2.4 Linear Inequalities and Problem Solving


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1 Section.4 Liner Inequlities nd Problem Solving 77.4 Liner Inequlities nd Problem Solving S 1 Use Intervl Nottion. Solve Liner Inequlities Using the Addition Property of Inequlity. 3 Solve Liner Inequlities Using the Multipliction nd the Addition Properties of Inequlity. 4 Solve Problems Tht Cn Be Modeled by Liner Inequlities. Reltionships mong mesurble quntities re not lwys described by equtions. For exmple, suppose tht slesperson erns bse of $600 per month plus commission of 0% of sles. Suppose we wnt to find the minimum mount of sles needed to receive totl income of t lest $1500 per month. Here, the phrse t lest implies tht n income of $1500 or more is cceptble. In symbols, we cn write income Ú 1500 This is n exmple of n inequlity, nd we will solve this problem in Exmple 8. A liner inequlity is similr to liner eqution except tht the equlity symbol is replced with n inequlity symbol, such s 6, 7,, or Ú. Liner Inequlities in One Vrible x 3x + 5 Ú 4 y 6 0 3(x  4) 7 5x 3 5 c c c c is greter thn is less is greter is less thn or equl to thn thn or equl to Liner Inequlity in One Vrible A liner inequlity in one vrible is n inequlity tht cn be written in the form x + b 6 c where, b, nd c re rel numbers nd 0. In this section, when we mke definitions, stte properties, or list steps bout n inequlity contining the symbol 6, we men tht the definition, property, or steps lso pply to inequlities contining the symbols 7,, nd Ú. 1 Using Intervl Nottion A solution of n inequlity is vlue of the vrible tht mkes the inequlity true sttement. The solution set of n inequlity is the set of ll solutions. Notice tht the solution set of the inequlity x 7, for exmple, contins ll numbers greter thn. Its grph is n intervl on the number line since n infinite number of vlues stisfy the vrible. If we use open/closedcircle nottion, the grph of 5x x 7 6 looks like the following In this text intervl nottion will be used to write solution sets of inequlities. To help us understnd this nottion, different grphing nottion will be used. Insted of n open circle, we use prenthesis. With this new nottion, the grph of 5x x 7 6 now looks like nd cn be represented in intervl nottion s 1,. The symbol is red infinity nd indictes tht the intervl includes ll numbers greter thn. The left prenthesis indictes tht is not included in the intervl. When is included in the intervl, we use brcket. The grph of 5x x Ú 6 is below nd cn be represented s [,. The following tble shows three equivlent wys to describe n intervl: in set nottion, s grph, nd in intervl nottion.
2 78 CHAPTER Equtions, Inequlities, nd Problem Solving Set Nottion Grph Intervl Nottion 5x x , 5x x 7 6 1, 5x x 6 1 , ] 5x x Ú 6 [, 5x 6 x 6 b6 b (, b) 5x x b6 b [, b] 5x 6 x b6 b (, b] 5x x 6 b6 b [, b) Helpful Hint Notice tht prenthesis is lwys used to enclose nd . CONCEPT CHECK Explin wht is wrong with writing the intervl 15, ]. EXAMPLE 1 Grph ech set on number line nd then write in intervl nottion.. 5x x Ú 6 b. 5x x 616 c. 5x x 36 Solution. [, b. 1 , c. 0.5 (0.5, 3] Grph ech set on number line nd then write in intervl nottion.. 5x x b. 5x x Ú 36 c. 5x 1 x 6 46 Answer to Concept Check: should be 15, since prenthesis is lwys used to enclose Solving Liner Inequlities Using the Addition Property We will use intervl nottion to write solutions of liner inequlities. To solve liner inequlity, we use process similr to the one used to solve liner eqution. We use properties of inequlities to write equivlent inequlities until the vrible is isolted.
