Chapter 1 Lesson 1.1.1 1-4.
a.
b.
c.
12 3
d.
6 4
–2
7
12 –3
3
–5
e. 4
4 7
8
1 8 2 8 12
–2
–4 –6
1-5.
a. 1 by 24 (perimeter = 50), 2 by 12 (perimeter 28), 3 by 8 (perimeter = 22), and 4 by 6 (perimeter = 20) b. Largest is a thin rectangle (perimeter = 50), smallest is like a square, (perimeter = 20)
1-6.
a. True
b. True
c. False
d. True
Lesson 1.1.2 1-7.
a. (2, 2), (3, 2), (3, 3), (-4, 3), (-6, -5), (9, -3) c. (0, 0) d. Trees he cannot see: (9, 6) and (12, 8); answers vary in terms of what those trees have in common; when looking at the coordinates, answers may include: the ycoordinates are 1.5 times the x-coordinates, the x-coordinates are multiples of 3, while the y-coordinates are multiples of 2. e. It is blocked by the trees at (3, –1) and (6, –2). f. (–8, 3), (–8, 1), (–9, 2), (–7, 2) g. (1, 2), (2, 1), (3, 1), (4, 2), (4, 3), (3, 4), (2, 3) h. 103 trees will still be healthy
1-10.
a. –9
b. –1
c. –7
1-11.
1 by 18, 2 by 9, 3 by 6
1-13.
a. Diagrams vary, but one way to describe it is as a whole.
1-12.
d. 4
e. 6
a. (72 + 89) ÷ 2 = 80.5%
f. 1 b. ! 83.67%
b. Diagrams vary; one-half, one-fourth, one and one-half
1-14.
Answer Key
c. Diagrams vary, 33 13 %
d. answers vary: 10%, one tenth
a. (-3, 1) and (-2, 3)
b. (2, -1) and (3, -2)
1
Lesson 1.1.3 1-17.
a. answers vary b. Yes; the taller the person is, the longer his or her reach. c. The independent quantities were represented by the x-axis, while the dependent _quantities were represented using the y-axis. d. A trend line can generalize the trend in the data.
1-18.
a. The graph is in the first quadrant because negative lengths do not exist; the range of the data determines the kind of graph. b. Counting by 10’s makes the graph a reasonable size. c. In this situation, including the origin with the graph is not suggested. It is easier to see the trend line when the data are not bunched together, and this can be done by changing the range of the graph to exclude the origin. d. The graph should include the maximum height (that of Yao Ming) on the x-axis and the height of the tunnel on the y-axis.
1-21.
a.
b.
66 6
11
–1
4
17
e. 1 2
1-22.
1 4
c.
–4
–3
3
f.
g.
1 3 3 13 4 12
1
–8
–2
–1
8
–5
1 4
1 2
d.
6
7
h.
x2 x
ab
x
a
2x
b a+b
a. –8
b. 29
c. –8
d. –22
e. –80
f. 12
g. 0
h. 3
i. –5
j. 30
k. 0
l. 10
1-23.
a. 5
b. 12
c. 16.5
1-24.
1-25.
a.
1-27.
18.75 inches
1-28.
84%
1-30.
a. i.
8 , etc. 10
b. 0.8; no
c. 80%
1-29.
1 2
ii.
1
2 12
2
a. 430
1-26.
b. –200
4 by 4; P = 16
MT ! 45 sq. un., CA ! 51 sq. un. iii.
10 2
5 7
5
iv. 1
15 3 15 13
– 48 8
–6 2
b. answers vary 2
Algebra Connections
Lesson 1.1.4 c. The 100th figure has 10,200 tiles
1-31.
b. answers vary
1-32. Figure 19
1-35.
a. 15, 21; add increasing amounts.
b.
c. 81, 243; multiply by 3.
d. -2, -7; subtract increasing amounts.
