Linear Inequalities A linear inequality in one variable is an inequality such as

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1 7.4 Liner Inequlities A liner inequlity in one vrile is n inequlity such s x 5 2, y 3 5, or 2k Archimedes, one of the gretest mthemticins of ntiquity, is shown on this Itlin stmp. He ws orn in the Greek city of Syrcuse out 287 B.C. A colorful story out Archimedes reltes his rection to one of his discoveries. While tking th, he noticed tht n immersed oject, if hevier thn fluid, will, if plced in it, descend to the ottom of the fluid, nd the solid will, when weighed in the fluid, e lighter thn its true weight y the weight of the fluid displced. This discovery so excited him tht he rn through the streets shouting Eurek! ( I hve found it! ) without othering to clothe himself! Archimedes met his deth t ge 75 during the pillge of Syrcuse. He ws using snd try to drw geometric figures when Romn soldier cme upon him. He ordered the soldier to move cler of his circles, nd the soldier oliged y killing him. Liner Inequlity in One Vrile A liner inequlity in one vrile cn e written in the form x v c, where,, nd c re rel numers, with 0. (Throughout this section we give the definitions nd rules only for, ut they re lso vlid for,, nd.) Solving Liner Inequlities An inequlity is solved y finding ll numers tht mke the inequlity true. Usully n inequlity hs n infinite numer of solutions. These solutions, like the solutions of equtions, re found y producing series of simpler equivlent inequlities. Equivlent inequlities re inequlities with the sme solution set. Such inequlities re found with the ddition nd multipliction properties of inequlity. Addition Property of Inequlity For ll rel numers,, nd c, the inequlities v nd c v c re equivlent. (The sme numer my e dded to oth sides of n inequlity without chnging the solution set.)

2 7.4 Liner Inequlities 353 As with equtions, the ddition property cn e used to sutrct the sme numer from oth sides of n inequlity. EXAMPLE 1 Solve x x 7 12 x Add 7 to oth sides. x 5 Using set-uilder nottion, the solution set of this inequlity is written x x FIGURE 8 It is customry to express solution sets of inequlities with numer line grphs. The solution set in Exmple 1 cn e shown on numer line s in Figure 8. The set of numers less thn 5 is n exmple of n intervl on the numer line. To write intervls, we use intervl nottion. For exmple, using this nottion, the intervl of ll numers less thn 5 is written, 5. The negtive infinity symol does not indicte numer. It is used to show tht the intervl includes ll rel numers less thn 5. As on the numer line, the prenthesis indictes tht 5 is not included in the solution set. Exmples of other sets written in intervl nottion re shown in the following chrt. In these intervls, ssume tht. Note tht prenthesis is lwys used with the symols nd. Type of Intervl Intervl Set Nottion Grph x x, Open intervl x x, x x, x x, Hlf-open intervl x x x x,, x x, Closed intervl x x, We sometimes use, to represent the set of ll rel numers.

3 354 CHAPTER 7 The Bsic Concepts of Alger FIGURE 9 EXAMPLE 2 Solve the inequlity 14 2m 3m. Give the solution set in intervl form, nd grph the solution set s well. 14 2m 3m 14 2m 2m 3m 2m Sutrct 2m. 14 m Comine like terms. The inequlity 14 m (14 is less thn or equl to m) cn lso e written m 14 (m is greter thn or equl to 14). Notice tht in ech cse, the inequlity symol points to the smller numer, 14. The solution set in intervl nottion is 14,. The grph is shown in Figure 9. Errors often occur in grphing inequlities where the vrile term is on the right side. (This is proly due to the fct tht we red from left to right.) To gurd ginst such errors, it is good ide to rewrite these inequlities so tht the vrile is on the left, s discussed in Exmple 2. An inequlity such s 3x 15 cn e solved y dividing oth sides y 3. This is done with the multipliction property of inequlity, which is little more involved thn the corresponding property for equtions. To see how this property works, strt with the true sttement 2 5. Multiply oth sides y, sy, Multiply y True This gives true sttement. Strt gin with 2 5, nd this time multiply oth sides y Multiply y Flse The result, 16 40, is flse. To mke it true, chnge the direction of the inequlity symol to get True As these exmples suggest, multiplying oth sides of n inequlity y negtive numer forces the direction of the inequlity symol to e reversed. The sme is true for dividing y negtive numer, since division is defined in terms of multipliction. Multipliction Property of Inequlity For ll rel numers,, nd c, with c 0, () the inequlities v nd c v c re equivlent if c w 0; () the inequlities v nd c w c re equivlent if c v 0. (Both sides of n inequlity my e multiplied or divided y positive numer without chnging the direction of the inequlity symol. Multiplying or dividing y negtive numer requires tht the inequlity symol e reversed.)

