23z 4 z 12 6z 8 z 12. Linear Equations and Inequalities in One Variable

Transcription

1 Chapter 1 Linear Equations and Inequalities in One Variable Section 1.1 Practice Exercises 1. a. equation. b. solution c. linear d. first e. solution; set f. solution g. conditional h. contradiction i. empty set; { } or Ø j. identity 1 6 x 8 y xy 6x 8y 10. 8xyxyx1xy xy1xy. z z1 6z8z1 z 0 7. x 1 x 0 Linear. ab a 1a 17a ab 6. 6w w 6w1w1 6w x 6 0 x x 0 Linear 9. x 7 9 Nonlinear 10. x x Nonlinear 11. x x 0 Linear 1. a. x is not a solution 1.. 7x 0 7x. 0 Linear 1. a. y is not a solution.

2 Section 1.1 Linear Equations in One Variable b is a solution. c is not a solution. d is not a solution. b c. is a solution is not a solution. 1 1 d. 1 is not a solution. 1. x 719 x x 1 1 Check: y 8 y 8 y Check: x x Check: 18. t t Check: z z z 0. 1 b 1 1 b b 1 1

3 Chapter 1 Linear Equations and Inequalities in One Variable 1 0z z Check: = 8 Check: a 8 a 8 a Check: x 1 8 x x 1 Check: t..t t 1.1 Check: y y 10.9 y 10.9 Check: p.9.8 p p Check: a.9.a.9.. a Check:

4 Section 1.1 Linear Equations in One Variable 7. 6q 6 6q 6 6q 66 Check: 6q q w 11 w 1111 w 0 Check: w 0 w y 17 y y y y 1 1 Check: z 8 6z 8 6z 108 6z z Check: b b b 7 b b Check: y1 61y 11 y y y 1 1 Check: x 6 x x18x x1818x18 x x1 xx xx1 x y y 1y y0 1yy yy0 8y 0 8y 0 8y 7

5 Chapter 1 Linear Equations and Inequalities in One Variable 8y y Check: Check: tt 6 t 1t0 t 1t0 t1t 1t1t0 16t 0 16t 0 16t 16t t Check: a10a 6a1810a8 6a8a8 6aa8aa8 8a 88 8a a 0 8a pp1 1p10 p6 p9p6 pp9pp6 7p 96 7p p 7p 7 7 p Check: bb 9b1 8b16b 9b9 11b16 9b9 11b9b16 9b9b9 0b b b 0b 0 0 8

6 Section 1.1 Linear Equations in One Variable a 0 0 Check: z1 z z1 6z 10 z 6 z z16z zz16zz 6z 16 6z z 18 6z z Check: b Check: w10w7w1 w10w7w7 w107 w7 6w07w7 6w6w07w6w7 0 w w 7 7 w Check: y y yy 6y6y9y 1y y 1y1y y1y 16y 16y. 1ww6w1 1ww0 w 1w7w 1ww7ww 1 w 1 w 9

7 Chapter 1 Linear Equations and Inequalities in One Variable 8 16y 8 16y y Check: x x x x 8x0 6 1 x 8x1 1 x8x 8x8x1 1 x x 1 1 x 0 x 0 x 0 0 Check: = 6 1=0 6 1= x x x x 1 x x w 19 w w Check: p 6p7 p1 8 p6p7 p1 p1 p1 p p1 p p1 p 11 p 1111 p 0 Check: p 0 p y y y 10 y 10 0

8 Section 1.1 Linear Equations in One Variable 8x x18x 8x7x18 8x7x7x7x18 1x 18 1x 18 1x 0 1x x y09y y9y09y9y y 0 y 000 y 6 y 6 y p p p p1 p p p p p 10 p106p p10 p107p10 p7p10 7p7p10 p 1010 p p 0 p 0 p q q 6 9 q 10 7 q q q 9 1q0 1q6 6 1q0 1q0 1q1q0 1q1q0 9q 0 0 9q q 0 9q q x7 x 6x x7 x 6x x x x x10 0 0x 18 6x x7186x x 6x7186x6x 1x x y y1 y y y1 y 0 0 yy110y 8y16y610y 8y16y6 8yy16yy6 7y y

9 Chapter 1 Linear Equations and Inequalities in One Variable 1x 9 7y 81 1x x 7y y 1. q 6 q q q 6 q q6 q6q0 q 16q8 q6 q0 9q 0 9q 0 9q 9q 9 9 q 6 6. a9 a a a9 a a a a a 6a181a0a60 1a 0 1a 0 1a 1a 1 1 a. 6.w 1..8 w w 1 8 6w w 6 6w w x.6 x x x x6 10x x x6 10xx 6 8x 6 8x x 67. m m m m m m0 100m m m m m 6. n n n n n n0 10n n n 0 10n n

10 Section 1.1 Linear Equations in One Variable 7. A conditional equation is an equation that is true for some values of the variable but false for other values of the variable. 9. x x x1x1 x1x1 0 0 This is an identity. xx 61. is a real number 11x x x1 11xx1 x1 7x1x1 7xx1xx1 x 11 x 1111 x 0 x 0 x 0 This is a conditional equation x 8x 7x8x 10x 10x8 10x 10x 10x10x8 8 This is a contradiction. 8. A contradiction has no solution, and an identity has all real numbers as solutions. 60. x 6 x x x6 xx 6 0 This is a contradiction. 6. x 7 x 107 x x x 0 x 0 x 0 This is a conditional equation x x x x8x91 x8x8 0 0 This is an identity. xx is a real number 6. b 9 71 b x x

11 Chapter 1 Linear Equations and Inequalities in One Variable b 80 x 8 b 80 b x x x x 6 1x 6 1x 1 1 x x x 7 9x 7 9x 9 9 x c 1c 10c1c 1c1c c c c 6 c 6 c 70. w16w7 w6w16w6w7 w 1 7 w w 8 w 8 w b8b8 1 b6 1 bb 0 0 The equation is an identity. The solution set is {b b is a real number}. 7. x x x 7 x 10x x 7 x 10 x 7 x x 10 x x z zz z7 z7 0 0 zz is a real number 7. x x 1 xx1 1 x 11 x 1111 x 0 x 0 x 6 6

12 Section 1.1 Linear Equations in One Variable 7. c c c 1 8 c c c ccc8 c 8 c c 76. d d d d d d ddd 1 7d 1 7d d xx6 x x 6xxxx6 x 6 x 6 x x x z z z z zz zz z p pp1 7p1pp1 p1p1 pp1 pp1 1 1 ppis a real number z z8z 6z1 6z8 6z 6z1 6z6z b b 6b 1bb6 b b18 6b x x x 6 x 6 6

13 Chapter 1 Linear Equations and Inequalities in One Variable 1b7 6 b 1bb7 6 bb 10b b b 66 10b b 6.6 x 1x x61x xx xxxx xx is a real number 8. 1 x x x 9 x 9 The equation is a contradiction. The solution set is { } w8 w 9 8w 8w 9 The equation is a contradiction. The solution set is { }. 8. y y 1 8 y y y y1 y78y y8y78y8y 7y+7= 7y 777 7y 7y 7 7 y x x 6 x x x x x 1x6 x 1x6 x1x 1x1x6 16x 6 16x 6 16x 8 16x x 6

14 Section 1.1 Linear Equations in One Variable 87. y 9 y 10 y y y9 1 10y y610y yy610yy 6 8y 6 8y 8 8 y x x x x x 6 6 x xx18x0 x18 x0 xx18xx0 x 180 x x 1 x 1 x x0.08x0.160 x x 0.08x x 8x8x160 x 0x10 1x 0x 1x10 1x1x x 10 x 10 x w w w w w w w80 6w9000 w w w x0. x x x x 1 x 6xx1 6xxxx1 x 1 x 1 x b b b 1 1 b 7 b 7 b 7

