CHAPTER 1 Equations and Inequalities

Transcription

1 CHAPTER Equations and Inequalities Section. Linear Equations... Section. Mathematical Modeling...6 Section. Quadratic Equations... Section.4 The Quadratic Formula...9 Mid-Chapter Quiz Solutions...4 Section. Other Types of Equations...47 Section.6 Linear Inequalities... Section.7 Other Types of Inequalities...6 Review Eercises...6 Chapter Test Solutions...67 Cumulative Test Solutions...69

2 CHAPTER Equations and Inequalities Section. Linear Equations Skills Review ( ) ( ) ( + ) The equation ( ) is an identity because (by the Distributive Property) it is true for every real value of.. The equation ( ) + 4 is conditional since there are real number values of for which the equation is not true.. The equation ( ) + + is conditional since there are no real number values of for which the equation is true. 7. +? (a) is not a solution. (b)? is not a solution.? (c) is a solution.? (d) is not a solution. 9. +? + (a) 6 6 is a solution.? (b) () () () is a solution.? (c) 0 4 is not a solution.? (d) is not a solution.

3 Section. Linear Equations. 4 4? (a) ( ) ( ) is a solution. 4? (b) is not a solution. 4 (c) is undefined is not a solution. 4? (d) is not a solution. 4. ( )( ) (a)? is not a solution. (b)? is not a solution. (c) ( + )( )? is not a solution. (d) ( + )( )? is a solution.. (a) 6?? 9 6 is a solution. (b) 9 9 is undefined is not a solution. (c) is undefined. is not a solution. (d) ( ) is undefined. is not a solution ( + ) 7 ( ) [ ]??

4 4 Chapter Equations and Inequalities ( z + ) ( z + 4) 0 4 4( z + ) 4( 4)( z + 4) 4( 0) 6( z + ) ( z + 4) 0 6z + 0 z 4 0 z 6 z ( 0 ) 4( 0.) + 4( 0.7)( 0 ) ( ) No solution u u u u ( u) ( u ) 400 6u u u 0 u ( ) ( + ) ( 0) ( 4) ( )( + ) + ( )( + ) ( )( + ) ( )( ) 6 ( ) + 4( ) A check reveals that is an etraneous solution, so there is no solution ( ) ( + ) ( + )( ) 6

5 Section. Linear Equations ( ) 4( ) A check reveals that is an etraneous solution, so there is no solution. 49. ( + ) + ( + ) ( + ) 4( + ) The equation is an identity; every real number is a solution.. ( + ) 4( + + ) No solution 4. When you check in the original equation, you get division by zero, which is undefined. So, is an etraneous solution and the equation has no solution. 7. Etraneous solutions may arise when a fractional epression is multiplied by factors involving the variable. For eample, in Eercise 47, we multiplied by ( ). If or are actually zero, an etraneous solution is introduced. In Eercise 47, an etraneous solution was since multiplying the equation by ( ) actually multiplied the equation by zero. 9. Equivalent equations are equations that have the same solutions as the original equation. Usually, as part of the process of solving an equation, we rewrite the equation (using the properties and rules of algebra) into an equivalent form. For eample, to solve Eercise 49, as we would rewrite Net we might combine terms on the left so we would have equivalent equation We would continue the solution process to the equation 0 and finally ( 4.40)( 7.98) (.40)( 7.98) Use the table feature in ASK mode or evaluate using the scientific calculator part of a graphing utility. (Answers will vary.) (a) (b) The second method introduced an additional round-off error (a) (b) The second method introduced an additional round-off error.

6 6 Chapter Equations and Inequalities 7. y 944.7t + 9,898, 8 t, t + 9,898, t,0 t t The per capita personal income was $,000 in inches 77. y 9.t + 0., 8 t t t 679. t 9. t Credit etended to consumers was $ trillion in y 0.t +.8, 7 t t t. t 0. t In 00 the value of the federal minimum wage was $4.60 in 996 dollars. Section. Mathematical Modeling Skills Review ( + ) 6 ( ) [ + + 6] 6 + ( + 7) ( ) 0( ) + [ + 0 0] + 7 ( 0) y y

7 Section. Mathematical Modeling 7 Skills Review continued 9. z z z + z z z + ( z )( z + ) z z + z 6 z z 6 0 z ( ). Model: (first number) + (second number) Labels: first number n, second number n + n + n + n + Epression:. Model: (rate) (time) Labels: rate 0 mph, time t Epression: 0t. Model: 0% (amount of solution) Labels: amount of solution (in gallons) Epression: Model: (width) + (length) Labels: width, length (width) + 6 Epression: 9. Model: (shipping fee) + (unit cost)(number of units) Labels: (shipping fee) $00, unit cost $, number of units Epression: r. 9. n + n 7. Model: sum (first number) + (second number) Labels: sum, first number n, second number n + Equation: n + ( n + ) 4 n n 6 Answer: first number n 6, second number n Model: difference (one number) (another number) Labels: difference 48, one number, another number Equation: Answer: one number 8, another number 7. Model: product (first number) (second number) Labels: product (first number) n, first number n, second number n + Equation: n n( n + ) n n + n n Answer: first number n, second number n + 4. Model: (Total of paychecks) (coworker's paycheck) + (your paycheck) Labels: Total of paychecks $848, coworker's paycheck, your paycheck (coworker's paycheck) + % (coworker's paycheck) Equation: Answer: coworker's paycheck $400, your paycheck. $448

8 8 Chapter Equations and Inequalities. Model: (Total profit) (January profit) + (February profit) Labels: Total profit $9,000, January profit, February profit (January profit) + % (January profit) Equation: 9, , ,96.8 Answer: January profit $6,96.8, February profit.0 $66, Model: (980 Star Wars) (percent change)(977 Star Wars) + (977 Star Wars) Labels: 980 Star Wars $90,7,960; percent change p, 977 Star Wars $460,998,007 Equation: 90,7,960 p(460,998,007) + 460,998,007 70,76,047 p 460,998,007 p % Answer: The percent decrease in revenues was about 7%. 9. Models: (999 Star Wars) (percent change)(98 Star Wars) + (98 Star Wars) Labels: 999 Star Wars $4,088,9; percent change p, 98 Star Wars $09,09,079 Equation: 4,088,9 p(09,09,079) + 09,09,079,879,6 p 09,09,079 p % Answer: The percent increase in revenues was about 9.4%.. Model: (00 Star Wars) (percent change)(00 Star Wars) + (00 Star Wars) Labels: 00 Star Wars $80,6,; percent change p, 00 Star Wars $0,67,8 Equation: 80,6, p(0,67,8) + 0,67,8 69,86,97 p 0,67,8 p 0.4.4% Answer: The percent increase in revenues was about.4%.. Model: (006 size) (percent change)(past size) + (Past size) Labels: 006 size 6, percent change p, Past size 7 Equation: 6 p(7) p 7 p % Answer: The percent increase in size was about 8.6%.. Model: (006 size) (percent change)(past size) + (Past size) Labels: 006 size 6, percent change p, Past size. Equation: 6 p(.) +.. p. p % Answer: The percent increase in size was about 7.4%.