3 Section.4 Liner Inequlities nd Problem Solving 79 Addition Property of Inequlity If, b, nd c re rel numbers, then 6 b nd + c 6 b + c re equivlent inequlities. In other words, we my dd the sme rel number to both sides of n inequlity, nd the resulting inequlity will hve the sme solution set. This property lso llows us to subtrct the sme rel number from both sides. EXAMPLE nottion. Solve: x Grph the solution set nd write it in intervl Solution x x Add to both sides. x 6 7 Simplify. The solution set is 5x x 6 76, which in intervl nottion is 1 , 7. The grph of the solution set is Solve: x Grph the solution set nd write it in intervl nottion. Helpful Hint In Exmple, the solution set is 5x x This mens tht ll numbers less thn 7 re solutions. For exmple, 6.9, 0, p, 1, nd re solutions, just to nme few. To see this, replce x in x with ech of these numbers nd see tht the result is true inequlity. EXAMPLE 3 Solve: 3x + 4 Ú x  6. Grph the solution set nd write it in intervl nottion. Solution 3x + 4 Ú x  6 3x x Ú x x Subtrct x from both sides. x + 4 Ú 6 x Ú 64 x Ú 10 Combine like terms. Subtrct 4 from both sides. Simplify. The solution set is 5x x Ú 106, which in intervl nottion is [10,. The grph of the solution set is Solve: 3x + 1 x  3. Grph the solution set nd write it in intervl nottion. 3 Solving Liner Inequlities Using the Multipliction nd Addition Properties Next, we introduce nd use the multipliction property of inequlity to solve liner inequlities. To understnd this property, let s strt with the true sttement nd multiply both sides by.
4 80 CHAPTER Equtions, Inequlities, nd Problem Solving Multiply by. True The sttement remins true. Notice wht hppens if both sides of re multiplied by Multiply by . Flse The inequlity is flse sttement. However, if the direction of the inequlity sign is reversed, the result is true True These exmples suggest the following property. Multipliction Property of Inequlity If, b, nd c re rel numbers nd c is positive, then * b nd c * bc re equivlent inequlities. If, b, nd c re rel numbers nd c is negtive, then * b nd c + bc re equivlent inequlities. In other words, we my multiply both sides of n inequlity by the sme positive rel number nd the result is n equivlent inequlity. We my lso multiply both sides of n inequlity by the sme negtive number nd reverse the direction of the inequlity symbol, nd the result is n equivlent inequlity. The multipliction property holds for division lso, since division is defined in terms of multipliction. Helpful Hint Whenever both sides of n inequlity re multiplied or divided by negtive number, the direction of the inequlity symbol must be reversed to form n equivlent inequlity. EXAMPLE 4 intervl nottion x 3 8 Solve nd grph the solution set. Write the solution set in b. .3x Helpful Hint The inequlity symbol is the sme since we re multiplying by positive number. Solution. 1 4 x # 1 4 x 4 # 3 8 x 3 Multiply both sides by 4. Simplify. The solution set is e x ` x 3 f, which in intervl nottion is , 3 d. The grph of the solution set is w
5 Section.4 Liner Inequlities nd Problem Solving 81 Helpful Hint The inequlity symbol is reversed since we divided by negtive number. b. .3x x x 73 Divide both sides by .3 nd reverse the inequlity symbol. Simplify. The solution set is 5x x 736, which is 13, in intervl nottion. The grph of the solution set is Solve nd grph the solution set. Write the solution set in intervl nottion.. 5 x Ú 4 15 b. .4x CONCEPT CHECK In which of the following inequlities must the inequlity symbol be reversed during the solution process?. x 7 7 b. x c. x x 6 7 d. x To solve liner inequlities in generl, we follow steps similr to those for solving liner equtions. Solving Liner Inequlity in One Vrible Step 1. Step. Step 3. Step 4. Step 5. Cler the inequlity of frctions by multiplying both sides of the inequlity by the lest common denomintor (LCD) of ll frctions in the inequlity. Use the distributive property to remove grouping symbols such s prentheses. Combine like terms on ech side of the inequlity. Use the ddition property of inequlity to write the inequlity s n equivlent inequlity with vrible terms on one side nd numbers on the other side. Use the multipliction property of inequlity to isolte the vrible. Helpful Hint Don t forget tht 5 x mens the sme s x Ú 5. Answer to Concept Check:, d EXAMPLE 5 Solve: 1x x x. Solution 1x x x x x x 5  x 7x x + x 7x x 5 8x x x 0 8 8x 8 5 x, or x Ú 5 Apply the distributive property. Combine like terms. Add x to both sides. Combine like terms. Add 15 to both sides. Combine like terms. Divide both sides by 8. Simplify.