15 Chapter 1 Linear Equations and Inequalities in One Variable x 1 x b1. 0. b 0.0b 0.b1. 0.b0. 0.0b 0.b1. 0.b The equation is a contradiction. The solution set is { } y 8 y 7 1 y y y 8 8 y 8 7y0y 7yy0yy y 0 y 0 y 18 y 18 9 y 9 b b 9. a a a a a a 70a70 70a 70a70a70 70a70a x x x 8 1 x8x x 1 6x8x0 6x810x0 6x10x8 10x10x0 16x x x 16x x h h h h 0.1 h 10 The family used 10 kwh h h h h 10 h 1 The student is taking 1 credit-hours. 8

16 Problem Recognition Exercises: Equations Versus Expressions 99. a. y1 y yy6 b. y y 1 0 y 8 y 80 y 8808 y 8 8 c. To simplify an expression, clear parentheses and combine like terms. To solve an equation, isolate the variable to find a solution a. w8 w w168w b. w w 8 0 w16 w 160 w w 16 w 16 w c. To simplify an expression, clear parentheses and combine like terms. To solve an equation, isolate the variable to find a solution. Problem Recognition Exercises 1. Expression. Expression x 68x x8x6 x yy8 yy8 7y. Equation. Equation 7b1b 10t 7t 7bb1bb 10t7t 7t7t b 1 17t b t b 17t 0 b b t t 0 0. Expression 6. Expression a a 10x 8 x

17 Chapter 1 Linear Equations and Inequalities in One Variable a 1a 7 10a 9 0x 0 0 8x 8x Equation 8. Equation 7 w w8 1 7w w8 1 7ww ws8 1 1w w 8 1 1w 6 1w w 9. Equation 10. Equation xx18x7 1 y 1y 0y 1y 0yy 1yy y 1 y 1 y y y 6 a 8a 1a19 6x80x 8x7 1 18a16a6 1a19 1x1 8x7 a6 1a19 1x8x1 8x8x7 a1a6 1a1a19 6x 17 a x 1171 a x 19 a 6x 19 a a x Expression 1 7 v v v v v v Expression 7 11 t u t u t t u u t u t u 8 0

18 Problem Recognition Exercises: Equations Versus Expressions 1. Equation 0x 8 7x8 7x9 7x0 7x9 7x7x0 7x7x Equation 7 1 y y y 6 8 y y 6 8 y 0y1 1y18 0y1y1 1y1y18 8y 118 8y y 9 8y y Expression 0.9c.9 0.1c 0.17c Equation p p p1 0.p1 0.p0.p1 0.p0.p1 11 p p is a real number 1. Equation 78w1w8w 8w8b8 8 { } 16. Equation 1 z z z 1 z z 10 z 80zz10 80zzzz10 8z z 10 8 z z z 18. Expression 0.k k 1.006k Equation 0.u1. 0.7u u0. 0.u1. 0.u1. 0.u0.u1. 0.u0.u uu is a real number 1

19 Chapter 1 Linear Equations and Inequalities in One Variable Section 1. Practice Exercises 1. a. consecutive. The smallest positive integer that could be b. even; odd used to clear the fractions is the LCD of, c. 1; ;, and 6, or 0. d. x 1 e. x f. x ; x g. Prt; interest h. $100 i. 0.8 L; 0.1( x 8) j. d t ; d r. 7a 11 7a 11 7a 1 7a a 7 7. x 719 x 1719 x 19 x x x 6 6. z 61 z 6616 z 1 z z 6. y 1 y 1 1 y 191 y 18 y 18 y p p 8 8 p 8 p x 1 x

20 Section 1. Applications of Linear Equations in One Variable p 68p 1 p8p68p8p1 p 61 p 6616 p 18 p p 18x 0 118x 01 8x 19 8x x x 10. n t 7 1. y 1. Let x = the smaller number x + = the larger number (larger number) (smaller number) = 8 x x 8 xx 8 x 8 x 8 x x 10 1 The smaller number is and the larger is Let x = one number x = the other number (one number) + (other number) = 1 x x 1 xx1 x 1 x 1 x 18 x 18 x 9 x 9 6 One number is 9 and the other is Let x = the number x + = the sum x = the difference (sum) = (difference) x x xx xx x 16. Let x = the number x + = the sum x 1 = the difference (twice the sum) = (difference) x 6 1 x x 7

21 Chapter 1 Linear Equations and Inequalities in One Variable x x 6 x 6 x The number is. x x1 x6 x1 xx6 xx1 The number is Let x = the first page number x + 1 = the consecutive page number (first) + (second) = x x1 x 1 x 11 1 x x x 111 x The consecutive page numbers are 111 and Let x = the first raffle ticket number x + 1 = the consecutive raffle number (first) + (second) = 808, x x1 808, x 1 808, x , 1 x 808, x 808, x 0,7 x 1 0,7 1 0,8 The consecutive raffle ticket numbers are 0,7 and 0, Let x = the first odd integer x + = the consecutive odd integer (first) + (second) = 18 x x 18 x 18 x 18 x 10 x 10 x 7 x 77 The two consecutive odd integers are 7 and Let x = the first integer x + 1 = the second consecutive integer x + = the third consecutive integer (first) + (second) + (third) = 7 x x1 x 7 x 7 x 7 x 60 x 60 x 0 x x 018 The three consecutive integers are 0, 19, and 18.

22 Section 1. Applications of Linear Equations in One Variable 1. Let x = the smaller even integer x + = larger consecutive even integer ( times small) = ( 16 minus times larger) x 16 x x 16 x8 x 1 x x x 1 xx 7x 1 7x x x 0 The two consecutive even integers are and 0.. Let x = the smaller odd integer x + = larger consecutive odd integer ( times smaller) = ( times larger minus 7) x x 7 x x10 7 x x6 xx xx6 x 6 x 6 x 66 The two consecutive odd integers are 6 and 6.. Let x = first odd integer x + = second consecutive odd integer x + = third consecutive odd integer ( times sum) = ( more than times third) x x x x x x 6 0 6x1x 6xx1 xx x 1 x x 1 x 1 x 1 The three consecutive odd integers are 1,, and.. Let x = the smaller even integer x + = second consecutive even integer x + = third consecutive even integer ( times smallest)=(10 more than times largest) x x 10 x x810 x x18 xx xx18 x 18 x 18 x x 6 x 68 x 6 10 The three consecutive even integers are 6, 8, and 10.. Option 1: Principal amount borrowed: P 1,000 Interest rate: r 0.08 Duration of loan: t 6. Option 1: Principal amount borrowed: P 7000 Interest rate: r 0.08 Duration of loan: t

23 Chapter 1 Linear Equations and Inequalities in One Variable x Prt x 1,000(0.08)() x 100 Option : Principal amount borrowed: P 1,000 Interest rate: r Duration of loan: t x Prt x 1,000(0.077)() x 81.0 She would pay $100 for yr at 8.% and $81.0 for yr at 7.7%; the 8.% option for yr requires less interest. x Prt x 7000(0.08)() x 1680 Option : Principal amount borrowed: P 7000 Interest rate: r 0.08 Duration of loan: t x Prt x 7000(0.08)() x 1190 He would pay $1680 for yr at 8% and $1190 for yr at 1 8 %; the for yr requires less interest. 1 8 % option 7. Let x = the amount of sales (earnings) = (sales amt)(comm. rate) x x x x ,000 x She needs to sell $60,000 to earn $ Let x = the sales amount (earnings) = (sales amt 00)(comm. rate) x x x x x She sold $ worth of merchandise. 9. Let c = the sales before tax (total cash) = (sales) + (sales tax) (sales tax) = (tax rate)(sales) 19.8 c0.080c c c c t Let c = the cost before tax (total bill) = (cost) + (sales tax) (sales tax) = (tax rate)(cost) 1, c0.07c 1, c 1, c ,980 c The cost before sales tax was $1,980. 6