9 Section. Mathematical Modeling 9 7. Model: (Smaller lunch) (percent change)(larger lunch) + Labels: Smaller lunch 660, percent change p, Larger lunch 440 Equation: 660 p(440) p 440 p % Answer: The percent decrease in calories is about 4.% 9. Model: (Salary) (percent increase)(salary previous year) + (Salary previous year) Labels: S Salary second year S Salary third year S Salary fourth year 4 (a) S 0.08(,000) +,000 7,800 Your salary for the second year is $7,800. (b) S 0.078(7,800) + 7,800 40, Your salary for the third year is $40, (c) S (40,748.40) + 40, ,78.7 Your salary for the fourth year is $44, Model: (Number of world Internet users) (percent change)(number of users in previous year) + (Number of users in previous year) Labels: N N N number of world Internet users in 00 number of world Internet users in 004 number of world Internet users in 006 (a) N (0.48)(00) + 00 N 79 The number of world internet users in 00 was about 79 million. (b) N (0.6)(79) + 79 N The number of world Internet users in 004 was about 86.8 million. (c) N (0.8)(86.8) N The number of world Internet users in 006 was about 09.9 million. (d) 09.9 p(00) p 00 p % The percent increase in the number of users from 00 to 006 was about 8.6%.

10 0 Chapter Equations and Inequalities 4. Model: (Media type usages) (percent)() Labels: H Hours spent watching TV H Hours spent listening to radio or recorded music H Hours spent using the Internet H 4 Hours spent playing non-internet video games H Hours spent reading print media H Hours spent using other media 6 H (0.44)() 64. In 009, the average person will spend 64. hours watching TV. H (0.)() 7.6 In 009, the average person will spend 7.6 hours listening to the radio or recorded music. H (0.06)(). In 009, the average person will spend. hours using the Internet H 4 (0.0)() 06.6 In 009, the average person will spend 06.6 hours playing non-internet video games. H (0.)() 9.0 In 009, the average person will spend 9.0 hours reading print media. H 6 (0.04)() 4. In 009, the average person will spend 4. hours using other media. 4. Model: perimeter (width) + (length) Labels: perimeter, length (in feet).(width). Equation: 7 + (. ) 7 Answer: width feet, length.. feet 47. Model: Interest (principal)(rate)(time) Labels: Interest $000, principal $00, rate 0.07, time Equation: 000 (00)(0.07) Answer: About.7 years (test #) + (test #) + (test #) + (test #4) 49. Model: average 4 Labels: average 90, test # 87, test # 9, test # 84, test # Equation: Answer: You must score 97 or more on test 4 to have an average of at least 90%.

11 Section. Mathematical Modeling. Model: (Sale Price) (list price) (discount percent)(list price) Labels: list price, 00 sale price, % discount Equation:00 (0.) $ Answer: The list price was $ percent discount discount amount. Model: 00 original price Labels: discount amount $0 original price $49 p percent discount p 0 Equation: p Answer: The satellite radio system was discounted 0.%.. Model: (Sale price) (List price) (discount percent)(list price) Labels: Sale price.60, List price (Whole price)(0.60) + Wholesale price, discount percent 0., wholesale price w w + w w + w Equation:.60 (0.60) 0. ( 0.60).60.60w 0.(.60w).60.w.60 w. 8 w Answer: The whole sale price of the power drill is $8. 7. (Reduced salary) (Original weekly salary) (discount percent)(original weekly salary) 4 0.(4) $6. The reduced salary is $ (time) (distance) ( rate) 0 miles 0 miles (rate) 8 minutes 7 hour ( 0) t hours 0 ( 7 ) The entire trip will take about hours.

12 Chapter Equations and Inequalities 6. distance rate time d 40 mph t d mph t (distance between cars) (second distance) (first distance) d d t 40t t t hour The cars will be miles apart after hour, or 0 minutes. 6. time distance rate meters 8 t.0 0 meters per second t.8 seconds It will take.8 seconds for a radio wave to travel from Houston to the surface of the moon. height of tree height of lamp post 6. tree's shadow lamp post's shadow 6. feet tall The tree is 6. feet tall. 67. Total epenses ( Monthly epenses),40 T $78,080 If the monthly epense rate continues, the total epenses for the year will be $78, Model: (Interest from 6.%) + (Interest from 7.%) (Total interest) Labels: Amount invested at 6.%, Amount invested at 7.%,000, Interest from 6.% 0.06, Interest from 7.% 0.07(,000 ), Total interest 00 Equation: (,000 ) ,00 Answer: The amount invested at 6.% is $0,00 and the amount invested at 7.% is,000,000 0,00 $ Model: (Amount earned by stock A) + (Amount earned by stock B) (Total amount earned) Labels: Amount invested in stock A, Amount invested in stock B 000, Amount earned by stock A 0.098, Amount earned by stock B 0.06 Equation: ( 000 ) , Total amount earned 89.0 Answer: The amount invested in stock A is $00 and the amount invested in stock B is $800.

