﻿ ﻿Check Point Answers - Idiot's Guides - Algebra I

## Idiot's Guides: Algebra I (2015)

### APPENDIX B. Check Point Answers

Chapter 1

1. is b) a rational number and d) a real number.

2. so it is a) an integer, b) a rational number, and d) a real number.

3. is c) an irrational number and d) a real number.

4. so it is b) a rational number and d) a real number.

5. distributive property

8. closure

9. identity (for multiplication)

10. associative property (for multiplication)

11. 4 + − 17 + − 3 + 29 = 4 + − 20 + 29 = 4 + 9 = 13

12. −7·5·−8·12 = −7·−8·5·12 = 56·60 = 3,360

13. 8·−6·3·−4 = 8·3·−6·−4 = 24·24 = 576

14. − 14 + 8 + − 31 + 27 + 6 + − 4 = − 8

15. − 25 (18 + − 14) + 8(− 9 + 12) = − 25 (4) + 8(3) = − 100 + 24 = − 76

16. 7(− 39 + 43) + 14·− 2 = 7(4) + 14·− 2 = 28 + − 28 = 0

17.

18. 4·(− 0.6)·(l.2)·(− 10)2 =− 288

19. −1.4 + 8.3 + − 3.1 + 2.7 + − 4.5 = 2

20.

21. (23)·(22) = 25 = 32

22.

23. (32)3 = 36 = 729

24.

25.

26. 6−3·65 = 62 = 36

27.

28.

29.

30.

31. 4 − 3·4 + 3 = 4 − 12 + 3 = − 8 + 3 = − 5

32. (4 − 3)·(4 + 3) = 1·7 = 7

33. 4 − 3·(4 + 3) = 4− 3·7 = 4 − 21 = − 17

34. (4 − 3)·4 + 3 = 1·4 + 3 = 4 + 3 = 7

35. 52 − 2·(7 − 3) = 25 − 8 = 17

36. (66 − 54) ÷ 3 + 10 ÷ 5 − (6 − 22) = 4 + 2 − 2 = 4

37. 5 − 2 + 2·(8 − 5) ÷ 2 = 3 + 3 = 6

38. 17 − (8 − 3)·(2 + 1) ÷ 5 = 14

39. 8(2(4 + 3)2 − 20 + 12) ÷ 4 = 720 ÷ 4 = 180

40. 7 + 32 ÷ 2 − 6 = 5.5

Chapter 2

1. −4x

2. y + 18

3.

4. 9x − 3

5. 19 + (− 3)x = 19 − 3x

6. (ab)· − 5 = −5(ab)