24 Section 1. Applications of Linear Equations in One Variable The total merchandise was $ and the sales tax was $ Let c = the cost before markup (price) = (cost) + (markup) (markup) = (markup rate)(cost).08 c0.0c c c c The price before markup was $.90.. Let p = the original price (sale price) = (price) (markdown) (markdown) = (markdown rate)(price) 9. p0.p p p p The original price was $.00.. % % Account Account Total Amount Invested x 1,00 x 100 Interest Earned 0.0x 0.0(1,00 x) 70 (int at %) + (int at %) = (total int) 0.0x0.0 1,00 x x6 0.0x x x x 0 0.0x x 0.0 x 800 1,00 x 1, $800 was invested at % and $000 was invested at %. 7

25 Chapter 1 Linear Equations and Inequalities in One Variable. 9% 10% Account Account Total Amount Invested x 1,000 x 1,000 Interest Earned 0.09x 0.10(1,000 x) 1 (int at 9%) + (int at 10%) = (total int) 0.09x0.10 1,000 x x x x x x x x ,000x 1, $6800 was invested at 9% and $800 was invested at 10%.. 11% 6% Loan Loan Total Amount Borrowed x 18,000 x 18,000 Interest Paid 0.11x 0.06(18,000 x) 180 (int at 11%) + (int at 6%) = (total int) 0.11x ,000 x x x x x x x x ,000 x 18, ,000 $6000 was borrowed at 11% and $1,000 was borrowed at 6%. 8

26 Section 1. Applications of Linear Equations in One Variable 6. % 8% Loan Loan Total Amount Borrowed x 6000 x 6000 Interest Paid 0.0x 0.08(6000 x) (int at %) + (int at 8%) = (total int) 0.0x x 0.0x x 0.0x x x 0.0x x x She borrowed $00 from her parents and $100 from the credit union. 7. % % Account Account Total Amount Invested x x 000 Interest Earned 0.0x 0.0(x 000) 70 (int at %) + (int at %) = (total int) 0.0x0.0 x x0.0x x x x x x 1,000 9

27 Chapter 1 Linear Equations and Inequalities in One Variable x 000 1, $1,000 was invested at % and $8000 was invested at %. 8..% 6% Account Account Total Amount Invested x x _ Interest Earned 0.0x 0.06(x+000) 110 (int at.%) + (int at 6%) = (total int) 0.0x0.06 x x0.06x x x x x x 8000 x ,000 $8000 was invested at.% and $1,000 was invested at 6%. 9. 1% 10% 1% nitrogen nitrogen nitrogen Amount of fertilizer x x + Amount of nitrogen 0.1(x) 0.10() 0.1(x + ) (amt of 1%) + (amt of 10%) = (amt of 1%) x 0.1x x x x0.1x x0.1x x x

28 Section 1. Applications of Linear Equations in One Variable 0.01x x x 8 8 oz of 1% nitrogen fertilizer should be used. 0. 8% 18% 1% Solution Solution Solution Amount of Solution x 80 x + 80 _ Amount of Saline 0.08x 0.18(80) 0.1(x + 80) (amt of 8%) + (amt of 18%) = (amt of 1%) x 0.08x x1. 0.1x x0.1x1. 0.1x0.1x x x x.8 0.0x x cc of 8% saline solution should be used. 1. 0% 7% 60% antifreeze antifreeze antifreeze Amount of fertilizer x x + Amount of nitrogen 0.0() 0.7(x) 0.60(x + ) (amt of 0%) + (amt of 7%) = (amt of 60%) x x x 0.60x x0.60x x0.60 x x x

29 Chapter 1 Linear Equations and Inequalities in One Variable 0.1x x x L of the 7% antifreeze solution should be used.. 0% 70% % fruit juice fruit juice fruit juice Amount of Punch x 10 x + 10 _ Amount of Fruit juice 0.0x 0.70(10) 0.(x + 10) (amt of 0%) + (amt of 70%) = (amt of %) x 0.0x x7 0.x. 0.0x 0.x7 0.x0.x. 0.0x x x. 0.0x x 0 0 gal of the 0% punch should be used.. 18% 10% 1% Solution Solution Solution Amount of Solution x 0 x 0 _ Amount of Alcohol 0.18x 0.10(0 x) 0.1(0) (amt of 18%) + (amt of 10%) = (amt of 1%) x 0.18x x 0.10x 0.08x 0.08x 0.08x 1

30 Section 1. Applications of Linear Equations in One Variable 0.08x x 1. 0 x L of 18% solution and 7. L of 10% solution must be mixed...% 10% % Solution Solution Solution Amount of Solution x 600 x 600 _ Amount of bleach 0.0x 0.10(600 x) 0.0(600) (amt of.%) + (amt of 10%) = (amt of %) x 0.0x x x x x x x x x ml of.% solution and 00 ml of 10% solution must be mixed.. 1% Pure 17.% Super Grow Super Grow Super Grow Amount of Solution x x _ Amount of Super Grow 0.1( x) 1.00x 0.17() (amt of 1%)+(amt of pure) = (amt of 17.%) x x xx.6

31 Chapter 1 Linear Equations and Inequalities in One Variable 0.88x x x x x oz of pure Super Grow must be added. 6. 0% Salt 8% Salt % Salt Solution Solution Solution Amount of Solution x 0 x + 0 _ Amount of Salt 0 x 0.08(0) 0.0(x + 0) (amt of 0%) + (amt of 8%) = (amt of %) x 0x x x x x x 60 oz of water must be added. 7. Distance Rate Time To FL (x + 60) x + 60 Return.x x. (dist to FL) = (dist back to Atlanta) x60.x x10.x x 10 x.xx 10 0.x 10 0.x x x

32 Section 1. Applications of Linear Equations in One Variable The plane flies 00 mph from Atlanta to Fort Lauderdale and 0 mph on the return trip. 8. Distance Rate Time Down (x + 1) x + 1 _ Back 6x x 6_ (dist down) = (dist back) x1 6x x6x xx 6xx x x 1 x x 111 Her rate is mph to the lake. 9. Distance Rate Time Car A x x _ Car B (x + ) x + _ (dist car A) + (dist car B) = (total dist) x x 19 xx819 x 819 x x 18 x 18 x 6 x 60 The cars are traveling at 6 mph and 0 mph. 0. Distance Rate Time Car A x x _ Car B (x ) x _ (dist car A) + (dist car B) = (total dist)

33 Chapter 1 Linear Equations and Inequalities in One Variable x x 190 xx10190 x x x 00 x 00 x 0 x 0 The cars are traveling at 0 mph and mph. 1. Let x = the first integer 0 x = the second integer (ten times first) = (five times second) 10x 0 x 10x 10 x 10x x 10 xx 1x 10 1x x 10 0 x The integers are 10 and 0.. Let x = the first integer 10 x = the second integer ( times first) = ( less than 8 times second) x8 10x x808x x778x x 8x778x8x 11x 77 11x x 7 10 x 10 7 The integers are 7 and.. New Price Original Price Markdown 89. x0.x 89. (1 0.) x x 89. x 0. x 199 The original price was $199.. New Price Original Price Markdown 1.60 x0.x 1.60 (1 0.) x x 1.60 x 0.6 x The original price was $. 6

34 Section 1. Applications of Linear Equations in One Variable. Distance Rate Time Boat A x x _ Boat B (x) x _ (dist boat B) (dist boat A) = (dist between) x x 60 6xx 60 x 60 x 60 x 0 x 0 0 The boat s rates are 0 mph and 0 mph. 6. Distance Rate Time Canoe A.x x._ Canoe B.(x) x._ (dist canoe B) (dist canoe A) = (dist between). x.x. 7x.x..x..x... x 1. x 1. The canoe s rates are 1. mph and mph. 7. % 6% Account Account Total Amount Invested x x _ Interest Earned 0.0x 0.06(x) 76 (int at %) + (int at 6%) = (total int) 7