13 Section. Mathematical Modeling 7. interest ( interest rate) i 9.% $,000 i r $8000 principal + ( interest in second account) total interest interest in first account $04.40 i + i , r r 0.4.4% 8000 An interest rate of.4% yields the same interest amount as the variable rate fund. 7. cost ( fied costs) + ( variable cost)( number of units) $90,000 $,000 + $8.7 7, The company can manufacture about 87 units. 77. Volume π rh diameter r 4 r 60. π h 60. h 48 feet long 4π 79. ( final concentration)( amount) ( sol. concentration)( amount) + ( sol. concentration)( amount) ( 7% )( gal) ( 40% )( ) + ( 00% ) gal distance 8. rate time miles hours 60. miles per hour 8. A bh A bh A b h 8. V lwh V wh l 87. V πr h V πr h 89. S C + RC S C + R S + R C 9. A P + Prt A P Prt A P r Pt A a b h A h a + b A ah h b 9. ( + ) 9. + ( ) L a n d L a + nd d nd L + d a L + d a n d

14 4 Chapter Equations and Inequalities 97. A π rh A h π r f R R n f R R 99. ( n ) ( ) + n f R R ( ) + ( ) ( ) R n f n fr R R ( n ) fr + ( ) R n f R is the reciprocal of. R 0., ,800 +,00 6,00 Williams' average $8,700 0, ,00 +,600 64,000 8,600 +, ,400 60,000 Walters' average $0,000 8,00 + 8,700 +, 000 9,800 Gilbert's average $9,9, ,00 + 0,000,00 Hart's average $7,8, , , ,00 +,000 8,600 January's average $7,0 8, ,00 +, , ,00 00,00 February's average $0,00 March's average,00 +, ,400 +, ,000 07,00 $, Williams' average $,0 Gonzalez's average $,867 Walters' average $,400 Gilbert's average $7,467 Hart's average $8,00 Reges' average $4,967 Sanders' average $,6 July's average $4,4 August's average $,7 September's average $,00 0. The time the diver takes to descend from feet to 0 feet and the depth of 0 feet are red herrings.

15 Section. Quadratic Equations Section. Quadratic Equations Skills Review () ( + 7) 6. 4 ( + )( ) 7. 6 ( ) 4 ( ) 4 ( ) ( 7)( + ) ( 7)( ) ( + 9)( ) ( )( + ) ( 6 )( ) ( ) ( )( ) or or ( ) or or ( ) or + 0 or or 0 6 or ( )( ) ( )( ) or + 0 or 6 ± 4 7 ± 7 ±.6

16 6 Chapter Equations and Inequalities 7. ± ± ± ± 6.4 or ( ). + + ± ±.46 or ± ± 8 ± ( ) ( ) ± ± ± ± 78 ± ± ± ( ) or 4 0 or ( )( ) ( ) ± or or ( )( ) or 0 or ( )( ) or 0 or ( 9)( ) 9 0 or 0 9 or ( ) ( )( ) or 0 or

17 Section. Quadratic Equations 7 9. ( ) ( ) ± ± + or or 6. ( ) ( ) Answers will vary. Sample answer: Algebra Argument: ( + ) ( + )( + ) Definition of an eponent F.O.I.L Simplify So, for any real. Graphing utility argument: Let y + and y + 4. Use the table feature with a value of other than zero. The table will show y is not the same as y. OR Use a scientific calculator to show that if then + 49 and so ( + ) is not the same as Area ( length)( width) 6 ( w + 4)( w) w + w 0 ( w + 48)( w 4) w or w 4 0 w 48 or w 4 Etraneous Solution: length w feet, width w 4 feet The building has a length of 48 feet and a width of 4 feet. Area (base)(height) 4 ( b)( b) 8 b ± 8 b b is etraneous. Solution: base height.8 feet The sign has a base length of about.8 feet and a height of about.8 feet. 69. Area ( length)( width) 00 ( + 0)( + 0) ( + 0) 0 4( + 0)( ) or 0 0 or Etraneous Solution: width feet The width of the path is feet.

18 8 Chapter Equations and Inequalities 7. s 6t + v0t + s0 v0 0 and s0 00 s 0 when the rock hits the ocean. 0 6t t 00 t t ± t.4 is etraneous The rock hits the ocean in about.4 seconds. s 6t + v0t + s0 v0 0 and s0 0 meters.808 feet s 0 when the diver hits the water. 0 6t t.808 t.00 t ± is etraneous t.4 The diver will be in the air about.4 seconds. s 6t + v0t + s0 s 000, v0 0, and s0, t +,000 6t 0,000 t t t 6 ± 6 t is etraneous The ball reaches a height of 000 feet 67 4 seconds faster than the skydiver. a + b c The sides of the isosceles right triangle are about 4.4 centimeters in length. 79. a a + b c a , ,09 a a 4,0 9 The flying distance from Atlanta to Buffalo by way of Chicago a + b 976 miles. A 6 cm c 70 mi. b 8 mi. B a C 8. Since the angle is 4, the triangle is isosceles ,000 4, The whale shark is about meters deep. 8. R p ( ),080, ,000 +,600,000, ,080,000 0 ( ) 60, ,000 To produce a revenue of $,080,000, a total of 60,000 units must be sold. 8. C.6t + 70, t 0 (t 0 corresponds to 000.),000.6t t 4.0 t.60 t The average monthly cost will reach $,000 in P 694.9t + 679, 0 t 9 (t 0 corresponds to 800, t corresponds to 80, etc.) (a) 0, t , t.0 t 8.74 t The resident population would have reached 0,000,000 in 987. t 8.74 represents ( ) (b) Because the lengths of the bars that represent the actual data are close to the lengths of the bars that represent the data values given by the model, the model is a good representation of the resident population through 890. (c) When t 0.6: P 694.9( 0.6) ,9 00,9,000 people. Because the population given by the model is close to the actual population in 006, the model is a good representation of the resident population through m 4

19 Section.4 The Quadratic Formula For this model t corresponds to , 6,6 The model is not a good predictor for the population in 00, since 6,6,000 is larger than 49,84,000. T t + t 0..9, 7 (t 7 corresponds to 7 A.M.) 8 0.t t t.96 t The temperature was 8 F at about P.M. When t 9 (7 P.M.): T 0.( 9) F Because the temperature is etremely high, the answer is unreasonable. So, the model should not be used to predict the temperature at 7 P.M. 9. H t + t 0.74, 0 (t 0 corresponds to 000.) 0.74t t 9.46 t.08 t The height of the tree was about inches in 00. Section.4 The Quadratic Formula Skills Review. 9 4()( ) 9 ( 44) ()() or + 0 or ( )( ) ( )( ) or + 0 or ( )( ) or + 0 or ( )( + 7) 0 0 or or ( )( ) 0 0 or 0 or ( )( ) or 4 0 or 4

