses must lie if they re to e within \$500 n within 5% of the specifie uget mount n () using the more stringent constrint, etermine whether the given epense is t vrince with the uget restriction. Item Buget Epense 47. Utilities \$ \$ Insurnce \$15, \$14, Mintennce \$20, \$22, Tes \$ \$

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