7. 19 − x

8. 2x + 4 = 8 + y

9. 3h + 1.50 = 12

10. n + (n + 15) = 42

11. 2 factors: 3 and x

12. 3 factors: -6, y and y

13. 3 factors: 23, x and y

14. 6 factors: -48, x, x, x, y, and y

15. 3 terms

16. 2 terms

17. 1 term

18. 2 terms

19. 3 terms

20. 5 terms

21. (x + 3) + (9 + x) = 2x + 12

22. (2x − 1) + (3x + 9) = 5x + 8

23. (x2 + 5) + (x2 + 6x) + (8x + 3) = 2x2 + 14x + 8

24. (2x + 7) + (8x + 4) + (− x2 − 1) = −x2 + 10x + 10

25. (2x − 4) + (6x + 1) + (3 − x) = 7x

26. (y + 5) + (2y − 9) + (4y + 10) + (y − 6) = 8y

27. (8t − 4) + (− t − 1) + (13t − 7) + (− 2t − 9) = 18t − 21

28. (x2 + 4) + (− x2 − 1) + (−3 − 5x) + (10x + 9) = 5x + 9

29. (3x2 + 2x + 4) + (x2 − 3x + 10) = 4x2x + 14

30. (4x2 − 5x − 6) + (9 + 7x − 4x2) = 2x + 3

31. 2(x − 4) + 7(x + 4) = 2x − 8 + 7x + 28 = 9x + 20

32. −3(y + 8) + 8(y − 3) = −3y − 24 + 8y − 24 = 5y − 48

33. 6(2t + 5) + 9(t − 2) = 12t + 30 + 9t − 18 = 21t + 12

34. −7(3w − 5) + 4(w + 5) = − 21w + 35 + 4w + 20 = − 17w + 55

35. x(x + 4) − (x + 4) = x2 + 4xx − 4 = x2 + 3x − 4

36. −8x(x − 3) + 6(x − 4) = −8x2 + 24x + 6x − 24 = −8x2 + 30x − 24

37. 5a(2a − 7) − 2(8 − a) = 10a2 − 35a − 16 + 2a = 10a2 − 33a − 16

38. −4y (7 − 3y) − 4(3 − 7y) = −28y + 12y2 − 12 + 28y = 12y2 − 12

39. 3x (2x + 3y) + 3y (2x + 3y) = 6x2 + 9xy + 6xy + 9y2 = 6x2 + 15xy + 9y2

40. 2x2 (x2 − 7x + 9) + 3x (x2 − 7x + 9) + 2 (x2 − 7x + 9) = = 2x4 − 11x3x2 + 13x + 18

41. 2x2 + x − 28

42. x3 − 2x2x − 6

43. 2x3 + 15x2 + 13x + 3

44. 6x3 − 17x2 − 12x + 32

45. (x + 3)·(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6

46. (x + 4)·(x − 2) = x2 − 2x + 4x − 8 = x2 + 2x − 8

47. (2x + 1)·(x + 7) = 2 x2 + 14x + x + 7 = 2x2 + 15x + 7

48. (4x − 1)·(2x + 3) = 8x2 + 12x − 2x − 3 = 8x2 + 10x − 3

49. (5x − 2)·(2x − 3) = 10x2 − 15x − 4x + 6 = 10x2 − 19x + 6

50. (x2 + x)·(x − 3) = x3 − 3x2 + x2 − 3x = x3 − 2x2 − 3x

Chapter 3

1. Function. No rule.

2. Function. It’s hard to see but this does follow a rule. Multiply the input by -3 and then add 4.

3. Function. Output = Input squared, minus 1.

4. Not a function. No rule.

5. Function. The outputs are the positive square roots of the inputs.

6. Domain: {−2, −1,0,1,2}, Range: {3,4,5,6,7}, Rule: Output = input + 5

7. Domain: {−2, −1,0,1,2}, Range: {4}, Rule: Output always equals 4, no matter what the input.

8. Domain: {0,2,4,6,8}, Range: {−3,1,5,9,13}, Rule: Output = twice the input minus 3.

9. Domain: {10,20,30,40,50}, Range:
{1,11,21,31,41}, Rule: Output = input minus 9.

10. Domain: {−5,6,8,11,21}, Range:
{−4,−1,−7,12,−5}, Rule: None

11. f (3) = 4 − 3·3 = 4 − 9 = − 5

12. f (−5) = 4 − 3(−5) = 4 + 15 = 19

13. f (0) = 4 − 3(0) = 4 − 0 = 4

14. f (−1) = 4 − 3 (−1) = 4 + 3 = 7

15. f (−100) = 4 − 3 (−100) = 4 + 300 = 304

16. If g(x) = 8x − 2 find
g (−4) = 8 (−4) − 2 = − 32 − 2 = − 34

17. If h(x) = x2 + 5 find h(0) = 02 + 5 = 0 + 5 = 5

18. If (x) = x2 − 3x + 7 find
p(−1) = (−1)2 − 3(−1) + 7 = 1 + 3 + 7 = 11

19. If v(t) = 25 − 8t find
v(5) = 25 − 8·5 = 25 − 40 = − 15

20. If a(t) = 9.8 find a(20) = 9.8 (Constant function)

21.

22.

23.

24.

25.

26. v(t) = 5t − 3

27. f (x) = − 3x + 7

28. g(x) = −x2 + 5

29. a(t) = 1.5t + 2

30. p(x) = 3x + 10

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41. function

42. not a function

43. function

44. not a function

45. function

46. not a function

47. function

48. function

49. function

50. not a function

Chapter 4

1. 3x + 3 = 132

2. 3y + 3 = y + 3

3. 2t − 10 = 5 + t

4. 2x − 8 = 2x − 1

5. −3a + 23 = 6a − 14

6. 2x + 3 = 5x + 3

7. 4v − 12 = − 2v + 4

8. 4x − 17 = 9x + 23

9. 6t − 15 = 6t − 15

10. −3a − 19 = 4 − 3a

11. x = 7

12. t = 20

13. y = 27

14. 48 = t

15. a = −25

16. w = 11

17. x = −9

18. t = 349

19. y = −120

20. z = 0

21. a = 7

22. z = −4

23. n = −7

24. y = 1

25. t = −5

26. x = 80

27. z = 2.5

28. w = −4

29. x = 3

30. y = −217

31. − 4 = x

32. 7 = t

33. y = 12

34. v = 45

35. 7 = x

36. no solution

37. 53 = x

38. identity

39. z = 0

40. −11.5 = x

41. The number is 7.

42. The number is 4.

43. The two consecutive numbers are 26 and 27.

44. 5,000 storage containers were manufactured.