20 40 Chapter Equations and Inequalities b 4ac ( 4) 4( 4) 0 One real solution b 4ac > 0 Two real solutions ± 4 a ± b b ac () () ± 7 ± b 4ac 4 < 0 No real solutions b 4ac 4 8 > Two real solutions b ± b 4ac a ± 4 ±, b ± b 4ac a ± ± 6, b ± b 4ac a ± ( ) 4 ± ± b ± b 4ac a ± () () ± 4 4 ± b ± b 4ac a ± 9 ( ) 4 9 ± ± b ± b 4ac a ± ± ± b b ac ± 4 a ± ± 8 ± 8

21 Section.4 The Quadratic Formula b ± b 4ac a 8 ± ( 49) 8 ± t + t + t 8t 0 b b ac t ± 4 a ± ( ) ± 6 6 ± y y y y y + 0 ± 4 a b b ac ( ) () ± 4 ± 6 ± ( )..7 ± , ( ) ± , ( ) 0.0 ± No real solution ± () ()() ± ± 4 9 ± ± + 4 OR ( ) () () ± 4 ± 7 7 ± Let one integer 00 the other integer ( ) Verbal models will vary. 49. Let an integer + net integer ( ) ( )( ) OR Verbal models will vary.

22 4 Chapter Equations and Inequalities. C , ± 0. 0 ± Choosing the positive value of, we have 00 units.. C ± ( ) ( 0.00) 0.04 ± Choosing the positive value for, we have units Number of rows 7 Number of seats/row + ( ) ( ) () 6 ± ± 4 8, seats/row 8 The original number of seats in each row was nine. 7. Volume The original piece of material was inches by 4 inches. 9. (a) s 6t + v t + s The initial velocity is 0 miles per hour, or 88 feet per second. So, 88 v and the initial height 88 is 0 s 984. s 6t + t (b) When 4: (c) t t + t s feet ( ) 0, ± 4( 6)( 984) 88 ± 6,86.4 t 8.8 or It will take the coin about 8.8 seconds to strike the ground. Moon:.7 s t + v0t + s0 Let v 40, s and set s t ( ) 40 ± Earth: t t.4 s t + v0t + s0 Let v 40, s and set s t + 40t + 0 6t + 40t + ± t 6 ( ) t 4.9 seconds t.6 seconds On the moon, it would take about 4.9 seconds to hit the surface while on Earth it would take only about.6 seconds.

23 Section.4 The Quadratic Formula 4 6. Moon: s.7t + v0t + s0 Let v0 7, s0 6 and set s t + 7t + 6 t t.4 t 0. seconds ( ) 7 ± The rock will take longer to reach the ground on the moon. 6. Distance between A and C Distance between C and B y Total distance + y y ,000 0 ( ) s t + v0t + s0 Earth: 6 Let v0 7, s0 6 and set s t + 7t + 6 t 7 t t.9 seconds ( ) 7 ± ± , ± ± 00 4 The other two distances are miles and miles. 67. S 8.t 6.t + 87., 6 t (t 6 corresponds to 996.) (a) t 6.t t 6.t 6.9 t ( 8.) 6. ± Choosing the positive solution, t.7, you can estimate that total sales were about $4 billion in 00. (b) t 6.t t 6.t 8.9 t ( 8.) 6. ± Choosing the positive solution t 4.9, you can predict that total sales were about $6. billion in 00. (c) t 6.t t 6.t ( 8.) 6. ± t Choosing the positive solution, t 7.48, you can predict that total sales will reach $9.4 billion in 007. So, the model agrees with the original prediction.

24 44 Chapter Equations and Inequalities 69. L t + t + t , 7 (t corresponds to :00 P.M.) t +.9t t +.9t 9.9 t ( ).9 ± t 4 or t 9 Because t 9 is not in the domain, choose t 4. The patient s blood oygen level was 9% at about 4:00 P.M. 7. de ( hours)( r + 0 mph) ds ( hours )( r mph) d ( r ) + d 440 E S r r 900r,9, ,9, ,847 ± ± r 8 6 Thus, the eastbound plane is moving at r mph and the southbound at r 0 mph. 7. C + (a) , 0 ( ) ± The air temperature is about 6.80 C. (b) When 0: C When 0: C Oygen consumption increased by a factor of.. 7. Revenue p ( ) 0, ,000 0 ( 0) ( 0) 4( 0.000)( 0,000) ( 0.000) ± ,7 units or units To produce a revenue of $0,000, approimately 79 or approimately 94,7 units must be sold. 77. Answers will vary.

25 Mid-Chapter Quiz Solutions for Chapter 4 Mid-Chapter Quiz Solutions. ( ) ( ) ( ) 4 ( )( 4) + ( )( 4) ( ) ( 4) + ( )( 4) ( ) ( + ) 6( + ) Contradiction: No solution ( + ) ( ) One method would be to use the scientific calculator portion of a graphing utility to check for a true statement. Another method would be to use a table feature (.004)(.8) (.004)(.8)( 00) ( Total cost) ( Fied costs) + ( Unit cost)( Number of units) 00,000 0, , ,000 The company can manufacture 0,000 units.

26 46 Chapter Equations and Inequalities 9. R p , , , ± ,000 7 ± or 70,96 To produce a revenue of $00,000 about 4044 units or about 70,96 units must be sold ( )( + ) 0 0 or + 0 or... ± ± ± 7 ± 7. or () ± 4 ± 4 ± 6 ± 6.4, ± ( 7) 4( ) () 7 ± , ( 4.0) ± ( 4.0) 4( 0.) () 4.0 ± or b 4ac ( 4) 4( 9) 6 < 0 No real solutions b 4ac One repeated real solution 8. Answers will vary. One method would be to use the definition of an eponent, multiply and simplify: Another method would be to use the table feature of a graphing utility and let y + and y As real values of are selected y will always equal y. A third method would be to use the scientific calculator portion of a graphing utility and demonstrate that with real values of.

