15.4: Chapter 5

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Your Turn

5.1

1. 18 plus 11 OR the sum of 18 and 11

2. 27 times 9 OR the product of 27 and 9

3. 84 divided by 7 OR the quotient of 84 and 7

4. $p$ minus $q$ OR the difference of $p$ and $q$

5.2

1. $43 + 67$

2. $\left( {45} \right)\left( 3 \right)$

3. $45 \div 3$

4. $89-42$

5.3

1. $n - 20$

2. $\left( {n + 2} \right) \bullet 6$ OR $6\left( {n + 2} \right)$

3. ${n^3} - 5$

4. $60h + 40$

5.4

1. $5y = 50$

2. $\frac{1}{2}n = 30$

3. $3n - 7 = 2$

4. $2x + 7 = 21$

5.5

1. $24 - \left( {17 - 6} \right) = 13$

2. $\left( {3 \bullet 6} \right) + 13 = 31$

3. $\left( {12 - 6} \right) \div \left( {5 - 3} \right) = 3$

4. $5 \bullet \left( {{3^2} + 5} \right) = 70$

5.6

1. $9$

2. $- 6$

5.7

1. $5{x^{ 2}}-3x - 7$

5.8

1. $- 9x + 4$

5.9

1. $2y + 10$

2. $- 2a - 2b + 8$

3. $112 + 16x$

4. $6x + 18$

5. $3a + 4$

5.10

$2{x^2} - 11x + 12$

5.11

$4{x^2} + x - 2$

5.12

1. $x = 3$

5.13

1. $y = 28$

5.14

1. Henry has 42 books.

5.15

1. Total cost is $48.85

5.16

1. Answers will vary. For example: You can rent a paddleboard for $25 per hour with a water shoe purchase of $75. If you spent $200, how many hours did you rent the paddle board for?
You rented the paddle board for 5 hours.

5.17

1. $12 = - 1$, which is false; therefore, this is a false statement, and the equation has no solution.

5.18

1. $- 12 = - 12$, which is true; therefore, this is a true statement, and there are infinitely many solutions.

5.19

1. $t = \frac{I}{{\text{P}}r}$

5.20

1. \(h = \frac

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5.21

1. $A number line ranges from 0 to negative 3, in increments of 1. A close parenthesis is marked at 2.5. The region to the left of the parenthesis is shaded on the number line. Text reads, (negative infinity, 2.5)$

5.22

1. $0 \leq x \leq 2.5$
Graph:
$A number line ranges from negative 1 to 3.5, in increments of 0.5. An open square bracket is marked at 0.0 and a close parenthesis is marked at 2.5. The region within the parentheses is shaded on the number line. Text reads, (0, 2.5)$

5.23

1. $A number line ranges from negative 7 to negative 4, in increments of 1. A close square bracket is marked at negative 5. The region to the left of the square bracket is shaded on the number line. Text reads, m is less than or equal to negative 5, (negative infinity, negative 5)$

5.24

1.
$p \geq 2$ $A number line ranges from 0 to 3, in increments of 1. An open square bracket is marked at 2. The region to the right of the square bracket is shaded on the number line. Text reads, (2, infinity).$

5.25

1. Taleisha can send/receive 106 or fewer text messages and keep her monthly bill no more than $50.

5.26

1. You could take up to 17 credit hours and stay under $2,000.

5.27

1. Malik must tutor at least 23 hours.

5.28

1. $a$ = 1 U.S. dollar, and $b$ = 1.21 Canadian dollars, the ratio is 1 to 1.21; or 1:1.21; or $\frac{1}1.21$.

5.29

1. With $a$ = 170 pounds on Earth, and $b$ = 64 pounds on Mars, the ratio is 170 to 64; or 170:64; or $\frac17064$.

5.30

1. $219.51

5.31

1. 501.6 pounds

5.32

1. 720 cookies

5.33

1. The constant of proportionality (centimeters divided by inches) is 2.54. This tells you that there are 2.54 centimeters in one inch.

