1.1 Exercis es - College of the Redwoodsmsenux2.redwoods.edu/IntAlgText/chapter1/chapter1exercises.pdf · 1 Cop yri gh ted materi al. See: h ttp: //msen ux. redw o o ds. edu/In tAl

Section 1.1 Number Systems 11

Version: Fall 2007

1.1 Exercises

In Exercises 1-8, find the prime factor-ization of the given natural number.

1. 80

2. 108

3. 180

4. 160

5. 128

6. 192

7. 32

8. 72

In Exercises 9-16, convert the given dec-imal to a fraction.

9. 0.648

10. 0.62

11. 0.240

12. 0.90

13. 0.14

14. 0.760

15. 0.888

16. 0.104

In Exercises 17-24, convert the givenrepeating decimal to a fraction.

17. 0.27

Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/1

18. 0.171

19. 0.24

20. 0.882

21. 0.84

22. 0.384

23. 0.63

24. 0.60

25. Prove that!

3 is irrational.

26. Prove that!

5 is irrational.

In Exercises 27-30, copy the given ta-ble onto your homework paper. In eachrow, place a check mark in each columnthat is appropriate. That is, if the num-ber at the start of the row is rational,place a check mark in the rational col-umn. Note: Most (but not all) rows willhave more than one check mark.

27.

N W Z Q R0"2"2/30.150.2!

5

12 Chapter 1 Preliminaries

Version: Fall 2007

28.

N W Z Q R10/2!

"60.9!

20.37

29.

N W Z Q R"4/3

120!

111.36/2

30.

N W Z Q R"3/5!

101.62510/20/511

In Exercises 31-42, consider the givenstatement and determine whether it istrue or false. Write a sentence explainingyour answer. In particular, if the state-ment is false, try to give an example thatcontradicts the statement.

31. All natural numbers are whole num-bers.

32. All whole numbers are rational num-bers.

33. All rational numbers are integers.

34. All rational numbers are whole num-bers.

35. Some natural numbers are irrational.

36. Some whole numbers are irrational.

37. Some real numbers are irrational.

38. All integers are real numbers.

39. All integers are rational numbers.

40. No rational numbers are natural num-bers.

41. No real numbers are integers.

42. All whole numbers are natural num-bers.

Section 1.1 Number Systems 13

Version: Fall 2007

1.1 Answers

1. 2 · 2 · 2 · 2 · 5

3. 2 · 2 · 3 · 3 · 5

5. 2 · 2 · 2 · 2 · 2 · 2 · 2

7. 2 · 2 · 2 · 2 · 2

9. 81125

11. 625

13. 750

15. 111125

17. 311

19. 833

21. 2833

23. 711

25. Suppose that!

3 is rational. Thenit can be expressed as the ratio of twointegers p and q as follows:

!3 = pq

Square both sides,

3 = p2

q2,

then clear the equation of fractions bymultiplying both sides by q2:

p2 = 3q2 (1)

Now p and q each have their own uniqueprime factorizations. Both p2 and q2 havean even number of factors in their primefactorizations. But this contradicts equa-tion (1), because the left side would havean even number of factors in its primefactorization, while the right side wouldhave an odd number of factors in its primefactorization (there’s one extra 3 on theright side).Therefore, our assumption that

!3 was

rational is false. Thus,!

3 is irrational.

27.

N W Z Q R0 x x x x"2 x x x"2/3 x x0.15 x x0.2 x x!

5 x

29.

N W Z Q R"4/3 x x

12 x x x x x0 x x x x!

11 x1.3 x x6/2 x x x x x

31. True. The only di!erence betweenthe two sets is that the set of whole num-bers contains the number 0.


Version: Fall 2007

33. False. For example, 12 is not an in-

teger.

35. False. All natural numbers are ra-tional, and therefore not irrational.

37. True. For example, ! and!

2 arereal numbers which are irrational.

39. True. Every integer b can be writ-ten as a fraction b/1.

41. False. For example, 2 is a real num-ber that is also an integer.

Section 1.2 Solving Equations 27

Version: Fall 2007

1.2 Exercises

In Exercises 1-12, solve each of the givenequations for x.