27 Section. Other Types of Equations s 6t + v0t + s0 Since v 0 0 and s 0 00, we have s 6t + 00 When it hits the ground, the height, s, is zero. 0 6t t 00 t 00 6 Since t represents time, it must be nonnegative. 00 t 4. seconds 6 4 The rock will hit the ground in about 4. seconds. 0. Volume ( length)( width)( height) 84 ( 6) Since must be nonnegative, we have 8. Dimensions of bo: 8 inches 8 inches 6inches 6 in. Section. Other Types of Equations Skills Review. + 0 ( )( ) 0 ( ) ( ) ( )( ) or or + 0 ± 0 + or 0 or ( ) or ( )( ) ( )( ) 6 0 or or or 0 or ( )( ) ± a + 0 or 0 b b 4ac or ( ) ()() () ± 4 ± ± ( )( ) or + 0 or ± 4 a b b ac ( ) () 4 ± ± 8 4 ± ±

28 48 Chapter Equations and Inequalities. 0 ( ) 0 ( ) ( 9)( 4) 0 ( 9) No real solution ( ) ± ( )( )( ) ( )( ) ( )( )( )( ) No real solution ( ) ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( ) No real solution ( )( ) ( )( )( )( ) No real solution No real solution ( )( ) ± ±.7 0 ±

29 Section. Other Types of Equations is not a solution ( )( ) ( + )( ) ( 9 + ) 0 is not a solution 9 6 ± 6 ± 64 69, / ± 6 ± ± 69 9 No real solution , 0, + 0 b ± b 4ac a ± 4 ( ) ± ± , b ± b 4ac a ± ( ) 4( 0) ( ) ± 9, , 0, + ( ) ,, ( )( ) ( ) + ( )

30 0 Chapter Equations and Inequalities. ( )( ) ± OR The quadratic equation was not written in standard form. As a result, the substitutions in the quadratic formula are incorrect. The solution should be: ( 7) ( 7) 4( 4) () ± is the only solution is not a solution.. Only and are solutions. ( )( ) OR ( )( ) Only and 0 are solutions ( ).. ± ± ± ( ).8 6 ± Number of students Cost per student f 700 f 700 f ( f 7.0)( + 6) ( + 6) 700 ( 400 )( + 6) ,400 0 The original number was 4. ( ) 90 ± , ± 0 0

31 Section. Other Types of Equations r A P + n r r r (.40) r r % The annual interest rate is about 6%. nt r A P + n ( ) r r r + nt (6) 6 6 r r % You are paying an annual interest rate of about.98%. C ,000 There were 4,000 passengers that flew in the month. y , ( ) ( 0.874) ± or 97.4 The solution 97.4 does not make sense in this situation. So, the person is about 6.9 years old. 7. p , 0 9, , ,760 units When the price is set at $.9, the demand for the product is 67,760 units. This model is only valid for 0 9,900 because it does not make sense in the contet of the problem to have a demand less than 0 or a price less than By the Pythagorean Theorem, we have: 77. h h Thus and Thus h. feet. The stays are attached about. feet up the mast. h ( ) The least acceptable weight is 78.8 ounces and the greatest acceptable weight is 8. ounces. 79. Let t time to paint the house then t portion painted by you, t 8 portion painted by your friend. t t + 8 8t + t 70 t 70 t hours Working together, it will take about 8.8 hours to paint the house.

32 Chapter Equations and Inequalities Section.6 Linear Inequalities Skills Review Because 7 is to the left of, is larger than 7., corresponds to. The interval is bounded.. [ ], corresponds to >.. The interval is unbounded., corresponds to <.. The interval is unbounded. Because is to the left of, is larger than 6 6. Because π( π.4) larger than π is to the left of, is π 6 6 Because is to the left of 6, 6 is larger than.. The statement is nonnegative can be represented by The statement z is strictly between and 0 can be represented by < z < The statement P is no more than can be represented by P. 8. The statement W is at least 00 can be represented by W When : 0 When : When : 0 0 () When : 7. < 4 indicates all points to the left of 4; graph (c). 9. < indicates all points from, but not including up to and including ; graph (f ).. < 4 indicates all points less than 4 units from 0 in both directions; graph (g).. > 0 > 0 is a solution of the inequality. (a) (b) 7 < 0 is not a solution of the inequality. (c) (d) > 0 is a solution of the inequality. < 0 is not a solution of the inequality. 9. > indicates all points more than units from ; graph (b).

33 Section.6 Linear Inequalities 7. 0 < < 4 4 (a) 0 < < 0 < < 4 is a solution of the inequality. 4 (b) (c) (d) 0 0 < < 0 < 0 is not a solution of the inequality < < 0 < < 0 is not a solution of the inequality < < 0 < < is a solution of the inequality (a) 0 is a solution of the inequality. (b) 0 is a solution of the inequality. (c) is a solution of the inequality. (d) is not a solution of the inequality (a) + 0 is not a solution of the inequality. (b) is a solution of the inequality. (c) + 0 is a solution of the inequality. (d) is a solution of the inequality.. If > 6, then >.. If 8, then > 0 4 > < If 4 > 0, then <. 9. If 6, then ( 9) < 8 < < < < < 4 < < 40 ( 0) ( 40) > 0 0 >

34 4 Chapter Equations and Inequalities 4. ( ) < < + < < ( ) + 7 < < < + 8 < >. > < or > < 6 > < < + < 8 < < < 7 4 < 8 < < + < 9 < < 6 < < < < 4 < < 9 < < 9 < < 0 0. > + > > > 0. < 6 6 < < > 6 < 6 or > 6 < > < > or < 9 < < 9 < 0 < < 8 > > or