35 Chapter 1 Linear Equations and Inequalities in One Variable 0.0x0.06 x x0.1x x x x 00 x $00 was invested at % and $9000 was invested at 6%. 8..% % Account Account Total Amount Invested x x _ Interest Earned 0.0x 0.0(x) 88 (int at.%) + (int at %) = (total int) 0.0x0.0 x x0.08x x x x 000 x Sienna deposited $8000 in her money market account. 9. Black Orange Tea Pekoe Tea Total Pounds of Tea x x _ Cost of Tea.0x.00( x).0() (cost black) + (cost orange) = (cost blend) x.0x x1 x x x

36 Section 1. Applications of Linear Equations in One Variable 0.80x 0.80x x. x.1.. lb of black tea and 1. lb of orange pekoe tea are used in the blend. 60. Almonds Cashews Total Pounds of Nuts x 16 x 16 _ Cost of Nuts.98x 6.98(16 x).7(16) (cost almonds) + (cost cashews) = (cost mix) x.98x x x x x x 0 x 0 x x lb of almonds and 6 lb of cashews must be mixed. 61. New Price Original Price Price Drop 0,100 x0.06x 0, x 0,100 x 0.9 x 1,000 The median price the previous year was $1, Let x = Associate s salary Bachelor s salary = Associate s salary + Increase 106 x 0.x x 106 x 1. x 760 The median weekly salary for an individual with an associate s degree is $760. 9

37 Chapter 1 Linear Equations and Inequalities in One Variable Section 1. Practice Exercises 1. a. l w c. supplementary b. o 90 d. o 180. x x x xx6 6x61 x16x7 x6x16x6x7 x 17 x 1171 x 1 x x. yy yy y110y 1y 1 1y y y 1 1. z z z7 zz z7 z6 z7 1z18 z7 1zz18 zz7 11z z z 11 z t t t t t18tt0t t0t0 tt0tt0 0 0 tt is a real number. a8a7a8a 10a 10a8 10a10a 10a10a Let w = the width of the rectangle l = w = the length of the rectangle Plw 177 w w 177 ww 177 6w w 9. 60

38 Section 1. Applications to Geometry and Literal Equations l w 9. 9 The court s dimensions are 9. ft by 9 ft. 8. Let w = the width of the rectangle l = w = the length of the rectangle P lw 11 w w 11 w8 w 11 6w w w w 0 l w The frame s dimensions are 0 in. by 6 in. 9. Let x = the length of the one side x + = the length of the second side x + = the length of the third side P abc xx x x 6 6 x x 6 x x 6 8 x 6 10 The lengths of the sides of the triangle are 6 m, 8 m, and 10 m. 10. Let x = the length of the one side x + 1 = the length of the second side x + = the length of the third side P abc 1 x x 1 x 1 x 1 x 1 x x x 11 x 6 The lengths of the sides of the triangle are ft, ft, and 6 ft. 11. a. Let l = the length of the run A lw b l l 100 l 00 l 8 l The dimensions are 8yd by 1. yd. P lw P 8 1. P 16 P 1 The perimeter is 1 yd. 61

39 Chapter 1 Linear Equations and Inequalities in One Variable 1. Let x = the length of a side of the trapezoid x = the length of the top of the trapezoid (fence backyard) = (side) + (side) + (top) (fence front yard) = 1 (side) + 1 (side) + (top) (fence backyd) + (fence frontyd) = (total fence) 1 1 x x x x x x 60 xx 60 6x 60 x 10 The sides of the back garden are 10 ft, 0 ft, and 10 ft. The sides of the front garden are ft, 10 ft, and ft. 1. Let w = the width of the pen w 7 = the length of the pen P lw 0 w7 w 0 w1 w 0 6w w 1 1 6w 9 w w The width is 9 ft and the length is 11 ft. 1. Let s = the length of a side of the square P s 18 s 9 1 s The lengths of a side of the square is 1 ft. 1. Let x = the measure of the two equal angles (x + x) = the measure of the third angle xx xx 180 xxxx180 6x 180 x 0 x x The measures of the angles are 0º, 0º, and 10º. 16. Let x = the measure of the largest angle 1 x = the measure of the smallest angle x = the measure of the middle angle 1 x x x180.x Let x = the measure of one angle x = the measure of the other angle x x 90 6x 90 x 1 x 1 7 6

40 Section 1. Applications to Geometry and Literal Equations.x 180.x 0 x 8 x x 8 7 The measures of the angles are 8º, 1º, and 7º. 18. Let x = the measure of one angle x 1 = the measure of the other angle x x1 180 x x x 19 x 8 x The measures of the supplementary angles are 8º and 1º. 0. x x x6 x0 90 1x x x x x x The measures of the angles are 6º and º.. x x x 1x x 180 x 10 The measures of the complementary angles are 1º and 7º. 19. x x x 180 x 0 7x x The measures of the angles are 19º, and 1º. 1. x x. 90 x 7.90 x x 8. x 7. x x The measures of the angles are 60º, and 0º.. x x x x x x 1 6

41 Chapter 1 Linear Equations and Inequalities in One Variable x 10 x The measures of the angles are 7º and 1º.. x x x x x x 186 x 6 10x x 6 0x The measures of the angles are 60º, º, and 116º. 6. x x 8 90 x x 90 x 090 x x 10 x x 6 x The measures of the angles are 6º and 6º. x 18 x 18 6 x x 18 The measures of the angles are 6º, 91º, and º.. x x 7 90 xx190 x 90 x 90 x 11 x x x The measures of the angles are º and 8º. 7. a. d rt d r t b. 00 r 161. mph.099 The average speed was 161. mph. 6

42 Section 1. Applications to Geometry and Literal Equations 8. a. d rt d t r 00 t. hr a. I Prt I t Pr b. The total time was. hr. b. 100 t 7 years a. F ma F m a b.. m. kg A lw for l A lw w w A l w. C R for R 1 C R 1 R C 1. I Prt for P I Prt rt rt I P rt. abc P for b abcac Pac b Pac 6. ymxb for x ybmxbb ybmx y b mx m m y b x m W K K for K. 1 1 W K K K K W K K W W K K W 1 K K W F C for C 9 F C F 9C160 F160 9C F160 9C 6

43 Chapter 1 Linear Equations and Inequalities in One Variable F160 9C 9 9 F 160 C 9 F C 9 F 9 8. C F for F C F 9 9 C F 9 C F 9 F C 9. 1 K mv for v 1 K mv K mv K mv m m K v m 0. I Prt for r I Prt Pt Pt I r Pt. a b c for b a b a c a b c a 1. v v at for a 0 vv v v at vv at 0 v v0 at t t v v a t. w pv v w p v v for v 1 p p w v v 1 p w v v v v p w v v 1 p w pv1 v p

44 Section 1. Applications to Geometry and Literal Equations. A lw for w A lw l l A w l 6. P L W for L PW LW W PW L P W L P W L P L W 8. 1 V r h for h 1 V r h V r h V r h r r V h r 0. xy xxy x yx. ax by c for y ax by ax c ax by c ax by c ax b b c ax y b 7. 1 V Bh for B 1 V Bh V Bh V Bh h h V B h 9. xy6 xxy6x yx6 1. x y 0 x xy 0x y 0x y 0 x 0 x y y x 67

45 Chapter 1 Linear Equations and Inequalities in One Variable. x y x xy x y x y x y x. x 7y 1 x x7y1x 7yx1 7y x y x x y 1 6x 6xy 16x y 6x1 y 6x1 1 y x. xy6 xxy6x yx6 y x6 yx 6. xy8 xxy8x yx8 y x8 y x 7. 9x y 9x y 7xy1 7x 7xy17x y7x1 y 7x1 7 1 y x 8. 1 x y 1 x y 1xy1 1x1xy1 1x y1x1 y x y1x1 9. x y0 x x y0 x y x y x y x 68