1. 45x+ 12 = 0

2. 76x! 55 = 0

3. x! 7 = !6x+ 4

4. !26x+ 84 = 48

5. 37x+ 39 = 0

6. !48x+ 95 = 0

7. 74x! 6 = 91

8. !7x+ 4 = !6

9. !88x+ 13 = !21

10. !14x! 81 = 0

11. 19x+ 35 = 10

12. !2x+ 3 = !5x! 2

In Exercises 13-24, solve each of thegiven equations for x.

13. 6! 3(x+ 1) = !4(x+ 6) + 2

14. (8x+ 3)! (2x+ 6) = !5x+ 8

15. !7 ! (5x! 3) = 4(7x+ 2)

16. !3! 4(x+ 1) = 2(x+ 4) + 8

17. 9! (6x! 8) = !8(6x! 8)

18. !9! (7x! 9) = !2(!3x+ 1)

19. (3x! 1)! (7x! 9) = !2x! 6


20. !8! 8(x! 3) = 5(x+ 9) + 7

21. (7x! 9)! (9x+ 4) = !3x+ 2

22. (!4x! 6) + (!9x+ 5) = 0

23. !5! (9x+ 4) = 8(!7x! 7)

24. (8x! 3) + (!3x+ 9) = !4x! 7

In Exercises 25-36, solve each of thegiven equations for x. Check your solu-tions using your calculator.

25. !3.7x! 1 = 8.2x! 5

26. 8.48x! 2.6 = !7.17x! 7.1

27. !23x+ 8 = 4

5x+ 4

28. !8.4x = !4.8x+ 2

29. !32x+ 9 = 1

4x+ 7

30. 2.9x! 4 = 0.3x! 8

31. 5.45x+ 4.4 = 1.12x+ 1.6

32. !14x+ 5 = !4

5x! 4

33. !32x! 8 = 2

5x! 2

34. !43x! 8 = !1

4x+ 5

35. !4.34x! 5.3 = 5.45x! 8.1

36. 23x! 3 = !1

4x! 1


Version: Fall 2007

In Exercises 37-50, solve each of thegiven equations for the indicated vari-able.

37. P = IRT for R

38. d = vt for t

39. v = v0 + at for a

40. x = v0 + vt for v

41. Ax+By = C for y

42. y = mx+ b for x

43. A = !r2 for !

44. S = 2!r2 + 2!rh for h

45. F = kqq0r2

for k

46. C = Q

mTfor T

47. Vt

= k for t

48. " = h

mvfor v

49. P1V1n1T1

= P2V2n2T2

for V2

50. ! = nRTVi for n

51. Tie a ball to a string and whirl itaround in a circle with constant speed.It is known that the acceleration of theball is directly toward the center of thecircle and given by the formula

a = v2

r, (1)

where a is acceleration, v is the speed ofthe ball, and r is the radius of the circle

of motion.

i. Solve formula (1) for r.ii. Given that the acceleration of the ball

is 12 m/s2 and the speed is 8 m/s, findthe radius of the circle of motion.

52. A particle moves along a line withconstant acceleration. It is known thevelocity of the particle, as a function ofthe amount of time that has passed, isgiven by the equation

v = v0 + at, (2)

where v is the velocity at time t, v0 is theinitial velocity of the particle (at timet = 0), and a is the acceleration of theparticle.

i. Solve formula (2) for t.ii. You know that the current velocity

of the particle is 120 m/s. You alsoknow that the initial velocity was 40 m/sand the acceleration has been a con-stant a = 2 m/s2. How long did ittake the particle to reach its currentvelocity?

53. Like Newton’s Universal Law of Grav-itation, the force of attraction (repulsion)between two unlike (like) charged parti-cles is proportional to the product of thecharges and inversely proportional to thedistance between them.

F = kCq1q2r2

(3)

In this formula, kC " 8.988#109 Nm2/C2

and is called the electrostatic constant.The variables q1 and q2 represent the charges(in Coulombs) on the particles (whichcould either be positive or negative num-bers) and r represents the distance (inmeters) between the charges. Finally, Frepresents the force of the charge, mea-sured in Newtons.