35 Section.6 Linear Inequalities 69. < 0 No solution 7. All real numbers no more than units from 0 yields 0 or. 7. All real numbers no less than units from 9 yields ( ) > + > 79. Rental fee for Company B > Rental fee for Company A > > 80 > 0 You must drive more than 0 miles in a week for the rental fee for Company B to be greater than that for Company A. 8. I Prt; P 00, t, I I > 90 00( r) > r > 90 r > r > 8.7% The simple interest rate must be greater than 8.7% The person must be in the program or less weeks. 8. Length + Girth in (a) The sides of the package s cross sections cannot be more than 6 inches R $99.0 $799 $498.0 C $80 $790 $ R $98 $ $897 (b) R > C 9.9 > > 80 0 The product will return a profit for 0 units. 89. C 0.m + 00 <,000 0.m < 900 m < 9,687. Less than 9,687. miles traveled will yield an operating cost that is less than $, According to the model, an IQ score greater than 4 would produce a grade point average of.0 or higher. 9. S 0.7t + 0.9, t 6 (t corresponds to 99.) S > 0.7t > 0.7t >.70 t > 7.7 According to the model, the average professional baseball player s salary will eceed $,000,000 sometime during Area Interval: in., in You could have been undercharged.7( 7.99).7( 7.99 ) $0. or you could have been overcharged.7( 7.99).6887( 7.99 ) $ h h h 7. Interval: [ 6.8, 7. ] C $470 $700 $6670

36 6 Chapter Equations and Inequalities 0. h h h 80 The minimum relative humidity is 0% and the maimum relative humidity is 80%. 0. B 6.98t.4, 8 t (t 8 corresponds to 998.) B > t.4 > t > 78.4 t >. The average price of a brand name prescription drug eceeded $7 sometime during D 76.4t + 6,66, t (t corresponds to 99.) (a) D > 8, t + 6,66 > 8, t > 44 t > 4.9 The demand for U.S. oil eceeded 8 million barrels a day in 99. (b) D >, t + 6,66 >, t > 44 t > 9. The demand for U.S. oil will eceed million barrels a day sometime during 009. Section.7 Other Types of Inequalities Skills Review y. > y < 6. 6z < 7 z > 9. + < 6 < < > 4 ( + ) 6 > ( + ) < + < or > 6. ( ) < ( ) < 7. < + 4 < < 4 < < < > < or > < or > > + 4 < or + 4 > < 6 or >

37 Section.7 Other Types of Inequalities < 0 0 ± Critical numbers: ± Test intervals: (, ), (, ), (, ) ( )( ) Critical numbers: 4, Test intervals: (, 4 ), ( 4, ), (, ) <, ( ) Critical numbers:, Test intervals: (, ), (, ), (, ) ( )( ) Critical numbers: ± Test intervals: (, ), (, ), (, ) Test: Is ( + )( ) 0? Solution set: [, ] > 4 4 > 0 + > 0 ( )( ) Critical numbers: ± Test intervals: (, ), (, ), (, ) Test: 0 Is 4 > 0? Solution set: (, ) (, ). ( ).. + < < + 4 < < 0 ( )( ) Critical numbers: 7, Test: intervals (, 7 ), ( 7, ), (, ) Test: Is ( + 7)( ) < 0? Solution set: ( 7, ) ( )( ) Critical numbers:, Test intervals: (, ), (, ), (, ) Test: Is ( + )( ) 0? Solution set: (, ] [, ) + < < 0 + < 0 ( )( ) Critical numbers:, Test intervals: (, ), (, ), (, ) Test: Is ( + )( ) < 0? Solution set: (, ) ( )( ) + > 0 Critical numbers: ± 0 Test intervals: (,, ) (,, ) (, ) Test: Is ( )( + ) > 0? Solution set: (, ) (, )

38 8 Chapter Equations and Inequalities < 0 + < 0 ( )( ) Critical numbers:, Test intervals: (, ), (,, ) (, ) Test: Is + < 0? Solution set: (, ) 4 [Note: Compare this problem to #6.] 4 6 < 0 ( ) < 0 Critical numbers: 0, Test intervals: (, 0 ), ( 0, ), (, ) Test: Is < 0? Solution set: (, 0) ( 0, ) ( ) Critical numbers: 0, ± Test intervals: (, ), (, 0 ), ( 0, ), (, ) Test: Is ( + )( ) 0? Solution set: [, 0] [, ] ( )( )( ) Critical numbers:, ± Test intervals: (, ), (, ), (, ), (, ) Test: Is ( )( + )( ) 0? Solution: [, ], [, ] > > 0 > 0 Critical numbers: 0, ± Test intervals: (,, ) (, 0 ), ( 0,, ) (, ) Test: Is > 0? Solution set: (, ) ( 0, ) < + 6 < ( + ) + < < 0 Critical numbers:, 4 Test intervals: (, ), (, 4 ), ( 4, ) 4 Test: Is < 0? + Solution set: (, ) ( 4, ) 0 > 4 4 > 0 4( ) > 0 > 0 Critical numbers:,,,,,, Test: Is > 0? Test intervals: Solution set: (, )

39 Section.7 Other Types of Inequalities > > > 0 ( + ) ( + ) ( + )( + ) + ( + )( + ) > 0 Critical numbers:,, Test intervals: (, ), (, ) (,, ) (, ) Test: Is 7( + ) ( + )( + ) > 0? Solution set: (, ) (, ) ( ) ( )( 4 + ) 0 ( )( 4 + ) 0 Critical numbers:,, 6 Test intervals: (, ),(, ), (, 6 ), ( 6, ) 4 4 Solution set: ( ) [ ) 6 Test: Is 0? ( )( ) 4, 6, Critical numbers: ± Test intervals: (, ), (, ), (, ) Test: Is 9 0? Domain: (, ] [, ) No critical numbers because b 4ac 0 4 ( 6) < 0. Test interval: (, ) Test: Is 6 + 0? Domain: (, ) or all real numbers Critical numbers: ±, 9, 9, 9, 9, Test intervals: Test: Is ? 9 9 Domain:, ( )( ) Critical numbers:,,,,,, Test intervals: Test: Is ( )( ) 0? Domain: (, ] [, ) No critical numbers since b ac Test interval: (, ) 4 4 < 0 Test: Is + 0? Domain: (, ) or all real numbers 47. You can take the cube root of a negative number and you will get a real number. For eample, if 7 + was equal to 8, then > 0 ( ) > 0 Critical numbers: 0, Test intervals: (, 0 ), ( 0, ), (, ) > 0? Test: Is Solution set: (, )