5.34

1. 30.5 hours (or 30½ hours, or 30 hours and 30 minutes)

5.35

1. $207.50

5.36

1. 125.9 miles

5.37

1. The scale is $1{\text{ inch}} = 91.25{\text{ miles}}$. The other borders would calculate as: eastern and western borders are 273.75 miles, and northern border is 365 miles.

5.38

1. 184 inches, or 15 feet, 4 inches.

5.39

1. $Five points are marked on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first, second, third, and fourth quadrants are labeled. The points are plotted at the following coordinates: a (negative 4, 2), b (negative 1, negative 2), c (3, negative 5), d (negative 3, 0), and e (negative 1.5, 2). Note: all values are approximate.$

5.40

5.41

1. $A line is plotted on an x y coordinate plane. The x and y axes range from negative 12 to 12, in increments of 2. The line passes through the following points, (negative 2, negative 7), (0, negative 1), (2, 5), and (4, 11). Note: all values are approximate.$

5.42

1. $A line is plotted on a coordinate plane. The horizontal axis ranges from 0 to 100, in increments of 10. The vertical axis ranges from 0 to 50, in increments of 10. A stamp of the USA flag is at the top-left. The line passes through the points, (3, 1.65), (20, 11), and (100, 55).$

Your friend will pay $1.65.

5.43

1. Yes

2. Yes

3. No

4. No

5. No

5.44

1. $y \geq - 2x + 3$

5.45

1. $A dashed line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 2.5. The line passes through the following points, (negative 10, negative 7.5), (0, negative 1), (5, 2.5), and (7.5, 3.75). The region above the line is shaded. Note: all values are approximate.$

5.46

1. $11x + 16.5y \geq 330$

2. $A line is plotted on an x y coordinate plane. The x-axis ranges from 0 to 35, in increments of 5. The y-axis ranges from 0 to 35, in increments of 5. The line passes through the following points, (0, 20), (15, 10), and (30, 0). The region above the line is shaded.$

3. Answers will vary.

5.47

1. ${x^2} + 4x + 3$

5.48

1. $2{x^2} - 5x - 3$

5.49

1. $\left( x + 2 \right)\left( x + 4 \right)$

5.50

1. $\left( {x - 7} \right)\left( {x - 9} \right)$

5.51

1. $A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens down and it passes through the following points, (negative 2, negative 4), (negative 1, negative 1), (0, 0), (1, negative 1), and (2, negative 4). Note: all values are approximate.$

5.52

1. $x = - 3,\,\,x = 2$

5.53

1. $x = 2$ or $x = - 5$

5.54

1. $x = 3$ or $x = - 5$

5.55

1. $x = 5$, $x = -5$

5.56

1. $p = \frac{7}{5}$, $p = - \frac{7}{5}$

5.57

1. $a = 5$, $a = - 3$

5.58

1. $y = 1$, $y = \frac{2}{3}$

5.59

1. 16, 15 and –16, –15

5.60

1. ${\text{width}} = 5{\text{ feet}}$, ${\text{length}} = 6{\text{ feet}}$

5.61

1. 22

2. 6

3. $3{t^2} - 2t + 1$

5.62

1. There 205 unread emails after 7 days.

5.63

1. This relation is a function.

2. This relation is not a function.

5.64

1. Both George and Mike have two phone numbers. Each $x$-value is not matched with only one $y$-value. This relation is not a function.

5.65

1. function

2. not a function

3. function

5.66

1. This graph represents a function.

5.67

1. This graph does not represent a function.

5.68

1. The domain is the set of all $x$-values of the relation: $\left\{ {1,2,3,4,5} \right\}$.

2. The range is the set of all $y$-values of the relation: $\left\{ {1,8,27,64,125} \right\}$.

5.69

1. The ordered pairs of the relation are: $\left\{ {\left( {-3,\,3} \right),\left( {-2,\,2} \right),\left( {-1,\,0} \right),\left( {0,\,-1} \right),\left( {2,\,-2} \right),\left( {4,\,-4} \right)} \right\}$.