46 Section 1. Applications to Geometry and Literal Equations x y0 1 x y 0 xy0 x yy0y y x 61. a. x z x z z x z x x z b. x a. b. x z x z z x z x x z or xz xz 10.(16) x x 1 x 1 x x x Expressions a, b, and c are equivalent. 6. z 1 1 z1 z1 1z 1 z1 z1 Expressions a, b, and c are equivalent. 6. x7 1 x7 x7 x7 y 1 y y y Expressions a and b are equivalent. 66. w w x y xy w 1 w x y 1 xy w x y w 1 w x y 1 xy w x y Expressions a, b, and c are equivalent. 67. t 6trt 1 for t 6r 1 t 6 r 1 6r 6r 1 t 6 r 69

47 Chapter 1 Linear Equations and Inequalities in One Variable 68. aca for a a c a c c c a c 69. ax 6x for x ax 6x 6x 6x ax 6x ax 6x ax 6x x a6 x a 6 a6 a6 x a 6 or x 6 a 70. cx dx 9 for x cx dx dx dx 9 x cd 9 x cd 9 x cd 1 x c d 1 cd cd 1 1 x or x cd dc 71. A PPrt for P AP 1rt A P 1 rt 1rt 1rt A P 1 rt 7. APPrt for r AP PPPrt AP Prt A P Prt Pt Pt A P r Pt 7. T mgmf for m T m g f T m g f g f g f T m g f 7. T mgmf for f T mgmgmgmf T mgmf 7. ax by cx z for x ax cx by cx cx z x ac by z 70

48 Section 1. Linear Inequalities in One Variable T mg mf m m T mg mgt f or f m m x ac byby zby x ac zby x a c z by ac ac z by by z x or x ac ca Section 1. Practice Exercises 1. a. linear; inequality. parenthesis b. negative c. Both statements are correct. 9. a. x 10 x 10 x 6 x n/a 10. a. x 6 x 6 x 8 x n/a b. x 10 x 10 x 6 x x x, b. x 6 x 6 x 8 x x x, 71

49 Chapter 1 Linear Equations and Inequalities in One Variable c. x 10 x 10 x 6 x x x, c. x 6 x 6 x 8 x x x, 11. y 6 y 666 y y y 1 1. y 11 y y 6 y 6 y a. y y 1 a. yy b., 1 b., 1. x x x 0 x 0 0 x x 10 a z z z 0 z 0 0 z z x x a. z z b. 10, b., 7

50 Section 1. Linear Inequalities in One Variable 1. 6z 16 6z 16 6z 1 6z z 6 a. 1 z z 6 b. 1, w 1 8w 1 8w 1 8w w 8 a. 1 w w 8 b. 1, t 8 t 1 t t 1 a p 1 p 0 p p 0 t t a. p p 0 b., 1 b. 0, y y 8y 9 8y 999 8y 1 8y y 8 0. x x x 1 x 111 x 6 x 6 x 1 7

51 Chapter 1 Linear Equations and Inequalities in One Variable a. 1 y y 8 b. 1, 8 a. x x 1 b. 1, a0. 0.a11 a a aa110 8aaaa110 a 110 a 110 a 10 a 10 a 1. 0.w w w w w79w w9w79w9w 11w 7 11w w 11 11w w 1 a. aa 1 a. w w 1 b., 1 b.,1. x 7 x 777 x 1 x 1 1 x x a.. w 69 w 6696 w 1 w 1 1 w w x x a. w w b., b., 7

52 Section 1. Linear Inequalities in One Variable. x x 6 18 x 0 9 x 10 a. 9 xx 10 b. 9, y 16 1 y 16 y 8 7 y 8 a. 7 y y 8 b. 7, 8 7. p 1 p 1 p 110 p p 9 p 9 p a. 8. k k k 0 k 0 k 18 k 18 k 6 p p a. kk 6 b., b. 6, 9. 0.t1.t10 t t t10t100 tt10 tt100 t x x 1 1 x 7

53 Chapter 1 Linear Equations and Inequalities in One Variable t t 110 t 110 t a. 1 1 x 6 x x 6 tt a. x x 6 b., b., 6 1. y6y1 y868y y8y10 y8y8y8y10 1y 10 1y 10 1y 1 1y y a. y y b.,. b b 1 1b8b10 b7b6 bb7bb6 b 76 b 7767 b 1 b 1 1 b a. 1 bb b. 1,. 7.k.1.7 k k 1 7 7k k 108 7k k or k 1.. 6h h h 19. 6h h. 76

54 a. k k 1. a. h h. b. 1., b.,. Section 1. Linear Inequalities in One Variable. x 81 x 8818 x 9 x 9 x 9 x 1 a a a a 8 a 8 a 8 a 0 x x a. aa 0 b.,1 b., b0. 0.b 1.b1.b0. 0.b1.b b b b b 0. a t 1. 0.t t. 0.t t 0. t 8 b b a. tt 8 b., 0. b. 8, 77

55 Chapter 1 Linear Equations and Inequalities in One Variable 9. c c c c c8c cc8cc 11c 11c c 11 or c 11 a. c c 11 b., q q q 6 q qq qqqq 7q 7q 7 7 q 7 or q 7 a. q q 7 b., 7 1. y y6 y8y6 y1y6 yy1yy6 y 1 6 y y 6 a. 6. k yy a. k k 6 b., 6 b., 6 66 k1 66k18k1 6kk1 6kkkk1 k 1 k 1 k 1 k 1 1 k k 6 78

56 Section 1. Linear Inequalities in One Variable.. p pp 6 x 1 x 6x 1x 6 x 6x 1x6 7x9 1x7x6 7x7x9 x 69 x 6696 x 1 8p6 pp9 7p6p 7pp6pp p 6 p 66 6 p 10 x 1 p 10 x p x x a. p p a. b., b.,. a a a a9a1a 6a 6a 6a 0 6a a 0 a q q aa a. qq 0 b., 0 b. 0, q8q11 6q 11 6q q 0 6q q 0 7. a x x a x x

57 Chapter 1 Linear Equations and Inequalities in One Variable 00 0 x x x 10 Nadia needs to score at least a 70% but less than 10% to get a B average x x x 100 Ty needs to score at least a 70% but less than 100% to get a C average. b x x 90 0 x x 0 0 x 10 It would be impossible for Nadia to get an A because she would have to earn 10% on her last quiz and it is impossible to earn more than 100%. b x x x x x 100 It would be possible for Ty to get a B if he scored 100% on his last exam, but it would be impossible to get an A. 9..a 1 1.a a 0.a 0.. a 8 Boys 8 years old or older will be on average at least 1 in. tall. 0..a 1 1.a a 10.a 10.. a Boys years old or older will be on average at least 1 in. tall. 1..a 1 6.a a 1.a 1.. a 6 Boys 6 years old or younger will be on average no more than 6 in. tall...a 1..a a..a... a 9 Boys 9 years old or younger will be on average at most. in. tall. 80

58 Section 1. Linear Inequalities in One Variable. a., x 0,000,000, x 0,000, x 1, x 1, x 7,000 Her sales must exceed $7,000. b., x 80,000,000, x 80,000, x, x, x 1,7,000 Her sales must exceed $1,7,000. c. The base salary is still the same; the increase comes solely from commission.. a t 10, t 10, t 000 t 000 t 1. It will take at least 1. years. b t 1, t 1, t 10,000 t 10,000 t 0.8 It will take at least 0.8 years.. R C 9.9x x 9.9x 18.0x x18.0x 1.x R C 9.9x 6,000 10x 9.9x 10x 6,000 10x10x 109.9x 6,000 81