40 60 Chapter Equations and Inequalities. 9 0 ( 9) ( ) Critical numbers: 0, ± Test intervals: (, ), (, 0 ), ( 0, ), (, ) Test: Is ( + )( ) 0? Solution set: (, ] [ 0, ]. ( ) ( ). + 0 Critical numbers:, Test intervals: (, ), (, ), (, ) Test: Is ( ) ( + ) 0? Solution set: [, ) + < < < ( ) Critical numbers: ±. Test intervals: (,. ), (.,. ), (., ) Solution set: (.,.) > The zeros are Critical numbers: 0.. ( ) ( 0.). ± Test intervals: (, 0. ), ( 0.,. ), (., ) Solution set: ( 0.,.) > > > 0.. Critical numbers:.9,.6 Test intervals: (,.6 ), (.6,.9 ), (.9, ) Solution set: (.6,.9 ) s 6t + v0t + s0 6t + 00 t, 0 t. 6t + 00t > 400 6t 00t < 0 8( t t + 0) < 0 8t t 0 < 0. seconds < t < 0 seconds 6. l + w 00 w 0 l lw 00 l( 0 l) 00 l + 0l 00 0 l ( ) l ( + ) 0.8 meters l + 6. meters. (Use the Quadratic Formula to find the critical numbers.) 6. (a) Profit Revenue Cost,60,000 ( ) ( + 0,000),60, ,800, ,800,000 0 Critical numbers: 00,000 or 90,000 90,000 units 00,000 units 8 ± ,800,

41 Section.7 Other Types of Inequalities 6 (b) p When 90,000, p $. When 00,000, p $0. $0 p $. (c) Revenue Cost ( ) + 0, , ,000 ( 0.000) 8 ± , or 8,967 Choose the larger value of. After revenue starts to decrease, the revenue is approimately equal to the cost when 8,967 units are sold. Producing more than 8,967 units will result in the cost being greater than revenue. So, the company would incur a loss instead of making a profit. A P + r A > ( r) ( r) > 00 + >. + r r > >.. r > % The interest rate must be greater than 4.%. P t + t + t , 6 ( t corresponds to 99. ) P > > 0.8t t 7 > 0 0.8t 80.0t ( ) 80.0 ± t 0.8 t.4 or t 4.66 Choose t.4 (because t 4.66 does not make sense in this situation). So, the world population will eceed 7,000,000,000 sometime during C t + t + t ,09, 6 ( t 6 corresponds to the 99/996 academic year. ) C >,000 t + t + > 4.9t t 6,69 > ,09, ± ,69 t 4.9 t 0. or t 8.94 Choose the positive solution t So, the average yearly cost of higher education at private institutions will eceed $,000 during the 008/009 school year. + R R R R + RR R R + R R + R R Since R, we have R + R R 0 + R R 0. + R Since R > 0, the only critical number is R. The inequality is satisfied when R ohms.

42 6 Chapter Equations and Inequalities Review Eercises for Chapter. ( ) Conditional?. (a) is not a solution.? (b) is a solution.? (c) is a solution.? (d) is not a solution ( ) ( ) ( ) ( + ) 0; Sum ( first integer) + ( second integer) + ( third integer) s n + ( n + ) + ( n + 4) 4 6n n n 6 n is the smallest of these integers. 9. Let amount of weight set A person weighing 0 pounds would set the weight at 0 pounds in order to pull 0 pounds.. Perimeter ( length) + ( width) 77 ( w) + w 77 6w 9. w width w 9. feet length w 9 feet. I Prt I After year, you will earn $0 in interest.. Sale Price List Price % of List Price 9 L 0.L 9 0.8L L 6. The outdoor grill was originally priced $6..

43 Review Eercises for Chapter 6 7. Distance Difference ( Rate of Faster Car) ( Time) ( Rate of Slower Car) ( Time) 0 0t 4t 0 t t hours 9. Total Revenue Monthly Revenue 7,8 6 $7,664 If the monthly revenue rate continues, the total revenue for the year will be $7,644.. ( ) + 0.0( 0 ) ( 0). 0% of 0 00% of 0% of 0 About. ( or ) quarts quarts will have to be replaced 9 with pure anti-freeze ( )( ) or + 0 or ( )( 8) 0 0 or 8 0 or 8 7. ± ± 4 ± 8.4 or () Use the table feature in ask mode with the variable equal to a solution. () Use the scientific calculator portion to evaluate the quadratic equation at a particular value. (Answers will vary.) 4. Area ( length)( height) 40 ( h + ) h 0 40 h + h h h 7 h 0 h Length h + 7 feet Height h feet h 40 ft ( h )( h ) ,000, ,000, ± ,000,000 ( 0.000) 00,000 units or 400,000 units To produce a revenue of $8,000,000, the company should produce 00,000 units or 400,000 units. b ac 4 4 > 0 Two real solutions h + 4( 0) () ± ± 4 ± 6 6 ± 6

44 64 Chapter Equations and Inequalities. y + 7 y y + 4y + 49 y y y + 9y ± () 9 ± 6 () () ± ± () 6 ± 4 ± (.7) (.7) 4(.6)(.9).6 ±.866 or ( 96) ( 96) 4( 4)( 47) 4 ± 96 ± 8, or On the moon 6. t + 00 t.7 t seconds On the Earth 6t t.4 seconds When the rock is dropped off a 00 foot cliff on the moon it takes about 8.6 seconds for it to hit the lunar surface. On Earth it would only take about.4 seconds. 9 0 ( ) 4 0 ( 4)( + ) 0 0, 4 or ( )( 4) ±, ± ( )( )( )( ) ( ) ( + 4) There are no real zeros since b 4ac < 0. No solution ± 7 or no real solution ± ± or or or ( + ) + ( + ) ( + )( + ),, ( 4) ( 4) 4 () ± 4 ± 76 4 ± 9 ± 9