2. The domain is the set of all $x$-values of the relation: $\left\{ {-3,-2,-1,0,2,4} \right\}$.

3. The range is the set of all $y$-values of the relation: $\left\{{-4,-2,-1,0,2,3}\right\}$.

5.70

1. The graph crosses the $x$-axis at the point (2, 0). The $x$-intercept is (2, 0). The graph crosses the $y$-axis at the point (0, −2). The $y$-intercept is (0, −2).

5.71

1. The $x$-intercept is (4, 0) and the $y$-intercept is (0, 12).
$A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The line passes through the following points, (0, 12), (4, 0), and (5, negative 3).$

5.72

1. $m = \frac-4{3}$

5.73

1. ${\text{slope}} = - 1$

5.74

1. slope $m = 2$ and $y{\text{-intercept}} = (0,-1)$

2. slope $m = - \frac{1}{4}$ and $y{\text{-intercept}} = (0,4)$

5.75

1. $A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The line passes through the following points, (negative 10, 7), (negative 3, 0), (0, negative 3), (1, negative 4), and (6, negative 9). Note: all values are approximate.$

5.76

1. $A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The line is vertical and it passes through the following points, (5, negative 2), (5, 0), and (5, 2). Note: all values are approximate.$

5.77

1. $A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The line is horizontal and it passes through the following points, (negative 2, negative 4), (0, negative 4), and (2, negative 4). Note: all values are approximate.$

5.78

1. (0, 20) is the $y\text{-intercept}$ and represents that there were 20 teachers at Jones High School in 1990. There is no $x\text{-intercept}$.

2. In the first 5 years the slope is 2; this means that on average, the school gained 2 teachers every year between 1990 and 1995. Between 1995 and 2000, the slope is 4; on average the school gained 4 teachers every year. Then the slope is 0 between 2000 and 2005 meaning the number of teachers remained the same. There was a decrease in teachers between 2005 and 2010, represented by a slope of –2. Finally, the slope is 4 between 2010 and 2020, which indicates that on average the school gained 4 teachers every year.

3. Answers will vary. Jones High School was founded in 1990 and hired 2 teachers per year until 1995, when they had an increase in students and they hired 4 teachers per year for the next 5 years. Then there was a hiring freeze, and no teachers were hired between 2000 and 2005. After the hiring freeze, the student population decreased, and they lost 2 teachers per year until 2010. Another surge in student population meant Jones High School hired 4 new teachers per year until 2020 when they had 80 teachers at the school.

5.79

1. 50 inches

2. 66 inches

3. The slope, 2, means that the height $h$ increases 2 inches when the shoe size(s) increases 1 size.

4. The $h$-intercept means that when the shoe size is 0, the woman’s height is 50 inches. $A line is plotted on an x y coordinate plane. The x and y axes range from negative 20 to 140, in increments of 20. The line passes through the points, (negative 30, 0), (0, 50), and (40, 130). Note: all values are approximate.$

5.80

1. $25

2. $85

3. The slope, 4, means that the weekly cost, $C$, increases by $4 when the number of pizzas sold, $p$, increases by 1. The $C$-intercept means that when the number of pizzas sold is 0, the weekly cost is $25.

4. Graph: $A line is plotted on an x y coordinate plane. The x-axis ranges from negative 60 to 210, in increments of 15. The y-axis ranges from negative 75 to 200, in increments of 15. The line passes through the points, (negative 15, negative 35), (0, 25), (13, 85), and (30, 145). Note: all values are approximate.$

5.81

1. No

2. Yes

5.82

1. No

2. No

5.83

1. $Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The first line passes through the points, (negative 9, negative 2), (negative 3, 0), (0, 1), and (9, 4). The second line passes through the points, (negative 4, 9), (0, 5), (5, 0), and (9, negative 4). The two lines intersect at (3, 2).$

5.84

1. (3, 2)