4 7.4 Liner Inequlities FIGURE 10 EXAMPLE 3 Solve 3x x. Give the solution set in oth intervl nd grph forms. 3x x 2x Distriutive property Add x. Add 10. Divide oth sides y 2. Be creful. Dividing y negtive numer requires chnging to. 2x 2 3x x 3x 10 8 x 3x 10 x 8 x x 2x x 9 2x 18 Divide y 2; reverse inequlity symol. Figure 10 shows the grph of the solution set,, 9. Most liner inequlities, such s the one in Exmple 3, require the use of oth the ddition nd multipliction properties. The steps used in solving liner inequlity re summrized elow. Solving Liner Inequlity Step 1: Step 2: Simplify ech side of the inequlity s much s possile y using the distriutive property to cler prentheses nd y comining like terms s needed. Use the ddition property of inequlity to chnge the inequlity so tht ll terms with vriles re on one side nd ll terms without vriles re on the other side. Step 3: Use the multipliction property to chnge the inequlity to the form x k (or x k, or x k, or x k). Rememer: Reverse the direction of the inequlity symol only when multiplying or dividing oth sides of n inequlity y negtive numer. Three-prt Inequlities Inequlities cn e used to express the fct tht quntity lies etween two other quntities. For exmple, sys tht 2 4 nd 4 7. This sttement is conjunction nd is true ecuse oth prts of the sttement re true. The three-prt inequlity 3 x 2 8

5 356 CHAPTER 7 The Bsic Concepts of Alger FIGURE 11 is true when the expression x 2 is etween 3 nd 8. To solve this inequlity, sutrct 2 from ech of the three prts of the inequlity, giving 3 2 x x 6. The solution set, 1, 6, is grphed in Figure 11. When using inequlities with three prts such s the one ove, it is importnt to hve the numers in the correct positions. It would e wrong to write the inequlity s 8 x 2 3, since this would imply tht 8 3, flse sttement. In generl, three-prt inequlities re written so tht the symols point in the sme direction, nd they oth point towrd the smller numer. EXAMPLE 4 Solve 2 3k 1 5 nd grph the solution set. Add 1 to ech of the three prts to isolte the vrile term in the middle k k k Divide y 3. Add _ k 2 FIGURE 12 A grph of the solution set, 13, 2, is shown in Figure 12. Applictions In ddition to the fmilir is less thn nd is greter thn, the expressions is no more thn, is t lest, nd others lso indicte inequlities. The tle elow shows how these expressions re interpreted. Word Expression Interprettion Word Expression Interprettion is t lest is t most is no less thn is no more thn Do not confuse sttement such s 5 is more thn numer with the phrse 5 more thn numer. The first of these is sentence, written s 5 x, while the second is n expression, written with ddition s x 5. Prolem Solving The next exmple shows n ppliction of lger tht is importnt to nyone who hs ever sked himself or herself, Wht score cn I mke on my next test nd hve (prticulr grde) in this course? It uses the ide of finding the verge of numer of grdes. In generl, to find the verge of n numers, dd the numers, nd divide y n. EXAMPLE 5 Christopher Michel hs test scores of 86, 88, nd 78 on his first three tests in geometry. If he wnts n verge of t lest 80 fter his fourth test, wht re the possile scores he cn mke on his fourth test?

6 7.4 Liner Inequlities 357 Let x Christopher s score on his fourth test. To find his verge fter 4 tests, dd the test scores nd divide y 4. Averge is t 80. lest x x x x x x 68 Add the known scores. Multiply y 4. Sutrct 252. Comine terms. He must score 68 or more on the fourth test to hve n verge of t lest 80. EXAMPLE 6 A rentl compny chrges $15.00 to rent chin sw, plus $2.00 per hour. Mmie Zwettler cn spend no more thn $35.00 to cler some logs from her yrd. Wht is the mximum mount of time she cn use the rented sw? Let h the numer of hours she cn rent the sw. She must py $15.00, plus $2.00h, to rent the sw for h hours, nd this mount must e no more thn $ Cost of renting 15 2h is no more thn 35 dollrs. 15 2h h 20 h 10 Sutrct 15. Divide y 2. Mmie cn use the sw for mximum of 10 hours. (Of course, she my use it for less time, s indicted y the inequlity h 10.) 35