1 1 1 , ; multiply by . 16 32 2
e. 53, 58; subtract 2, then add 5 repeatedly. 1-36.
a.
b.
45 –9
–5
9
–14
1-37.
–2
b
1-38.
0.67 is closer
1-39.
a. 15 by 18 = 270 ft2
1
3 5
a
d c
2
d.
6 2
7
-5 -4 -3 -2 -1 0
Answer Key
c.
–18
1 2
1 43 4
3 12
e 3
4
5
b. 28 by 23 = 644 ft2
c. 270 – 150 = 120 ft2
3
Lesson 1.2.1 1-40.
There are two possible solutions: living room with a base of 20 ft and a height of 9 ft or a base of 15 ft and a height of 12 ft.
1-41.
a. If the base is 10, the height must be 18 and the total area is (20)(24) = 480 ft2. If the height of the living room is 10, the base must be 18 and the total area is (28)(16) = 448 ft2. The results are different, so it does matter which side of the living room is 10 feet long. b. No; the total area should be 450 ft2.
1-44.
a.
b.
15 3
5
e. 1 2
c.
3 3
8
d. answers vary
1
5
4
f.
4 8 17 2
1 6
d.
10
1 2
2 7
g.
0.06 0.3 0.2
h.
11 1
0.5
11
b. 84.25%
3 –3
12
1 3
5 6
–1 –4
1-45.
a. 74.5%, No
c. It lowered her score a lot. Yes
1-46.
He owns 9 spy novels and 17 science-fiction novels.
1-47.
The less soda costs, the more sodas that are sold. But if the soda price rises to a certain value, no one will buy the soda.
1-49.
a. –9
b. 34
c. –27
d. –20
e. –85
f. 8
g. –5
h. –1
i. –54
j. 25
k. -258
l. 26
Lesson 1.2.2 1-50.
12 bulls-eyes
1-51.
a. Adele $320, Amanda $160, Alisa $160 c. 32 laps for Hector
b. 9 units by 15 units
d. 984 and 2568 students
e. 6 and 12
1-54.
Todd: 37 years old, Jamal: 27 years old
1-55.
36, 21, 18
1-56.
33 = 27 , 34 = 81 , 35 = 243 , 36 = 729 , 37 = 2187 , 38 = 6561 , 39 = 19683 a. The pattern “3, 9, 7, 1” repeats.
1-57. 4
a. 2, 5
b. 2, -4
c. 3, 5
b. 35, 39, and 313 d. –3, -3
1-58.
642 cm. Algebra Connections
Lesson 1.2.3 1-59.
a. 31 tomato paste, 13 noodles b. It cost $21.70 for tomato paste and $6.50 for the noodles, total cost was $28.20.
1-60.
30 million
1-61.
$14
1-62.
27, 51, 32 inches
1-63.
14 birds
1-65.
16 cm, 32 cm, and 28 cm
1-66.
a. A 5 unit square with a 4 unit square inside it.
1-67.
b. It grows by two tiles each time.
b. 9 sq. units
c. Methods may vary, but the 100th figure will have 201 tiles. 1-68.
a. steady increase in speed b. quick increase in speed then levels off at certain speed c. increasing acceleration; speed increases faster and faster
1-69.
Answer Key
! 450; 250; 50
5
Chapter 2 Lesson 2.1.1 2-4.
a. 4x 2 + 3x + y + 7 b. 3x 2 + 3xy + 6 c. not possible to simplify this expression, since none of the terms are alike d. y 2 + 7y + 8xy + 4x + 3
2-6.
4x + 7
2-8.
a. perimeter = 120 units, area = 647 sq. units b. perimeter = 70 units, area = 192 sq. units
2-9.
perimeter: 52, area: 168
2-10.
a. 11
2-7.
c. !30
b. 14
3x 2 + 8x + 9
d. 10
2-11. 33 m and 51 m
Lesson 2.1.2 2-12.
unit tile: 4, 5-piece: 12, x-tile: 2x + 2 , x2-tile: 4x , y-tile: 2y + 2 , y2-tile: 4y , xy-tile: 2x + 2y
2-13.
a. 4x + 2y + 6
b. 2x + 4
c. 2x + 4y + 2
d. 4x + 2y + 6
2-14.
a. 34 units
b. 10 units
c. 40 units
d. 34 units
2-15.