Section 1.2 Solving Equations 29

Version: Fall 2007

i. Solve formula (3) for r.ii. Given a force F = 2.0 # 1012 N, two

equal charges q1 = q2 = 1 C, find theapproximate distance between the twocharged particles.


Version: Fall 2007

1.2 Answers

1. ! 415

3. 117

5. !3937

7. 9774

9. 1744

11. !2519

13. !25

15. ! 411

17. 4742

19. 7

21. 15

23. !1

25. 40119

27. 3011

29. 87

31. !280433

33. !6019

35. 280979

37. R = PIT

39. a = v ! v0t

41. y = C !AxB

43. ! = Ar2

45. k = Fr2

qq0

47. t = Vk

49. V2 = n2P1V1T2n1P2T1

51. r = v2/a, r = 16/3 meters.

53. r " 0.067 meters.

Section 1.3 Logic 47

Version: Fall 2007

1.3 Exercises

Perform each of the following tasks inExercises 1-4.

i. Write out in words the meaning ofthe symbols which are written in set-builder notation.

ii. Write some of the elements of this set.iii. Draw a real line and plot some of the

points that are in this set.

1. A = {x ! N : x > 10}

2. B = {x ! N : x " 10}

3. C = {x ! Z : x # 2}

4. D = {x ! Z : x > $3}

In Exercises 5-8, use the sets A, B, C,andD that were defined in Exercises 1-4. Describe the following sets using setnotation, and draw the corresponding VennDiagram.

5. A %B

6. A &B

7. A & C.

8. C %D.

In Exercises 9-16, use both interval andset notation to describe the interval shownon the graph.

9.

3


10.

0

11.

$7

12.

1

13.

0

14.

1

15.

$8

16.

9

In Exercises 17-24, sketch the graph ofthe given interval.

17. [2, 5)

18. ($3, 1]

19. [1,')


Version: Fall 2007

20. ($', 2)

21. {x : $4 < x < 1}

22. {x : 1 # x # 5}

23. {x : x < $2}

24. {x : x " $1}

In Exercises 25-32, use both intervaland set notation to describe the inter-section of the two intervals shown on thegraph. Also, sketch the graph of the in-tersection on the real number line.

25.

1$3

26.

$6$3

27.

2

$4

28.

118

29.

$62

30.

15

31.

95

32.

$14

$6

In Exercises 33-40, use both intervaland set notation to describe the unionof the two intervals shown on the graph.Also, sketch the graph of the union onthe real number line.

33.

$10$8

34.

$3$2

35.

159


Version: Fall 2007

36.

514

37.

$53

38.

119

39.

109

40.

7$2

In Exercises 41-56, use interval nota-tion to describe the given set. Also, sketchthe graph of the set on the real numberline.

41. {x : x " $6 and x > $5}

42. {x : x # 6 and x " 4}

43. {x : x " $1 or x < 3}

44. {x : x > $7 and x > $4}

45. {x : x " $1 or x > 6}

46. {x : x " 7 or x < $2}

47. {x : x " 6 or x > $3}

48. {x : x # 1 or x > 0}

49. {x : x < 2 and x < $7}

50. {x : x # $3 and x < $5}

51. {x : x # $3 or x " 4}

52. {x : x < 11 or x # 8}

53. {x : x " 5 and x # 1}

54. {x : x < 5 or x < 10}

55. {x : x # 5 and x " $1}

56. {x : x > $3 and x < $6}


Version: Fall 2007

1.3 Answers

1.

i. A is the set of all x in the naturalnumbers such that x is greater than10.

ii. A = {11, 12, 13, 14, . . .}iii.

11 17

3.

i. C is the set of all x in the set of inte-gers such that x is less than or equalto 2.

ii. C = {. . . ,$4,$3,$2,$1, 0, 1, 2}iii.