5.85

1. $(3, 2)$

5.86

1. There is no solution to this system.

5.87

1. There are infinitely many solutions to this system.

5.88

1. Jenna burns 8.3 calories per minute circuit training and 11.2 calories per minute while on the elliptical trainer.

5.89

1. not a solution

2. a solution

5.90

1. The region containing (0, 0) is the solution to the system of linear inequalities.

5.91

1. The solution is the darkest shaded region.
$Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 10, in increments of 1. The first line passes through the points, (negative 7, 9), (0, 2), (2, 0), and (9, negative 7). The region below the line is shaded in dark blue. The second line passes through the points, (negative 9, negative 7), (0, negative 1), and (9, 5). The region above the line is shaded in light blue. The two lines intersect approximately at (1.8, 0.3). The region to the left of the intersection point and within the lines is shaded in gray.$

5.92

1. The solution is the lighter shaded region.
$Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 10, in increments of 1. The first (dashed) line is horizontal and it passes through y equals negative 1. The region below the line is shaded in gray. The second (solid) line passes through the points, (negative 2, negative 8), (0, negative 2), and (3, 7). The region to the left of the line is shaded in dark blue. The two lines intersect approximately at (0.5, 1). The region below the intersection point and within the lines is shaded in light blue.$

5.93

1. No solution.
$Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 10, in increments of 1. The first line passes through the points, (negative 6, negative 8), (0, 1), and (4, 7). The region above the line is shaded in gray. The second line passes through the points, (negative 2, negative 9), (0, negative 6), (4, 0), and (10, 9). The region below the line is shaded in blue.$

5.94

1. The solution is the lighter shaded region.
$Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 10, in increments of 1. The first line passes through the points, (negative 3, negative 8), (0, 1), and (2, 7). The region above the line is shaded in light blue. The second line passes through the points, (negative 1, negative 7), (0, negative 4), (2, 2), and (4, 8). The region above the line is shaded in gray. The region above the second line is shaded in both colors and appears dark blue.$

5.95

1. $\left\{ \begin{array}{l}240h + 160C \ge 800\\1.40h + 0.50C \le 5\\h \ge 0\\C \ge 0\end{array} \right.$

2. $Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 0 to 10, in increments of 1. The first line passes through the points, (0, 10), (2, 5), and (4, 0). The region above the line is shaded in light gray. The second line passes through the points, (0, 6), (2, 3), and (4, 0). The region below the line is shaded in dark blue. The region within the lines is shaded light blue.$

3. The point $(3, 2)$ is not in the solution region. Omar would not choose to eat 3 hamburgers and 2 cookies.

4. The point $(2, 4)$ is in the solution region. Omar might choose to eat 2 hamburgers and 4 cookies.

5.96

1. With $a =$ the number of bags of apples sold, and $b =$ the number of bunches of bananas sold, the objective function is $P = 4a + 6b$.

5.97

1. $T = 20x + 28y$

5.98

1. The constraints are $a + b \leq 20$ and $3a + 5b \leq 70$. The summary is: $P = 4a + 6b$, $a + b \leq 20$, and $3a + 5b \leq 70$.

5.99

1. The constraints are:
$15 \leq x \leq 22$
$13 \leq y \leq 19$
So the system is:
$T = 20x + 28y$
$15 \leq x \leq 22$
$13 \leq y \leq 19$

5.100

1. The maximum value for the profit $P$ occurs when $x = 15$ and $y = 5$. This means that to maximize their profit, the Robotics Club should sell 15 bags of apples and 5 bunches of bananas every day.