59 Chapter 1 Linear Equations and Inequalities in One Variable 1.x x 7.1 There will be a profit if more than 7 jackets are sold x 6, x 09. There will be a profit if more than 09 bicycles are sold. 7. a b ac bc 8. a b ac bc 9. a b ac bc for c a b ac bc for c 0 Section 1. Practice Exercises 1. a. union; A B. b. intersection; A B c. intersection x 1 x x x x (, ) d. a x b e. union. 6u 8 6u 6 6u u 1,1. z z 6 z 6 z,. 1 p 1 p 16 p p 16 16, 6. 1 w 1 w 1 w,1 7. a. M N, 1 8. a. P Q a, e, i,,, 1, 0,1,, b. P Q a, b, c, d, e, f, g, h, i, o, u b. M N 8

60 9. A C 7, 10. B C, Section 1. Compound Inequalities 11. AB,, 1., 0, A D 1. A B 1. A D 1. B C 7, 16. B D 0, 17. C D 0, 18. B D, 19. C D 7, 0. A C, 1. a., 1, 1, b., 1,,. a., 1, 1, b., 1,,. a. 9, 1, 1, b. 9, 1, 9,. a..,1.6.,.1.,1.6 b..,1.6.,.1.,.1. a., 0, 0, 6. a. 1, 0, 0,, 0,, b. 1, 0, 1, b. 7. y 7 9 and y y y,,, 8. a6 and a0 a 8 a6 8,, 68, 6 8

61 Chapter 1 Linear Equations and Inequalities in One Variable 9. t7 19 and t1 8 t 1 t 1 t 6 t, 6,, 6 0. p p1 and 9pp 7p1 6 p p p,,, 1..1k k1.9 and 0.k1.10.1k0.9 k k k k and k11 6k19 and k11 k9 1k11 19 k11 9 1k 0 k 0 k k,,,. 0.6w0.1 0.w1.1 and.w1. 0.w6. w w w w and w1 w11 and w1 w6 w1 11 0w1 6 w 1 0w 0 w 0 w 0,,,. p110 and p0 p1 10 p 0 p1 1 p p16 p1 p8 p7. a6 and a1 6 and 1 a a a10 1 a6 a a10 10 a a 10,,, 8

62 Section 1. Compound Inequalities 8, 7, 8,. x 1 and 1 x 6x 10 x 1 x1 6x 10 x 1 11x1 x x x 10 1 x 11, 10 1, 7. t and t The statement 6 x is equivalent to 6 x and x. However, no real number is greater than 6 and also less than y and y7 16 y 6 y y116 y y y y y y y y,, 8..8 y and y 1 0. The statement t 1 is equivalent to t and t 1. However, no real number is greater than and also less than The statement y is equivalent to yand y. However, no real number is less than and also greater than.. The statement w 1 is equivalent to w and w 1. However, no real number is less than and also greater than b 9 b 1 b 7, 7. 6k 90 k 9 1 k 1,. a a 6 6, x 0 6 x 0 6, 0 8

63 Chapter 1 Linear Equations and Inequalities in One Variable 7. y 1 6 y 8 y, t 1t 6 t 10,10 9. x8 7x x 10 7, 0. 1x x 1 1 x 1, x 0 96x 1 1 x,. x 7 1x 1 1 x 1 1, t.8 7t t 0. 0., x x 81 1, 81. y1 or y y y y y,, 6 x 0 or x 17 x 0 x 6 x 0 x, 0, 86

64 Section 1. Compound Inequalities z 8 or 8z z 8z16 z z,,, 8. t10 or 7 t t 1 t 8 t 6 t, 8, 6, 8 9. x 1 or x 11 x x 6 x 0 x 6 x 0 x 6 0, 6, 6, 60. p p 7 10 or 1 1 p p 1 p p1 p p,,, 61. v or v6 1 v v 7 v v7, 7,, 6. u10 or u 8 u1 u u 1 u 8 8 u u 8,,, 6. 0.w.w or 0.w0.1w1.6 w 0.w1.6 w9 w 9 w w 9,, 6. 1.a 0.a6 or.a1 9 1.a 0.7a 6 a a9 a10 a1, 1, a 87

65 Chapter 1 Linear Equations and Inequalities in One Variable 6. a. b. x 19 and x x x 0 x 8 x 10, 8 10, 10, 8 x 19 or x x x 0 x 8 x 10, 8 10,, 66. a. x x x and 1 7. x 0.8x7 x 7..x 7 x..x 11 x 0.8 x x 0.8,, 0.8, 0.8 b. x x x or 1 7. x 0.8x7 x 7..x 7 x..x 11 x 0.8 x x 0.8,, 0.8, 67. a. x x 8 6. or x 10. x6 x 1. x 8 1.,, 8 b. x x 8 6. and x 10. x6 x 1. x 8 1.,, 8 No Solution 68. a. b. r8 or r8 r 1 r r 6 r 1 6,,1 r 8 and r 8 r 1 r r 6 r 1 6,,1 No Solution 69. x 8 1 x 1 x x, 70. x 1 0 x 0 x x, 71. t t or 6 1t8 t t1 t 6 1t 8t t1 6 t 7. w w or w 6w w w w16w w w1 88

66 Section 1. Compound Inequalities 7 t 6 t 1 7 t t 1 7,, 1, 0 w 0 w 0 w 0 w, 0 0,, 0 7. x x 1x x or 1 x x x x x1 8 x x 8 x 7x1 8 x 8 7x 7 x x 1 x,1, 7. y7 1 y1 1 or y7 1 y y y y8 y y y1 1 y y 1,, 7. a. 800 x 10, a. 1 x 16 b. x800 or x 10,800 b. x 1 or x a. % x 8% 78. a.. x.6 b. x % or x 8% b. x. or x x 1 x 6 All real numbers between and x 1 or x 11 x or x x or x 1 All real numbers greater than or less than x x 1 All real numbers between 6 and x or x x6 or x 1 All real numbers less than 6 or greater than 1 89

67 Chapter 1 Linear Equations and Inequalities in One Variable 8. a x x 90 0.x 16. x 8 Amy would need 8% or better on her final exam. b x x x 16. x 8 If Amy scores at least % and less than 8% on her final exam, she will receive a B in the class. 8. a x x 90 0.x 6.6 x 91. Robert would need 91.% or better on his final exam. b x x x x 91. If Robert scores at least 66.% and less than 91.% on his final exam, he will receive a B in the class F F 9 0 F 0. 0 F F F F 10. º F.08º F F F F F F 1 0 F 1 68º F 8.º Section 1.6 Practice Exercises 1. a. absolute; { a, a} b. Subtract from both sides. c. y; y d. { }; { }. a a 6 and 1 a6 6 a6 1 a a a a,,, 90

68 Section 1.6 Absolute Value Equations. x 7x or x1x x x1 x 8 x x x,,. x 6 7x x x x 7, 7. x x 10 x 6 6 x 10 6, x x x 9 18 x 7 18, 7 7. p 7 p7 or p7 7, 7 8. q 10 q 10 or q10 10, x x 0 x 6 x 6 or x 6 6, 6 x x or x, 11. y 8 y 1. x 1 6 x 6 1. w 1 1. z 1 10 w w or w, z z or z, 1. y y or y, 16. y y or y, 91

69 Chapter 1 Linear Equations and Inequalities in One Variable 17. w w 18. w 8 w q 0 0. p 0 q 0 or q 0 q 0 0 p0 or p0 p x 8 x8 or x8 x 1 or x x or x,. x 1 6 x16 or x16 x or x x or x,. x x or x x 9 or x x or x,. 10 x 7 x7 10 or x7 10 x or x x 1 or x 1,. 7z 1 6 7z 1 7z 1 7z 1 or 7z19 or 7z19 7z10 or 7z z or z, w 7 w 9 w w 9 or 9 w18 or w18 w 1 or w 1 1, x x. 9