45. The garden measures 6 feet by 9 feet.

46. The price of a hot dog is \$3.50.

47. Carlos and Dahlia will have the same amount in savings in 10 weeks.

48. The average price per e-book was \$7.89.

49. Sophie bought 6 pounds of almonds.

50. The number is 8.

Chapter 5

1. x > 5

2. r ≤ 4% or r ≤ 0.04

3. h > 48

4. p ≥ 20

5. I ≤ 12,000

Sample answers for problems 6 through 10.

6. A number, x, is at least 16.

7. A number, y, is less than 12.

8. A number, t, exceeds 22.

9. A number, z, is no more than 100.

10. A number, a, is more than 0.

11. x > − 3

12. y < 6

13. x ≥ 4

14. t > − 4.5

15. y ≤ 1.5

16. x = − 4 is part of the solution

17. x = − 6 is not part of the solution

18. y = 1 is not part of the solution

19. x = 5 is not part of the solution

20. t = 10 is part of the solution

21. x > 5

22. t ≥ −8

23. y < − 4

24. z ≥ 7

25. a ≤ 9

26. x ≤ 4

27. −2 > x

28. x ≤ − 4

29. y > − 1

30. t ≤ 0

31. y > 3 or y < − 2

32. 1 ≤ z ≤ 3

33. 18 ≤ x or x < 7

34. 3 < t < 4

35. 17 < 4y + 5 ≤ 1 has no solution.

36. −8 ≤ t or t > − 5 is equivalent to t ≤ −8

37. −4 ≤ x ≤ 6

38. 5 < x or x < 3.5

39. −4 > t ≥ −5

40. 2 ≤ y < 4

Chapter 6

Answers for problems 1 through 5 are shown on the graph.

6. (4, 1) or (8,9)

7. (0, 4) or (0, -3)

8. (3, -1) or (7, -5)

9. (-6, -3) or (-8, -9)

10. (-5, 0) or (2, 0)

Sample tables are shown for problems 11 through 20.

11. f (x) = 3x − 4

12.

13. y = 2x − 3

14. g(x) = x + 5

15. y = −x + 5

16. f (x) = −2x + 7

17.

18. y = 4 − 3x

19. f (x = −2x)

20.

21. x-intercept: (3, 0), y-intercept: (0, -6)

22. x-intercept: (2, 0), y-intercept: (0, 6)

23. x-intercept: (-1, 0), y-intercept: (0, 4)

24. x-intercept: (5, 0), y-intercept: (0, 5)

25. x-intercept: (-4, 0), y-intercept: (0, 4)

26. x-intercept: (4, 0), y-intercept: (0, 8)

27. x-intercept: (2, 0), y-intercept: (0, -6)

28. x-intercept: (6, 0), y-intercept: (0, -6)

29. x-intercept: (-5, 0), y-intercept: (0, 5)

30. x-intercept: (8, 0), y-intercept: (0, -4)

31. Slope: , y-intercept: (0, -5)

32. Slope: , y-intercept: (0, -7)

33. Slope: , y-intercept: (0, 9)

34. Slope: , y-intercept: (0, -3)

35. Slope: , y-intercept: (0, -4)

36. Slope: , y-intercept: (0, 7)

37. Slope: , y-intercept: (0, 5)

38. Slope: , y-intercept: (0, -9)

39. Slope: , y-intercept: (0, 3)

40. Slope: , y-intercept: (0, 1)

41. direct variation

42. No, direct variation doesn’t have constant term.

43. No, xy = 12 is equivalent to . Direct variation cannot have a variable in the denominator.

44. direct variation

45. No, the graph of a direct variation equation passes through the origin.

46. k = 45

47. k = 2π

48. The equation is F = 50a.

49. Two pounds of potatoes cost \$2.98.

50. At 30 degrees Celsius, the volume will be 1,800 cubic centimeters.

Chapter 7

1.

2.

3.

4. y = 3x − 9

5. y = − 4x + 2

6.

7. y = −6x + 3

8.

9.