Check Your Understanding

1. $J = V + 2$, $V = J - 2$

2. $5x+8$, $2n+3m$

3. $(8x + 12x) \div \left( {4x - 2x} \right)$

4. $- 3$

5. ${x^2}-2xy + {y^2}$

6. $3x\left( {3{x^2} + x-2} \right)$

7. multiplication

8. division

9. It is a correct solution strategy.
Let
$\begin{array}{rcl}{x}&{ = }&{38}\\{8\left( {38-2} \right)}&{ \mathop = \limits^? }&{6\left( {38 + 10} \right)}\\{8\left( {36} \right)}&{ \mathop = \limits^? }&{6\left( {48} \right)}\\{288}&{ = }&{288 ✓}\\\end{array}$

10. It is a correct solution strategy.
Let
$\begin{array}{rcl}{x}&{ = }&{- \,2}\\{7 + 4\left( {2 + 5\left( { - \,2} \right)} \right)}&{ \mathop = \limits^? }&{3\left( {6\left( { -\,2} \right) + 7} \right)-\left( {13\left( { -\,2} \right) + 36} \right)}\\{7 + 4\left( {2-10} \right)}&{ \mathop = \limits^? }&{3\left( { - 12 + 7} \right)-\left( { -\,26 + 36} \right) }\\{7 + 4\left( { -\,8} \right)}&{ \mathop = \limits^? }&{3\left( { - 5} \right)-\left( {10} \right) }\\{7-32}&{ \mathop = \limits^? }&{ -\,15-10}\\{-\,25}&{ = }&{-\,25✓}\end{array}$

11. This is not a correct solution strategy. The negative sign is not distributed correctly in the second line of the solution strategy. The second line should read $8x + 7-2x + 9 = 22-4x + 4$.

12. \(

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= 1.7x + 3\)

13. $40.40

14. \(

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= 1.6y + 5\)

15. $40.20

16. The Enjoyable Cab Company, because the cab fare will be $ 0.20 less than what it would cost to take a taxi from the Nice Cab Company.

17. Luis is; there are infinitely many solutions. If this is solved using the general strategy, it simplifies to $0 = 0$. This is a true statement, so there are infinitely many solutions.

18. d $F = \frac{9}{5}C + 32$

19. b $K = C + 273$, although d $C = K – 273$ is an equivalent formula.

20. a $K = \frac{5}{9}\left( {F-32} \right) + 273$

21. b $R = \frac{9}{5}C + 492$

22. d $\left( { - \infty , - 1} \right)$

23. b $[ - 5,\infty )$

24. c $\left[ {\frac{3}{2},\infty } \right)$

25. a $( - 4,3)$

26. e $6x \geq 24$

27. b $-6x \gt 18$

28. c $4x - 3 > - 11$

29. c $- 3x + 14 > - 13$

30. b $8x < 764$

31. d $50x > \text{8,120}$

32. a True

33. a True

34. a True

35. b False

36. a True

37. $12{:}16$

38. $16{:}28$

39. 324.00 British pounds (None of these.)

40. 10 inches

41. Yes he can, but barely. At 37 miles per gallon, Albert can drive $499.5$ miles. While in theory he can make it, he probably should fill up with gasoline somewhere along the way!

42. $$40.66$

43. d $y = \frac{5}{3}$

44. $A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10 in increments of 2.5. The line passes through the points, (negative 2.5, negative 2.5) and (0, 5).$

45. b $y = \frac{1}{2}x + 4$

46. $A dashed line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10 in increments of 2.5. The line passes through the following points, (negative 2.5, negative 10) and (0, 2.5). The region above the line is shaded. Note: all values are approximate.$

47. b $y \leq -2x + 5$

48. d

49. b

50. c

51. d

52. b False

53. b False

54. a True

55. b False

56. \(x = \frac{5}{2} \pm \frac

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57. a True

58. b False

59. b False

60. a True

61. b False

62. a True

63. b False

64. b False

65. a True

66. b False

67. Elimination

68. Substitution

69. Substitution

70. Elimination

71. Substitution

72. Elimination

73. Elimination

74. Substitution

75. c

76. e

77. d

78. b

79. a

80. c

81. a

82. b

83. d

84. a

85. c

86. d

87. $Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from 0 to 20, in increments of 2. The first line passes through the points, (0, 17), (4, 3), and (5, 0). The second line passes through the points, (0, 12), (6, 6), and (12, 0). The two lines intersect at (2, 10). The region within the lines and below the intersection point is shaded. Note: all values are approximate.$

88. b

89. $$140$

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