Multiple solutions are possible, including: y = 5 , x = 1 ; y = 4 , x = 4 ; y = 3 , x = 7 ; y = 2 , x = 10 ; and y = 1 , x = 13 .
2-17.
a. y 2 + 6y + 5 b. not possible to simplify this expression, since none of the terms are alike c. 3xy + 6x + 3y + 6 d. 4m 2 + 5m + 2mn
2-18.
a. 7
2-19.
a. 10 feet by 15 feet b. bathroom = 63 sq. ft, kitchen = 105 sq. ft, living room = 150 sq. ft c. 318 sq. ft
2-20.
A(– 4, 3), B(2, 1), C(–2, 0), and D(–3, –3)
2-21.
6x 2 + 4x + 3xy + 6y + y 2
6
b. 14
c. –2
d. 74
Algebra Connections
Lesson 2.1.3 2-22.
Ideas should include subtraction, opposite, and negative.
2-27.
They are all correct. 2 – 4 is -2.
2-29.
a. 4x 2 + 6x + 13 b. 5y 2 + 8x + 19
c. 9x 2 + x + 44
2-30.
a. 14
d. 12
2-31.
a. a square
2-32.
82 beige, 107 red, and 246 navy blue
2-33.
a. – 45
b. 6.5
c. 74 b. 9 units
b. 10
c. 2
d. 5y 2 + 6xy + 30
e. 11
c. 81 square units
d. –254
d. 36 units
e.
!1 4
f. –2
Lesson 2.1.4 2-34.
Each situation deals with “equal opposites” so that the net effect is zero. They can be represented with equations, such as $15 – $15 = $0, 12 – 12 = 0 yards, and (-x) + x = 0, if x represents the level of the sand above sea level.
2-36.
a. One possible expression is 2x ! x + 1 ! 3 .
2-37.
answers vary
2-38.
a. !2x ! 2 + x ! (!2) = !x
2-39.
a. x ! 4
2-42.
Parts (a) and (b) are possible, but part (c) is not.
2-43.
a. 3 g: –12
2-44.
a. 2x ! 3 ! (x + 1) = x ! 4 c. !x ! (x + 2) = !2x ! 2
2-45.
4x + 8
2-46.
a. –4
–7 –11
Answer Key
c: –24
b.
28
b. x + 1 ! x ! 2 ! (x ! 3) = !x + 2
b. !2y + 6
b. 18 h: –1
b. answers vary
3x 2 + 14x + 15
2-41.
d: –30
e: –6
f: 8
b. y + 3 ! y ! 1 = 2
c.
–12 6
–2 4
1 2
d.
–8 –16
–15.5
1 2
1 10 7 10
1 5
7
Lesson 2.1.5 2-47.
answers vary
2-48.
left: x + 2 ! 2 ! (3) , right: x + 2 ! 3 ! (!2) a. left: x ! 3 , right: x + 1 b. right mat is greater because x plus 1 is greater than x minus 3, whatever the value of x
2-49.
a. left expression is greater c. right expression is greater
2-50.
a. both are equal
2-52.
a. 4x 2 + 3x ! 3
2-53.
b. right expression is greater b. 5x 2 ! 23x c. 4x + 3y
–64 8
b. left expression is greater
d. 2y 2 + 3xy + 27
–25 –8
–5
0
–12 5
4
0
2-54.
a. The steeper line is B
2-55.