10. y = 10x + 10

11. 6xy = 7

12. 3xy = − 13

13. 3x + y = 5

14. 4x + y = 11

15. 8x − 4y = 32

16. x − 2y = 2

17. 4x + 5 y = 3

18. 6x + 2y = − 19

19. 3x − 2y = 7

20. 3x + 5y = 28

21. y − 7 = −2(x + 5) point-slope form

22. 3x − 4y = 12 standard form

23. y = 2x − 6 slope-intercept form

24. x + y = 10 standard form

25. point-slope form

26. y = − 3x + 0 or simply y = −3x

27. y = 5x − 9

28.

29.

30. y = 1x + 3 or simply y = x + 3

31. y = −4x + 3

32. y = 5x + 16

33.

34.

35. y = − x + 5

36. y = 2x − 3

37.

38. y = x + 1

39.

40. y = 3x − 23

41.

42.

43.

44.

45.

46. , or y + 10 = − 3 (x − 5)
or y − 20 = − 3 (x + 5)

47. , or

48. , y + 5 = −4(x − 4) or y − 7 = − 4 (x − 1)

49. , or

50.

51. Use 2. y + 3 = 2 (x − 6) or y = 2x − 15.

52. . Use . .

53. Use m = 4. y + 3 = 4(x + 4) or y = 4x + 13.

54. Use . or .

55. Use .

56. Use . or .

57. Use .

58. . Use m = -2. y − 7 = −2(x + 1) or y = −2x + 5.

59. Use m = -3. or
.

60. Use m = -1. y + 3 = −1(x − 4) or y = −x + 1.

Chapter 8

1. |9 − n| = 6 or |n − 9 = 6

2. |t − (−4)| = 10 or |−4 − t| = 10

3. |x − 5| = 12 or |5 − x| = 12

4. 2|y − (− 4)| = 12 or 2| − 4 − y| = 12

5. |z − 12| = z + 4 or |z − 12| = z + 4

Sample sentences are given for problems 6 through 10.

6. The number 7 is 3 units from x.

7. The distance between t and -2 is 9 units.

8. The number y is 11 units from 8.

9. Five times the distance between x and 4 is 20.

10. The distance between z and 7 is 3 less than z.

11. x = 3 or

12. t = − 1 or t = −4

13. y = 9 or y = 7

14. x = 8 or x = 18

15. |11 + 8x| = − 17 has no solution. Absolute value is never negative.

16. or x = −2

17. y = 3 or y = 1

18. x = 2 or x = −4

19. or x = 2

20. or 1 = x

21. or x > 2

22.

23. −2.8 ≤ y ≤ 2

24. or z ≥ 2

25. −2.75 ≤ a ≤ 2

26. 6.5 > x > − 4

27. −1.75 ≥ p or p ≥ − 1

28. |9t + 5| ≤ −2 has no solution. Absolute value is never negative.

29. −21 < x < 11

30. |7y − 23| > 0 is true for all values of the variable. The solution set is all real numbers.

31. x = 36 or x = − 8

32. y = − 41 or y = 7

33. The number n is between 6 and 12, inclusive.

34. Paul’s salary is between \$86,500 and \$78,500.

35. The points are (0, -10) or (0, 2).

36. The possible points are (-25, 0) or (1, 0).

37. The points are (0, -7) or (0, 13).

38. 96.6 > t or t > 100.6

39. The range of acceptable voltages is 100 < V < 120.

40. The candidate should receive between 42% and 50% of the vote.

41. y = |x − 3|

42. y = |x| + 1

43. y = |x + 1|

44. y = |x| − 5

45. y = |2x|

46. y = −3|x|

47. y = |2x − 5|

48. y = |7 − 4x|

49. y = |3x − 8|

50. y = |12 − 5x|

Chapter 9

1. y = 12 Horizontal line

2. x = 5y Oblique line

3. y = 12 + x Oblique line

4. x = 7 Vertical line

5. x − 6 = 0 Vertical line

6. x = − 4

7. y = − 3

8. x + 4 = 6

9. y + 7 = 4

10. x = 1

11. y ≤ 2x − 1

12.

13. y < −3x + 5

14.

15. x < 3

16. y − 5 ≥ 0

17. x + y ≤ 8

18. 2xy > − 6

19. 4x − 3y < 12

20. x ≤ 2y + 8

21. y < |x − 7|

22. y ≥ |x + 3|

23. y > |x| − 4

24. y ≤ |x| + 1

25.

26. y ≥ |2x| − 3

27.

28.