$4 2

5. A % B = {x ! N : x > 10} ={11, 12, 13, . . .}

A

B

7. A & C = {x ! Z : x # 2 or x >10} = {. . . ,$3,$2$1, 0, 1, 2, 11, 12, 13 . . .}

A C

9. [3,') = {x : x " 3}

11. ($',$7) = {x : x < $7}

13. (0,') = {x : x > 0}

15. ($8,') = {x : x > $8}

17.

2 5

19.

1

21.

$4 1

23.

$2

25. [1,') = {x : x " 1}

1

27. no intersection

29. [$6, 2] = {x : $6 # x # 2}

$6 2

31. [9,') = {x : x " 9}

9


Version: Fall 2007

33. ($',$8] = {x : x # $8}

$8

35. ($', 9] & (15,')={x : x # 9 or x > 15}

159

37. ($', 3) = {x : x < 3}

3

39. [9,') = {x : x " 9}

9

41. ($5,')

$5

43. ($',')

45. [$1,')

$1

47. ($3,')

$3

49. ($',$7)

$7

51. ($',$3] & [4,')

4$3

53. the set is empty

55. [$1, 5]

$1 5

Section 1.4 Compound Inequalities 63

Version: Fall 2007

1.4 Exercises

In Exercises 1-12, solve the inequality.Express your answer in both interval andset notations, and shade the solution ona number line.

1. !8x! 3 " !16x! 1

2. 6x! 6 > 3x+ 3

3. !12x+ 5 " !3x! 4

4. 7x+ 3 " !2x! 8

5. !11x! 9 < !3x+ 1

6. 4x! 8 # !4x! 5

7. 4x! 5 > 5x! 7

8. !14x+ 4 > !6x+ 8

9. 2x! 1 > 7x+ 2

10. !3x! 2 > !4x! 9

11. !3x+ 3 < !11x! 3

12. 6x+ 3 < 8x+ 8

In Exercises 13-50, solve the compoundinequality. Express your answer in bothinterval and set notations, and shade thesolution on a number line.

13. 2x! 1 < 4 or 7x+ 1 # !4

14. !8x+ 9 < !3 and ! 7x+ 1 > 3

15. !6x!4 < !4 and !3x+7 # !5

16. !3x+ 3 " 8 and ! 3x! 6 > !6


17. 8x+ 5 " !1 and 4x! 2 > !1

18. !x! 1 < 7 and ! 6x! 9 # 8

19. !3x+ 8 " !5 or ! 2x! 4 # !3

20. !6x! 7 < !3 and ! 8x # 3

21. 9x! 9 " 9 and 5x > !1

22. !7x+ 3 < !3 or ! 8x # 2

23. 3x! 5 < 4 and ! x+ 9 > 3

24. !8x! 6 < 5 or 4x! 1 # 3

25. 9x+ 3 " !5 or ! 2x! 4 # 9

26. !7x+ 6 < !4 or ! 7x! 5 > 7

27. 4x! 2 " 2 or 3x! 9 # 3

28. !5x+ 5 < !4 or ! 5x! 5 # !5

29. 5x+ 1 < !6 and 3x+ 9 > !4

30. 7x+ 2 < !5 or 6x! 9 # !7

31. !7x! 7 < !2 and 3x # 3

32. 4x+ 1 < 0 or 8x+ 6 > 9

33. 7x+ 8 < !3 and 8x+ 3 # !9

34. 3x < 2 and ! 7x! 8 # 3

35. !5x+ 2 " !2 and ! 6x+ 2 # 3

36. 4x! 1 " 8 or 3x! 9 > 0

37. 2x! 5 " 1 and 4x+ 7 > 7

38. 3x+ 1 < 0 or 5x+ 5 > !8


Version: Fall 2007

39. !8x+ 7 " 9 or ! 5x+ 6 > !2

40. x! 6 " !5 and 6x! 2 > !3

41. !4x! 8 < 4 or ! 4x+ 2 > 3

42. 9x! 5 < 2 or ! 8x! 5 # !6

43. !9x! 5 " !3 or x+ 1 > 3

44. !5x! 3 " 6 and 2x! 1 # 6

45. !1 " !7x! 3 " 2

46. 0 < 5x! 5 < 9

47. 5 < 9x! 3 " 6

48. !6 < 7x+ 3 " 2

49. !2 < !7x+ 6 < 6

50. !9 < !2x+ 5 " 1

In Exercises 51-62, solve the given in-equality for x. Graph the solution set ona number line, then use interval and set-builder notation to describe the solutionset.