6 8 CHAPTER Equtions, Inequlities, nd Problem Solving The solution set written in intervl nottion is c 5, b nd its grph is e 5 Solve: 14x x x. Grph nd write the solution set in intervl nottion. EXAMPLE 6 Solve: 1x  6 Ú x Solution 1x  6 Ú x c 1x  6d Ú 51x x  6 Ú 51x  1 x  1 Ú 5x  53x  1 Ú 53x Ú 73x x Multiply both sides by 5 to eliminte frctions. Apply the distributive property. Subtrct 5x from both sides. Add 1 to both sides. Divide both sides by 3 nd reverse the inequlity symbol. Simplify. The solution set written in intervl nottion is ,  7 d nd its grph is Solve: 3 1x  3 Ú x  7. Grph nd write the solution set in intervl 5 nottion. EXAMPLE 7 Solve: 1x x + 1. Solution 1x x + 1 x x + 1 x x 7 x x Distribute on the left side. Subtrct x from both sides. Simplify is true sttement for ll vlues of x, so this inequlity nd the originl inequlity re true for ll numbers. The solution set is 5x x is rel number6, or 1 , in intervl nottion, nd its grph is Solve: 41x  6 4x + 5. Grph nd write the solution set in intervl nottion.
7 Section.4 Liner Inequlities nd Problem Solving 83 4 Solving Problems Modeled by Liner Inequlities Appliction problems contining words such s t lest, t most, between, no more thn, nd no less thn usully indicte tht n inequlity is to be solved insted of n eqution. In solving pplictions involving liner inequlities, we use the sme procedure s when we solved pplictions involving liner equtions. EXAMPLE 8 Clculting Income with Commission A slesperson erns $600 per month plus commission of 0% of sles. Find the minimum mount of sles needed to receive totl income of t lest $1500 per month. Solution 1. UNDERSTAND. Red nd rered the problem. Let x = mount of sles.. TRANSLATE. As stted in the beginning of this section, we wnt the income to be greter thn or equl to $1500. To write n inequlity, notice tht the slesperson s income consists of $600 plus commission (0% of sles). commission In words: Ú , of sles T T T Trnslte: x Ú SOLVE the inequlity for x x Ú x Ú x Ú 900 x Ú INTERPRET. Check: The income for sles of $4500 is , or Thus, if sles re greter thn or equl to $4500, income is greter thn or equl to $1500. Stte: The minimum mount of sles needed for the slesperson to ern t lest $1500 per month is $4500 per month. 8 A slesperson erns $900 month plus commission of 15% of sles. Find the minimum mount of sles needed to receive totl income of t lest $400 per month. EXAMPLE 9 Finding the Annul Consumption In the United Sttes, the nnul consumption of cigrettes is declining. The consumption c in billions of cigrettes per yer since the yer 000 cn be pproximted by the formul c = 9.4t where t is the number of yers fter 000. Use this formul to predict the yers tht the consumption of cigrettes will be less thn 00 billion per yer. Solution 1. UNDERSTAND. Red nd rered the problem. To become fmilir with the given formul, let s find the cigrette consumption fter 0 yers, which would be the yer , or 00. To do so, we substitute 0 for t in the given formul. c = = 43