a. 832 b. 416
1 2
–3 1
b. ! 3 years
c. 208 d. 83.2
! 16 1 6
! 13
c. ! $65,000 d. Company B 2-56. a. –23.8 b. 36 c. 1.5 d. –14
Lesson 2.1.6 2-57.
a. left f. right
2-58.
b. No; it depends on the value of x
2-59.
a. 2x ! 1
2-60.
a. 1=D, 2=A, 3=C, 4=B b. 6, 4, 24 sq. units d. Possible solutions: 10 by 3.6, 15 by 2.4, 12 by 3
2-61.
a. 17
2-62.
a. $26.28 ÷ 12 = $2.19 c. $8.76 ÷ $2.19 = 4 gallons
b. $2.19 · 15 gallons = $32.85
2-63.
a. 6
b.
8
b. right
c. left
e. neither (they are equal)
c. x 2 ! y ! 4
b. 4
b. 9
d. right
c. 45
d. 10
3 4
÷
c. the area " 36 e. curve e. –22
1 8
=
3!8 4 1
f. 4
=6 Algebra Connections
Lesson 2.1.7 2-64.
a. neither (they are equal) b. cannot be determined (depends on the value of x) c. left
2-67.
a. Not enough information c. The left expression is greater
2-68.
685
2-69.
Not quite. She correctly removed 2x from both sides and also flipped a 1 from the “–” region to the “+” region and removed a zero. However, on the left side, the –1 and the 1 in the “–” region do not make zero, so this is not a “legal” move.
2-70.
a. Figure 4 is a 5-by-5 square, and Figure 5 is a 6-by-6 square. b. An 11-by-11 square, a 101-by-101 square c. Figure 5 would have 21 tiles; Figure 8 would have 33 tiles; each figure has 4 more tiles than the figure before it.
2-71.
a. 4y + 2x + 2 ; 4(10) + 2(6) + 2 = 54 units b. y 2 + 2x + 1 = (10)2 + 2(6) + 1 = 113 square units
2-72.
a. 4
b. 1
c. 7
b. The right side is greater d. They are equal.
d. 2
e. 5
f. 6
g. 3
Lesson 2.1.8 2-73.
a. The right side is greater b. Cannot be determined, because it depends on the value of x
2-74.
a. x < 5
b. x > 5
2-75.
a. x = 3
b. x = !2
2-76.
a. x = 3 d. x = 4
b. x = 6 e. x = 2
2-77.
a. The right side is greater b. The left and right expressions are equal.
2-78.
27,13
2-80.
(-3, 2), (-6, 4), (9, 6) and various others
2-81.
a. ! 17.67
Answer Key
2-79. a. 2y ! 2x + 3
b. 18
c. x = 5
c. x = 2 f. no solution
b. 2x 2 + 3x + 6
c. 0
d. x ! y
c. 20 9
Lesson 2.1.9 2-82.
a. x = 3
b. no solution
2-83. a. –3
2-84.
a. –2
b. no solution
c. 3
2-86.
possible solution: 2 ! 2x ! (4 ! x) = 2 ! 3 ! (x ! 2)
2-88.
Sample solutions: (– 4, 2), (–6, 3), (4, –2)
2-89.
a. x = 2 y = 5 c. x = 3 y = 5
2-90.
a. 60
b. 0
c. all numbers
d. –3 2-87. 16 weeks
b. x = 2 y = !4 d. x = !3 y = !3
b. –32
c. 3598
d. –6
Lesson 2.2.1 2-91.
a. 19.2 cm and 25.6 cm
b. $105, $115.50
c. ! 44 minutes, 102 transparencies e. ! 20 cm, ! 1.4 cm
d. 300 tamales
2-93.
P = 88 m, A = 369 sq. m
2-94.
x + 1 ! 2 ! (!x + 3) = 6 ! (1 ! 1) , x = 5
2-95.
a. 3
b. –2
2-96.
a. 4x 2 + 5x ! 4y ! 3
b. !6x 2 ! 2x + 8
c. !3y 2 + 34x ! 5y
d. 0
2-97.