70 Section 1.6 Absolute Value Equations w 1 w 1 1 w or w 8 w 0 or 8 w 0 w 1 or w 8 w 1 or w 8 1, x 6x 6 6x 6 or 6x 6 x 0 or x 1 x 0 or x 6 0, y 1 y 1 6 y1 6 or y1 6 y or y y or y,. 1 x 7 x 7 1 x7 1 or x7 1 x 6 or x x or x,. b799 b70 b 7 0 b70 or b70 b b. x1 x10 x 1 0 x10 or x10 x x. x 6. x 7 x x x 9 6x90 or 6x90 6x 9 x 8. 7 k k 6 k60 or k60 k 6 k 9

71 Chapter 1 Linear Equations and Inequalities in One Variable k k or k k 18 or k 18 k 0 or k 16 k or k, h h or h h 9 or h 9 h 1 or h 6 h or h, 1. 6x 10 6x 1 6x 6x or 6x 6x or 6x x or x, x 7 1x 10 1x 1x or 1x x 1 or x 1 1 x or x,. x 8 x 8 x 8 or x 8 x 10 or x 6 x or x,. x x x or x x 0 or x x 0 or x 0,. w w ww or w w ww or ww w or 6w w8 or 6w w or 1 1 w, 6. y1 y7 y1y7 or y1 y7 y1y7 or y1y7 y1 7 or y1 7 y8 or y6 y8 or 6 6 y 8, 9

72 Section 1.6 Absolute Value Equations 7. y 7y y7 y or y 7y y7 y or y7y y 7 or 7 y or contradiction 8. 9a 9a1 9a9a1 or 9a 9a1 9a9a1 or 9a9a1 1 or 18a 1 contradiction or 18a 1 1 y a w1 w 1 6 w1 w 1 w1 w 1 or 6 6 w1 w 1 w1 w 1 or 6 6 w1 8w or w1 8w 8w8w or 8w8w or 16w contradiction or 16w w p p 8 6p 6p p or 8 8 p 6p 6p p or p 8 8 6p6p16 or 6p6p16 16 or 1p 1 contradiction or 1 1 p x x x x or x x x or xx x or x x xxis a real number. y y yy or y y y6 or yy y or yyis a real number..m1. 8.m6. 11.n9 7.n.1.m1. 8.m6 or.m1. 8.m6.m1. 8.m6 or.m1. 8.m6 11.n9 7.n.1 or 11.n9 7.n.1 11.n9 7.n.1 or 11.n9 7.n.1 9

73 Chapter 1 Linear Equations and Inequalities in One Variable m1. 6 or 1m1. 6 m7. or 1m.8 m1. or m0. n9.1 or 18.n9.1 n 11.1 or 18.n6.9 n.77 or n0.7 1., 0..77, 0.7. x x A positive number cannot equal a negative number. 6y 8y - A negative number cannot equal a positive number w 7w8 87w7w8 or 87w 7w8 1w16 or 8 7w7w8 8 w or ww 7 wwis a real number 8. z z zz or z z 6z8 or zz z or zz zzis a real number 9. x x t6 t1 0 x x - A positive number cannot equal a negative number. t6 t1 - A positive number cannot equal a negative number. 61. x x 6. x 6. x 9 Section 1.7 Practice Exercises 1. a. a ; a. a. x x 8 x 96

74 Section 1.7 Absolute Value Inequalities b. a ; b. x x 8 x (, ) c. ; (, ) c. x x 8 x (, ) d. includes; excludes. 7x 1 1 7x 7x 1 or 7x 1 7x or 7x x or x, x 1 6 x 1 0 x 10 x 1 x. 1 w 6 9 9w w 1, 1 6. y 1 and y 1 y y 1 y y,,, 7. m7 or m710 m m,,, 8. b7 or b b9 b6 b b6, 6, 9. a. b. x x or x, x x or x,, 10. a. b. a a or a, a a or a,, 97

75 Chapter 1 Linear Equations and Inequalities in One Variable c. x x, c. a a, 11. a. b. x 7 x 7 or x 7 x or x10,10 x 7 x7 or x7 x or x 10, 10, 1. a. b. w 6 w 6 or w 6 w8 or w 8, w 6 w6 or w6 w8 or w, 8, c. x 7 7 x 7 x 10,10 c. w 6 6w 6 8 w 8, 1. a. p 1. a. x 1 b. p All real numbers, b. x 1 All real numbers, c. p c. x 1 1. a. y a. z b. y 1 6 All real numbers, b. z All real numbers, c. y 1 6 c. z 98

76 Section 1.7 Absolute Value Inequalities 17. a. b. x 0 x 0 0 x 0 x 0 or x 0, 0 0, 18. a. b. p 0 p 0 p p 0 p0 or p0 p or p,, c. x 0 c. p a. b. k 7 0 k 70 k 7 7 k 7 0 k 7 0 or k 7 0 k 7 or k 7, 7 7, 0. a. b. x x 1 x x 1 All real numbers, c. k 7 0 c. x x 1 1. x 6 x 6 or x 6, 6 6,. x 6 6 x 6 6, 6. t t,. p p or p,, 99

77 Chapter 1 Linear Equations and Inequalities in One Variable. y 0 All real numbers, n All real numbers, 7. x 1 8. x 7 x 1 x1 or x1 x or x 6 x or x,, x 7 or x 7 x or x 9, 9, 9. k 7 0. h 9 1. w 1 w w 1 w 1 10 w 1 10,1. x x 6 x x 6 or 6 x1 or x1 x 1 or x 9, 1 9,. 1 9 y 9y 1 9y1 or 9y1 y or y y or y,,. m 7 1 m 7 m 7 1 1m 71 6m 8 m, 100

78 Section 1.7 Absolute Value Inequalities. x 1 1 x 1 x 1 0 x x x, 6. x 9 x 7 x 7 7 x 1 x 9 1, x 1 x x All real numbers, x x 1 x 1 1 All real numbers, 9. m 1 0. x m 1 0 m 1 0 x 0 x 0 1. p 0 0 p 0 p. y 1 y y 10 1 y 1 1. z 6 z 6 0 z6 0 or z6 0. c 1 c 1 0 c1 0 or c10 101

79 Chapter 1 Linear Equations and Inequalities in One Variable z6 or z6, 66, c1 or c c or c 1 1,,. y61 y6 10 y 6 y6 or y6 y or y8 y or y,, 6. 7 y y 1 1 y 1 y 1 or y 1 y or y1, 1, 7. 6t x 1 6t 6 8x 16 6t 8x 6t 8 t 8 t, 8 8x 1 x 1 x, x x x x x x x. 10., x x x. 1.6,. 1. x 7. x 10

80 Problem Recognition Exercises: Identifying Equations and Inequalities. x 1. x 6. x x x x w w w ,.01 The solution set isw 1.99 w.01 or equivalently in interval notation, 1.99,.01. This means that the actual width of the bolt could be between 1.99 cm and.01 cm, inclusive. 60. p p p , 0.6 The solution set is p 0.0 p 0.6 or equivalently 0.0, 0.6. This means that the percentage of votes received by the frontrunner was projected to be between 0% and 6%, inclusive. 61. b 6. d 6. a 6. c Problem Recognition Exercises 1. a. b. c. x 918 x 7 x 9 9 x 9 18 x9 18 or x9 18 x 7 or x 9 x 9 or x 9, x x x 7 x 9, 9. a. y 0 y y b. y 0 c. y 0 10