8 84 CHAPTER Equtions, Inequlities, nd Problem Solving Thus, in 00, we predict cigrette consumption to be bout 43 billion. Vribles hve lredy been ssigned in the given formul. For review, they re c = the nnul consumption of cigrettes in the United Sttes in billions of cigrettes nd t = the number of yers fter 000. TRANSLATE. We re looking for the yers tht the consumption of cigrettes c is less thn 00. Since we re finding yers t, we substitute the expression in the formul given for c, or 9.4t SOLVE the inequlity. 9.4t t t t 7 pproximtely INTERPRET. Check: Substitute number greter thn 4.6 nd see tht c is less thn 00. Stte: The nnul consumption of cigrettes will be less thn 00 billion more thn 4.6 yers fter 000, or pproximtely for = 05 nd ll yers fter. 9 Use the formul given in Exmple 9 to predict when the consumption of cigrettes will be less thn 75 billion per yer. Vocbulry, Rediness & Video Check Mtch ech grph with the intervl nottion tht describes it , b. 15,  c. 1, 5 d. 1 , # c 7 4, .5b b. .5, 7 4 d c. c .5, 7 4 b d. 7 4, .5b (, 114 b. 111, c. 311, ) d. 1 , 11 j c , 0.b b. 0., d c , 0. d d. c 0., b Use the choices below to fill in ech blnk. 1 , 0.4 (, , ) 10.4, (, The set 5x x Ú written in intervl nottion is. 6. The set 5x x written in intervl nottion is. 7. The set 5x x written in intervl nottion is. 8. The set 5x x written in intervl nottion is.
9 Section.4 Liner Inequlities nd Problem Solving 85 MrtinGy Interctive Videos See Video.4 Wtch the section lecture video nd nswer the following questions Using Exmple 1 s reference, explin how the grph of the solution set of n inequlity cn help you write the solution set in intervl nottion. 10. From the lecture before Exmple 3, explin the ddition property of inequlity in your own words. Wht equlity property does it closely resemble? 11. Bsed on the lecture before Exmple 4, complete the following sttement. If you multiply or divide both sides of n inequlity by the nonzero negtive number, you must the direction of the inequlity symbol. 1. Wht words or phrses in the Exmple 7 sttement tell you this is n inequlity ppliction (besides the lst line telling you to use n inequlity)?.4 Exercise Set Grph the solution set of ech inequlity nd write it in intervl nottion. See Exmple x x x x x x Ú x x x 7 x6 6. 5x 7 Ú x6 7. 5x  6 x x 5 Ú x 716 Solve. Grph the solution set nd write it in intervl nottion. See Exmples through x  7 Ú x x 6 6x x 6 10x x  7 7x x Ú x Ú x x x Ú 9. 4x Ú 8 Solve. Write the solution set using intervl nottion. See Exmples 5 through x + 7 Ú x x Ú 4x x 6 6x x x Ú x x x x x x 3  x x Ú x 5 x x 3 7 x Ú x  1 Ú 6x x x x x  14 MIXED Solve. Write the solution set using intervl nottion. See Exmples 1 through x x x x x x Ú x Ú x x x x x x  7 Ú x x + 1 x x + 0.6x Ú x  x x x  4 Ú 31x x x x x x x  6 Ú 61x y  19y + 51y x  1 Ú 4x x x x 55. 3x x x 6 71x x x +
10 86 CHAPTER Equtions, Inequlities, nd Problem Solving x x x x x x x x x x x x x x x x 3 Ú x x x x x x Solve. See Exmples 8 nd 9. For Exercises 69 through 76,. nswer with n inequlity nd b., in your own words, explin the mening of your nswer to prt (). Exercises 69 nd 70 re written to help you get strted. Exercises 71 nd 7 re written to help you write nd solve the inequlity. 69. Shurek Wshburn hs scores of 7, 67, 8, nd 79 on her lgebr tests.. Use n inequlity to find the scores she must mke on the finl exm to pss the course with n verge of 77 or higher, given tht the finl exm counts s two tests. b. In your own words, explin the mening of your nswer to prt (). 70. In Winter Olympics 5000meter speedskting event, Hns Holden scored times of 6.85, 7.04, nd 6.9 minutes on his first three trils.. Use n inequlity to find the times he cn score on his lst tril so tht his verge time is under 7.0 minutes. b. In your own words, explin the mening of your nswer to prt (). 