45 Review Eercises for Chapter 6 7. Let r rent per student per week if three students rent the condominium. Weekly rent: r 4( r 7) r 4r r 77. The weekly rent is 00 ) $900. t r A P + r r.07 + r r r % The annual percentage rate for the cash advance is about 8.0%. 79. ( ) 8. < + 8 < + 8 < + < < 4 < + < < < < < < + 0 < < + 0 < < < 8 8. R > C 89.9 > > 00 4 To maimize the profit, the company should produce 4 or more units. 87. ( )( ) + < 0 Critical number:, Test intervals: (,, ) (, ), (, ) Solution set: (, ) < < 0 ( ) Critical numbers: 0, ± Test intervals: (, ), (, 0 ), ( 0, ), (, ) Solution set: (, ) ( 0, ) + 4 < + < ( 4 ) 4 < < 0 6 Critical numbers:, 4 Test intervals: (, 6 ), ( 6, 4 ), ( 4, ) 6 Solution set: (, ) (4, ) >.. >.44 < < < > > (.7 6.) > > Critical numbers:.74,.6.6 < < Domain: 0, ) 99. Since you can take the cube root of any real number,, or all real numbers. the domain is

46 66 Chapter Equations and Inequalities ( )( ) Critical numbers: 6, 9, 6, 6, 9, 9, Test intervals: Domain: (, 6] [ 9, ) S 6t + v0t + s0 Let v0 4 and s0 6t + 4t 0. > 76 6t + 4t 76 > > 0 ( t t ) Critical numbers: ( ) () 67 ± ± 7 t 8 6 t 4.7 or t.6.6 seconds < t < 4.7 seconds 0. (a) y < < 8 (b) < < 0.4 ± or 8. ( ) Solution set: ( 0, 8.0) ( 8., ) The value of must be greater than 0 and less than 8, or greater than 8.. (c) Because is not part of the solution set, the player will not score a goal l l P l + w 60 l + w 60 l w 0 l w A 0 lw 0 l( 0 l) 0 0l l 0 0l l 6. or l.7 ( ) () 0 ± Solution set: [ 6.,.7 ] The length must be between 6. feet and.7 feet. 8 r > r + >. + r > r > r > % The interest must be greater than 9.%. R C,000, ,000,000, ,000,000, ,00,000 0 ± 0.000,679 or 67, ( ) ,00,000 Solution set: [,679, 67, ] p p ,679 $ , $4.4 The company should set the price between $4.4 and $8.66 to obtain a profit of at least $,000,000.. R t t + t , 6 ( t 6 corresponds to 996. ) (a) t 6 0 R

47 Chapter Test Solutions for Chapter 67 (b) R t 0.44t t 0.44t t t 0.7 or t 6.8 ( 0.099) 0.44 ± Choose the positive value for t. So, according to the model, Sonic s revenue per share will be at least $8.80 in 007. The model supports the prediction. (c) R t 0.44t t 0.44t t t.7 or t 8.8 ( 0.099) 0.44 ± Choose the positive value for t. So, according to the model, Sonic s revenue per share will be at least $.0 in 009. The model supports the prediction. Chapter Test Solutions ( ) 7. (a) Since you can take the cube root of any real number,. (b) the domain is Critical numbers: ± Test intervals: (, ), (, ), (, ) Test: Is ( )( + ) 0? Domain: [, ]. Total profit April profit + May profit 6, ,09..9,786 99, The profit in April was $,786 and the profit in May was $99, ( )( ) + 0 or 0 or or or 4 0 or 4 or ± ( ) 0 ± () ± () 4 ± 69

48 68 Chapter Equations and Inequalities 8. 0 ( ) 4 () ± ± ± (.) (.) 4(.4)(.).4 ±. ± 0.4 ± ± or 0 7 or 7 or ( + 4)( 7) 4 0 or or 7 4 is the only solution. (Note: 7 is etraneous.) ( )( ) 9 0 ( )( )( )( ) + + 0,,,. ( ) 9 9 ( ) ± 79 9 ± 7 9 ± 7 6 or 8 ± 6 No real solution 4. Revenue p ( ),000, ,000,000 0 ( 40) ( 40) 4( 0.000)(,000,000) ( 0.000) ± 40 ± ± ,4 units or ,79 units Since the equation is quadratic, it is possible to have two distinct solutions. In this case they would both have to be positive since represents the number of units sold.. + < + < 0 < 9 < or or 0 or

49 Cumulative Test Solutions for Chapters > > ( ) > > Critical numbers:, 7 Test intervals: (, ), (, 7 ), ( 7, ) Solution set: (, 7) < < ( ) 4 0 ( ) Critical numbers: 0, ± Test intervals: (, ), (, 0 ), ( 0, ), (, ) Solution set: (, ] [ 0, ], R C , , ,00, , ,00 800,000 + ( ) ± ,89 or 4, ,00,000 Solution set: [ 9,89, 4, ] The company must sell between 9,89 units and 4, units to obtain a profit of at least $800,000. C t + t + t , 0 ( t 0 corresponds to 000. ) C > t + 6.9t + 4 > t + 6.9t 67 > ± t 7.7 t.7 or t 7.9 Choose the positive value for t. So, average dormitory costs eceeded $4000 in Cumulative Test Solutions. 4( ) 4( 8 6 ) ( 6) ( 6) ( )( 6). 6. ( )( + ) ( ) , 4 y y y + y y + y y, 0, y 0 + y ( )( )( 6) +

50 70 Chapter Equations and Inequalities C.9t + 4, 0 t ( t 0 corrosponds to 000. ) (a) When t C : The average monthly retail sales in 00 was $0. billion. (b) C > 60.9t + 4 > 60.9t > 7 t > 9.8 The average monthly retail sales will eceed $60,000,000,000 in is a solution. 0 is a solution. ( )( ) ,.0 ( ).. ± or or 0 or ± < < 0 < < 0 < < 9 > > < < ( ) ( ) Critical numbers: 0, ± Test intervals: ( ) ( ) ( 0, ), (, ) Test: Is ( )( + ) 0?,,, 0, Solution set:, 0,, ) () 0 ± ± 4 ±.7, The only solution is ( )( ) ( )( )( )( ) ±, ± 4

51 Cumulative Test Solutions for Chapters P R C P 600,000 ( ) ( ,000) 600, , , , , , ,000 ( ) ± ,000 0,6 or 89,77 Solution set: [ 0,6, 89,77 ] To obtain a profit of at least $600,000, between 0,6 units and 89,77 units must be sold. 8. D t + t + t ,808, 0 ( t 0 corresponds to 000. ) D > 0,000 t + t + > 8.7t +.t,9 > ,808 0,000. ± ,9 t 8.7 t 8.89 or t 7.48 Choose the positive value for t. Per capita gross domestic product will f000 during 007.