29. y ≤ −2|x| + 5

30. y > −3|x + 4| + 5

Chapter 10

1. y = 2x + 3
y = 3x − 1

2.

3. x + y = 9
xy = 1

4. 3xy = 5
6x − 2y = 8

5. 2x − 5y = −44
y = 6 − x

6. y = 3x − 7
2x − 3y = 14

7.

8. 2x + y = −8
3x − 5y = 14

9. x + y = 6
y= − x − 4

10. 3x − 2y = 5
x + 4 y = −3

11. x = 1, y = 10

12. , y = −2

13. x = 3, y = −3

14. x = −2, y = −5

15. a = 1, b = 6

16. x = 0, y = 9

17. x = 5, y = 2

18. x = 7, y = 3

19. x = −1, y = 2

20. a = −1, b = 2

21. x = 10, y = 7

22. x = 7, y = 1

23. x = 2, y = 3

24. x = 3, y = 0

25. x = 3, y = −3

26. x = 4, y = −3

27. x = 7, y = 3

28. x = 5, y = 2

29. x = 3, y = 4

30. x = −1, y = 2

31. There are 7 quarters and 13 dimes.

32. There were 18 of the 14 cent stamps and 12 of the 17 cent stamps.

33. Use 20 pounds of \$6 coffee and 40 pounds of \$5.10 coffee.

34. Invest \$2,600 at 4% and \$3,400 at 7%.

35. Use 16 ml of 5% solution and 32 ml of 20% solution.

36. She traveled 6.4 hours at 50 mph and 1.6 hours at 20 mph.

38. There were 7 nickels and 13 dimes.

39. Pens cost 80 cents and pencils cost 30 cents.

40. There were 110 adults and 140 children.

41. y ≤ 2x + 1
y ≥ 4 − x

42. y < 7 − 3x
2x + y > 8

43. 3xy < 1
y ≥ 2 − 3x

44. 5x + 2y ≥ 12
3x + 7 < y

45. 3x + 4 ≥ 4y
x + 2y < 12

46. x + y ≥ 40
xy ≤ 8

47. xy ≤ 12
x + y ≥ 20

48. x + y ≤ 80
100x + 300y ≥ 7500

49. 15x + 20y ≤ 400
12x + 30y ≤ 300

50. 2x + 9y ≤ 25
3x + 4y ≤ 36

Chapter 11

1. is not a polynomial.
Division by a variable is not permitted, because they would be represented by negative exponents.

2. 3y7 − 4y3 + 8 is a polynomial.

3. is not a polynomial because there is a variable under a radical. Radicals are the equivalent of fractional exponents.

4.

5.

6.

7. is a binomial of degree 1.

8. is a trinomial of degree 2.

9. is a monomial of degree 3.

10. is a polynomial of degree 12.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

Chapter 12

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

Chapter 13

1.

2.

3.

4.

5.

6.

7.

8. is prime.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25. is prime.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

Chapter 14

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

Chapter 15

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16. No solution in the real numbers.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26. There are two rational solutions.

27. There are two irrational solutions.

28. There are no real solutions.

29. There is one solution.

30. There are two rational solutions.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

Chapter 16

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36. Vertex (2, 3)

37. Vertex (5, −3)

38. Vertex (1, −3)

39. Vertex (−4, 8)

40. Vertex (5,−4)

41. It will take 2.646 seconds to hit the ground.

42. feet high after one second.

43. The object is 76 feet high after 1.5 seconds.

44. After 2 seconds, the object is 261 feet high.

45. The maximum height the object reaches is 405 feet after 5 seconds.

46. There are two moments when the object is 81 feet off the ground, half a second after
it is thrown, and again on the way down, 9.5 seconds after it was thrown.

47. The object hits the ground just after 10 seconds.

48. After 0.5 seconds, the object is 66.25 meters from the ground.

49. The object is 20 meters high at approximately 2.14 seconds after it was released.

50. It takes approximately 2.65 seconds to hit the ground.

Chapter 17

1. All real numbers except 0.

2. All real numbers except 7.

3. All real numbers except -8.

4. All real numbers except 3.

5. All real numbers except 1 and -1.

6. All real numbers except -2.

7. All real numbers except -3.

8. All real numbers except 0 and -5.

9. All real numbers except -2 and -3.

10. All real numbers. (There is no value of x that can make the denominator equal 0.)

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

Chapter 18

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18. 1 = x
The solution of x = 1 must be rejected, so there is no solution.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

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