71. A clerk must use the elevtor to move boxes of pper. The elevtor s mximum weight limit is 1500 pounds. If ech box of pper weighs 66 pounds nd the clerk weighs 147 pounds, use n inequlity to find the number of whole boxes she cn move on the elevtor t one time. Strt the solution: 1. UNDERSTAND the problem. Rered it s mny times s needed.. TRANSLATE into n inequlity. (Fill in the blnks below.) Let x = number of boxes elevtor Clerk s number weight of + times mximum weight of boxes ech box weight T T T T T T T + x # Finish with: 3. SOLVE nd 4. INTERPRET 7. To mil lrge envelope first clss, the U.S. Post Office chrges 88 cents for the first ounce nd 17 cents per ounce for ech dditionl ounce. Use n inequlity to find the number of whole ounces tht cn be miled for no more thn $.00. Price of first ounce Strt the solution: 1. UNDERSTAND the problem. Rered it s mny times s needed.. TRANSLATE into n inequlity. (Fill in the blnks below.) Let x = number of dditionl ounces (fter first ounce) + number of price per 00 dditionl times dditionl mximum ounces ounce cents T T T T T T T + x # Finish with: 3. SOLVE nd 4. INTERPRET 73. A smll plne s mximum tkeoff weight is 000 pounds or less. Six pssengers weigh n verge of 160 pounds ech. Use n inequlity to find the luggge nd crgo weights the plne cn crry. 74. A shopping mll prking grge chrges $1 for the first hlfhour nd 60 cents for ech dditionl hlfhour. Use n inequlity to find how long you cn prk if you hve only $4.00 in csh. 75. A cr rentl compny offers two subcompct rentl plns. Pln A: $36 per dy nd unlimited milege Pln B: $4 per dy plus $0.15 per mile Use n inequlity to find the number of dily miles for which pln A is more economicl thn pln B.
11 Section.4 Liner Inequlities nd Problem Solving Northest Telephone Compny offers two billing plns for locl clls. Pln 1: $5 per month for unlimited clls Pln : $13 per month plus $0.06 per cll Use n inequlity to find the number of monthly clls for which pln 1 is more economicl thn pln. 77. At room temperture, glss used in windows ctully hs some properties of liquid. It hs very slow, viscous flow. (Viscosity is the property of fluid tht resists internl flow. For exmple, lemonde flows more esily thn fudge syrup. Fudge syrup hs higher viscosity thn lemonde.) Glss does not become true liquid until tempertures re greter thn or equl to 500 C. Find the Fhrenheit tempertures for which glss is liquid. (Use the formul F = 9 5 C + 3.) 78. Stibnite is silvery white minerl with metllic luster. It is one of the few minerls tht melts esily in mtch flme or t tempertures of pproximtely 977 F or greter. Find the Celsius tempertures t which stibnite melts. [Use the formul C = 5 1F  3.] Although beginning slries vry gretly ccording to your field of study, the eqution s = 184.7t + 48,133 cn be used to pproximte nd to predict verge beginning slries for cndidtes with bchelor s degrees. The vrible s is the strting slry nd t is the number of yers fter 000. (Source: Sttisticl Abstrct of the U.S.). Approximte the yer in which beginning slries for cndidtes will be greter thn $70,000. (Round your nswer up nd use it to clculte the yer.) b. Determine the yer you pln to grdute from college. Use this yer to find the corresponding vlue of t nd pproximte beginning slry for bchelor s degree cndidte Use the formul in Exmple 9 to estimte the yers tht the consumption of cigrettes will be less thn 50 billion per yer. b. Use your nswer to prt () to describe the limittions of your nswer. The verge consumption per person per yer of whole milk w cn be pproximted by the eqution w = .11t where t is the number of yers fter 000 nd w is mesured in pounds. The verge consumption of skim milk s per person per yer cn be pproximted by the eqution s = 0.4t where t is the number of yers fter 000 nd s is mesured in pounds. The consumption of whole milk is shown on the grph in blue nd the consumption of skim milk is shown on the grph in red. Use this informtion to nswer Exercises 81 through 90. Averge consumption per person per yer (in pounds) Whole Milk versus Skim Milk Whole milk Skim milk Yers fter Source: Bsed on dt from U.S. Deprtment of Agriculture, Economic Reserch Service 81. Is the consumption of whole milk incresing or decresing over time? Explin how you rrived t your nswer. 