The 24-ounce bag is best. It costs roughly 5¢ per ounce while the 36-ounce bag costs about 8¢ per ounce.
2-98.
a. ! $40
10
b. ! $55
c. ! $75
Algebra Connections
Lesson 2.2.2 2-99.
a. 42 tubes in 5 years, 16.8 tubes in 2 years, 11.9 years b. x ! 7.83 , y = 27
2-102.
a. 200 blue candies
2-103.
21
2-104.
a. 3x ! 1 ! (1) = x ! 3 ! (5) , x = !3
b. !11 = !11
2-105.
a. 2
b. 0
2-106.
a. 35g, 70g
2-107.
answers vary
b. $1495, $14950, $7475
c. $1845.75, 32 games
b. cookies: 200, 400; sugar: 35, 70, 7, 3.5, 17.5
Lesson 2.2.3 2-109.
a. Corresponding vertical sides represent the number of origami cranes, while the bottom pair represent minutes. b. ! 1428.6 minutes c. He’s trying to find how many cranes he can produce in 250 minutes, which is 7 · 25 = 175 cranes.
2-110.
answers vary
2-112.
32 eggs
2-113.
a. Car A
b. A (2, 120); B (4, 120)
2-114.
a. 3
b. 1
2-115.
a. 2x 2 + 12x ! 8
b. 8y 2 ! 10y + 1
c. 30xy + 4y ! 4x
d. 0.8x
Answer Key
c. 60; 30
c. –1.5
d. –1
11
Chapter 3 Lesson 3.1.1 3-2. a.
x (in)
y (out)
A
b.
x (in)
y (out)
C
Easy
Hard
L
N
Dark
Light
D
F
Hot
Cold
Q
S
Up
Down
W Y Rule. y is two letters after x d.
x (in)
y (out)
8
e.
y (out)
17
100
-2
-3
4
x (in)
y (out)
heptagon quadrilateral
Right Left Rule: y is the opposite of x x (in)
decagon Rule: y has two more sides than x x (in)
y (out)
51
4
16
4
3
-1
1
9
6
4
3 or -3
9
12
25
30
16
12
144
10
21
60
31
-6
36
Rule: Double x and add 1
12
c.
Rule: Halve x, and add 1
f.
Rule: Square x
Algebra Connections
3-3. x (in)
a.
y (out)
b.
5 8 0 -2 -4 -10 10 18 -2 -6 100 198 0.5 -1 Double x, then subtract 2 x (in)
d.
y (out)
6 0 11 5 2 -4 23 17 -7 -13 46 40 -4 -10 Subtract 6 from x
x (in)
c.
x (in)
3 -9 10 -30 -1 3 -2 6 0 0 12 -36 -5 15 Multiply x by –3 e.
x (in)
a. left. -4, right –7 (left is greater)
3-5.
a. -45
3-6.
a. It subtracts 6 from the input value
3-7.
$50, $250 and $85
3-8.
a. Car B, line is steeper.
c. –62
y (out)
f.
x (in)
y (out)
-8 -37 10 53 3 18 0 3 1 8 19 98 4 23 Multiply x by 5 and add 3
b. left. -5, right. -1 (right is greater) d. 9 b. 71
b. In 2 hours, both cars were 80 miles away from the game.
Answer Key
y (out)
0.5 1.25 -6 or 6 37 2 5 -10 101 -5 26 0 1 -7 or 7 50 Square x and add 1
2 6 4 20 10 110 -3 6 5 30 7 56 1 2 Multiply x by x + 1
3-4.
b. –6
y (out)
c. 160 miles
13
Lesson 3.1.2 3-10.
a. 13 ft.
b. 33 ft.
c. 5 ft.
d. about 200 ft.