51. !13 <x

2 + 14 <

13

52. !15 <x

2 !14 <

15

53. !12 <

13 !x

2 <12

54. !23 "

12 !x

5 "23

55. !1 < x! x+ 15 < 2

56. !2 < x! 2x! 13 < 4

57. !2 < x+ 12 ! x+ 1

3 " 2

58. !3 < x! 13 ! 2x! 1

5 " 2

59. x < 4! x < 5

60. !x < 2x+ 3 " 7

61. !x < x+ 5 " 11

62. !2x < 3! x " 8

63. Aeron has arranged for a demon-stration of “How to make a Comet” byProfessor O’Commel. The wise profes-sor has asked Aeron to make sure theauditorium stays between 15 and 20 de-grees Celsius (C). Aeron knows the ther-mostat is in Fahrenheit (F) and he alsoknows that the conversion formula be-tween the two temperature scales is C =(5/9)(F ! 32).

a) Setting up the compound inequalityfor the requested temperature rangein Celsius, we get 15 " C " 20. Us-ing the conversion formula above, setup the corresponding compound in-equality in Fahrenheit.

b) Solve the compound inequality in part(a) for F. Write your answer in setnotation.

c) What are the possible temperatures(integers only) that Aeron can set thethermostat to in Fahrenheit?

Section 1.4 Compound Inequalities 65

Version: Fall 2007

1.4 Answers

1. (!$, 14 ] = {x|x " 1

4}

14

3. [1,$) = {x|x # 1}

1

5. (!54 ,$) = {x|x > !5

4}

!54

7. (!$, 2) = {x|x < 2}

2

9. (!$,!35) = {x|x < !3

5}

!35

11. (!$,!34) = {x|x < !3

4}

!34

13. (!$,$) = {all real numbers}

15. (0, 4] = {x|0 < x " 4}

0 4

17. no solution

19.!!$,!1

2"# $13

3 ,$%

={x|x " !12 or x # 13

3 }

!12

133

21. (!15 , 2] = {x|! 1

5 < x " 2}

!15

2

23. (!$, 3) = {x|x < 3}

3

25. (!$,!89 ] = {x|x " !8

9}

!89

27. (!$, 1]#

[4,$) = {x|x " 1 or x #4}

1 4

29. (!133 ,!

75) = {x|! 13

3 < x < !75}

!133 !7

5

31. [1,$) = {x|x # 1}

1

33. no solution


Version: Fall 2007

35. no solution

37. (0, 3] = {x|0 < x " 3}

0 3



43. [!29 ,$) = {x|x # !2

9}

!29

45. [!57 ,!

27 ] = {x|! 5

7 " x " !27}

!57 !2

7

47. (89 , 1] = {x|89 < x " 1}

89

1

49. (0, 87 ) = {x|0 < x < 8

7}

0 87

51. (!7/6, 1/6) = {x : !7/6 < x <1/6}

!7/6 1/6

53. (!1/3, 5/3) = {x : !1/3 < x <5/3}

!1/3 5/3

55. (!1, 11/4) = {x : !1 < x < 11/4}

!1 11/4

57. (!13, 11] = {x : !13 < x " 11}

!13 11

59. (!1, 2) = {x : !1 < x < 2}

!1 2

61. (!5/2, 6] = {x : !5/2 < x " 6}

!5/2 6

63.

a) 15 " 59(F ! 32) " 20

b) {F : 59 " F " 68}

c) {59, 60, 61, 62, 63, 64, 65, 66, 67, 68}

Documents

1.1 Exercis es - College of the Redwoodsmsenux2.redwoods.edu/IntAlgText/chapter1/chapter1exercises.pdf · 1 Cop yri gh ted materi al. See: h ttp: //msen ux. redw o o ds. edu/In tAl