8. Is the consumption of skim milk incresing or decresing over time? Explin how you rrived t your nswer. 83. Predict the consumption of whole milk in 015. (Hint: Find the vlue of t tht corresponds to 015.) 84. Predict the consumption of skim milk in 015. (Hint: Find the vlue of t tht corresponds to 015.) 85. Determine when the consumption of whole milk will be less thn 45 pounds per person per yer. 86. For 000 through 010, the consumption of whole milk ws greter thn the consumption of skim milk. Explin how this cn be determined from the grph. 87. Both lines hve negtive slope, tht is, the mount of ech type of milk consumed per person per yer is decresing s time goes on. However, the mount of whole milk being consumed is decresing fster thn the mount of skim milk being consumed. Explin how this could be. 88. Do you think it is possible tht the consumption of whole milk will eventully be the sme s the consumption of skim milk? Explin your nswer. 89. The consumption of skim milk will be greter thn the consumption of whole milk when s 7 w.. Find when this will occur by substituting the given equivlent expression for w nd the given equivlent expression for s nd solving for t. b. Estimte to the nerest whole the first yer when this will occur. 90. How will the two lines in the grph pper if the consumption of whole milk is the sme s the consumption of skim milk? REVIEW AND PREVIEW List or describe the integers tht mke both inequlities true. See Section x 6 5 nd x x Ú 0 nd x x Ú  nd x Ú 94. x 6 6 nd x 65
12 88 CHAPTER Equtions, Inequlities, nd Problem Solving Solve ech eqution for x. See Section x  6 = x  1 = x + 7 = 5x x  4 = x  4 CONCEPT EXTENSIONS Ech row of the tble shows three equivlent wys of describing n intervl. Complete this tble by filling in the equivlent descriptions. The first row hs been completed for you. Ech inequlity below (Exercises ) is solved by dividing both sides by the coefficient of x. Determine whether the inequlity symbol will be reversed during this solution process x x x Ú x Solve: x  3 = Solve: x Solve: x Set Nottion 5x x 636 Grph 3 Intervl Nottion 1 , Red the equtions nd inequlities for Exercises 109, 110, nd 111 nd their solutions. In your own words, write down your thoughts When grphing the solution set of n inequlity, explin how you know whether to use prenthesis or brcket Explin wht is wrong with the intervl nottion 16, x x Explin how solving liner inequlity is similr to solving liner eqution x x Explin how solving liner inequlity is different from solving liner eqution (, , 4) Integrted Review LINEAR EQUATIONS AND INEQUALITIES Sections.1.4 Solve ech eqution or inequlity. For inequlities, write the solution set in intervl nottion x = 0. 4x x 4 Ú 4. 5x + 3 Ú + 4x 5. 61y  4 = 31y x x Ú y + 4 = 41y x 6 71x  5x x = x + 4 = x  x = x x x x 5  x 4 = x x  1 = 81x x x = b b  1 = 5b x x x 1. 3t = 5 + t x x = 81x  35x 3. x 6 + 3x y y 5 = y x x 7 31x x  17x 13x x +  x Ú 3 8 1x x x 7 5 1x
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