81 Chapter 1 Linear Equations and Inequalities in One Variable d. x 9 18 x9 18 or x9 18 x7 or x9 x9 or x, 9, d. y 0,. a. b. t 1 0 t 1 t 10 t 1 t 7 7 t 7, 7. a. b. x 9 x 7 x 9 9 x 9 x 7 x 9 9, c. t 1 0 t 1 t 7 7, c. x 9 x 7 x 9, 9. a. 8t t 8tt or 8t t 10t or 8t t 10t or 6t t 10 or 6t t or t, a. x and x 8 7 x and x 6 7, 6 b. 8t t 10t 1 1 t b. x 8 7 x 6 7, 6 10

82 Problem Recognition Exercises: Identifying Equations and Inequalities 7. a. b. x 911 or x 1 x 0 or 1 x x or x 1, x 9 11 and x 1 x 0 and 1 x x and x 1 1, 8. a. y or y y1 y or y6y1 y or 7 y, 7, b. y and y y1 y and y6 y1 y and 7 y 9. a. linear equation 10. a. linear equation b. 0.y y y 6 6 b. m 918 m 7 m a. absolute value inequality 1. a. absolute value inequality b. t 8 t 8 1 t 6 t 6, b. 1x 1 1. a. compound inequality 1. a. compound inequality b. 11 t 1 19 b. z 11 or z 9 1 t 18 z1 or z6 6 t 9 6, 9 z7 or z, 7, 1. a. absolute value equation 16. a. absolute value equation b. 1 y 1 1 y or y 1 1 y or y8 y or y16, 16 b. x 9x x 9 x or x 9x 6x 9 or x9x 6x 6 or x 1 x 1 or x 6 1, 6 10

83 Chapter 1 Linear Equations and Inequalities in One Variable 17. a. linear inequality 18. a. linear inequality b. b. p 9 p 9 p 1,1 8w w1 w 1 w w 1 1, 19. a. absolute value inequality 0. a. absolute value inequality b. x 9 b. 10 x x9 x9 10 x or x9 1 or x x 1 x or x6 x x1 or x, 1, x, 1. a. absolute value equation. a. absolute value equation b. c c b. 10n n 0 10n 0 10n 1 1 n 10. a. linear equation. a. linear equation b. w w1 1 w w w w w1w1 w 171 w w b. 1 1 y y y 6 y 1 6 yy6 11 y

84 Chapter 1 Group Activity. a. compound inequality 6. a. compound inequality b. x 7 9 and x 6 x 16 and x 1 x 8 and x 1 8,1 b. 1 x x and x x and 6 x 6, 7. a. linear equation 8. a. linear equation b. x7xx1 x107xx xx, b. 7y y1 y 7y yy 7y 7y Group Activity 1. False. True. True. False. True 6. True 7. True 8. True 9. True 10. False 11. False 1. True 1. False 1. True 1. False 16. True 17. False 18. False 19. False 0. True 1. True. True 107

85 Chapter 1 Linear Equations and Inequalities in One Variable. False. False Chapter 1 Section 1.1 Review Exercises 1. The empty set; no solution. All real numbers. x 7 x x A conditional equation. 7 y y y 8 8 A conditional equation x 0.x 7.0.6x0.6x 0.x0.6x 7. 0.x 7. 0.x x A conditional equation y1.1.y 0.1y0.1y1.1.y0.1y 1.1.1y 1.1.1y y 0. A conditional equation 7. m9m m 79m m9m 79m9m 6m 7 6m 1 6m m 6 6 A conditional equation 8. n6n 10n1 n9 10nn1 nn9 7n 1 9 7n n 1 7n n A conditional equation 108

86 9. x x1 1 x x x x x610x10 8x x x 8 8 A conditional equation 10. x x x9x x7x xx7xx 7 Chapter 1 Review Exercises This is a contradiction m18 m m m18 m 8 8 m m18 m m This is a contradiction. Section m m1 m m m m m m 1 1 m m 1 1 mm mm 1 1 mmis a real number This is an identity. 1. xx, 1, x 1. xx, 1. D rt Distance equals rate times time 16. I Prt Simple interest equals principal times rate times time. 17. a. Let x = the amount of tax 0.88,00 tax tax rate income x x,86 The tax is $, Let x = the number of men in college in 000 (recent year) = (000 number) + (increase) (increase) = (increase rate)(000 number) b. Let y = net income 109

87 Chapter 1 Linear Equations and Inequalities in One Variable net income income tax y 8,00,86 y 61, The net income is $61,. 7. x 0.08x x x x The number of men in college in 000 was approximately 6.7 million. 19. Let x = the number of alcohol deaths in 1999 (recent year) = (1999 deaths) + (increase) (increase) = (increase rate)(1999 deaths) 17,0 x 0.0x 17,0 1.0x 17,0 1.0x ,600 x The number of alcohol-related deaths in 1999 was 16, Let x = the first even integer x + = the second even integer x + = the third even integer (first) + (second) = (third) 6 x x x 6 x x xx xx x x x x x 0 The integers are,, and Let x = the length of the first piece 1 x = the length of the second piece (length of first) + (length of second) = (total) 1 x x 8 x 8 x x 1 1 x The lengths are ft and ft. 110

88 Chapter 1 Review Exercises. 6% 9% Account Account Total Amt Invested x x _ Int Earned 0.06x 0.09(x+000) 0 (int at 6%) + (int at 9%) = (total int) 0.06x0.09 x x0.09x x x x 0.1x x 100 x $100 was invested at 6% and $00 was invested at 9%.. 10% Acid % Acid 1% Acid Solution Solution Solution Amount of Solution x 1 x + 1 _ Amount of Alcohol 0.10x 0.(1) 0.1(x + 1) (amt of 10%) + (amt of %)=(amt of 1%) x 0.10x x0. 0.1x x0.1x0. 0.1x0.1x x x x x x L of 10% solution should be used. 111

89 Chapter 1 Linear Equations and Inequalities in One Variable. Distance Rate Time Linda 0.x x 0._ Lynn 0.(x + 1) x _ (distance Linda) + (distance Lynn) = 7. 0.x0. x x0.x7. 7. x x 0 x Linda drives 0 mph and Lynn drives mph. Section 1.. Let w = the width of the rectangle w + = the length of the rectangle P lw 0 w w 0 w w 0 w 0 w 6 w 9 w w 9 11 The width is 9 ft and the length is 11 ft. 6. x x 1 x x 1 x0 x xx0 xx x 0 x x x 7 x The measure of each angle is 7º. 7. x x x 90 x 90 x 0 x x y for y x xyx yx y x y x 11

90 Chapter 1 Review Exercises x The measures of the angles are 9º and 61º. 9. 6xy1 for y 6x6xy6x1 y6x A bh for b 1 A bh A bh A bh h h A b h 0. S rr h for h S r r r r h S r r h S r r h r r S r h r. a. C r for C r r r C r b Section 1.. 6x 6 6x 6 6x 8 6x x. 10x 1 10x x a. x x a. x x 11

91 Chapter 1 Linear Equations and Inequalities in One Variable b.,. x 7 19x 7x1 19x 7x16 19x 7x7x16 19x7x 16 6x 16 6x x or x 1 1 b., 6. x 10x x 10x0 x10x 10x10x0 7x 0 7x 0 7x 6 7x x 7 a. 8 x x 1 b. 8, 1 7. x 9 8 x x 7 x 7 x 67 x x a. 6 x x 7 b. 6, 7 8. x 8 x 8 x x x 9 x 9 9 x a. 67 x x b. 67, a. 9 x x b. 9, 11

92 Chapter 1 Review Exercises x x 90 x 0 x 0 x 9 Dave must earn at least 9% on his th test. Section C D is the set of all elements in both C and D. 1. X Y 10,1 C D is the set of elements in either set C or set D, or in both sets.. X Y,. Y Z,. Y Z 1,1. ZW, 1, 6. Z W 7. m11 and m1 11 m m16 11 m m 11,, 11, 11