3-11.
a. Line
b. Yes
c. 11
d. No
3-12.
a. 0th year, 5 ft; 1st year, 9 ft; 2nd year, 13 ft; 6th year, 29 ft b. y = 4x + 5 c. 205 ft
3-13.
a. x + 4 = 3x ! 2, x = 3
3-14.
a. 31
3-15.
a. 2x 2 ! x + 5 c. 2x 2 + 30y ! 3y 2 + 4xy ! 14 ! x
3-16.
a. -13 e. 16
b. 4 f. –3
3-17.
a. $22.50
b. $12.50
14
b. x + 2 = !x ! 1, x = ! 23
b. 2 b. !2y 2 + 4x ! 8 d. 30
c. 36 d. –4 g. They are different
Algebra Connections
Lesson 3.1.3 3-18.
b. i. x row should read 0, 1, 2, 3, 4. y row should read 3, 9, 15, 21, 27 ii. y = 6x + 3 c. 33 d. It will be a large “C,” with width 101 and height 203. Horizontal parts will each be 2 rows, 101 tiles long. It will have 603 tiles.
3-19.
a. The points are in a line. b. The points should be co-linear c. The points would follow the pattern.
3-20.
a. The line touches all the points. The equation describes the pattern from the table. b. It means the figures grow continuously between the tile patterns. Therefore, a figure exists for any figure number, such as Figure 1.5 or Figure 3.2. c. Because each given figure number is an integer and because there is no evidence to suggest that figures exist between them.
3-21.
a. She is incorrect, since 6(5) + 3 = 33 b. He is correct, since 6(16) + 3 = 99 . c. He is not correct since his rule only works with (1, 9). d. She is correct since 3(2x + 1) = 6x + 3 .
3-22.
Using the rule, the 100th figure will have 6(100) + 3 = 603 tiles. To use a graph, change the viewing window to include x = 100 and y = 625 .
3-23.
x + 1 ! 2 ! (!x + 3) = 6 ! (!1 + 1), x = 5
3-24.
a. -5
3-25.
a. 2
3-26.
1036
3-27.
a. 26
3-28.
See table below.
b. –10
c. –15
d. –2
e. -10
b. no solution
IN (x)
b. 4
2
c. 2
10
0
OUT (y) -6 -30 0 a. Multiply x by negative 3
7 -21
d. 43
-3
-5
-10
100
x
9 b. -3x
15
30
-300
-3x
3-29.
a. 2x + 1 ! (!x ! 2) = 2 ! (!2x);!!x = !1
b. 0 = 0
3-30.
a. 4
b. 0
3-31.
a. 3x 2 + 2x + 8, 5x 2 + 2, 8x 2 + 2x + 10
b. 7x 2 ! 7x ! 17
c. Answer Key
3 4
f. 3, -3
x + 23 x = 17 12 x 15
Lesson 3.1.4 3-33. IN (x)
-4
-3
-2
-1
0
1
2
3
4
OUT (y)
-7
-5
-3
-1
1
3
5
7
9
IN (x)
-4
-3
-2
-1
0
1
2
3
4
OUT (y)
-9
-7
-5
-3
-1
1
3
5
7
3-34.
b. a slanted line that goes down from left to right c. Expected response: One line goes up while the other goes down. They both cross the y-axis at the same point. 3-35. IN (x)
-4
-3
-2
-1
! 12
0
1 2
1
2
3
4
OUT (y)
16
9
4
1
1 4
0
1 4
1
4
9
16
b. (0, 0) 3-36.
The line should pass through the points (0, 2) and (1, 3).
3-37. IN (x)
2
10
6
7
-3
-8
-10
100
x
OUT (y)
5
21
13
15
-5
-15
-19
201
2x+1
a. Multiply x by two and add one b. y = 2x + 1 3-38.
a. 2
b. 2.5
3-39.
a. 37
b. 5.5
3-40.
a. ! 71.4 or 72 tomatoes
16
c. 3
d. 12 b. ! 81 people
Algebra Connections
Lesson 3.1.5 3-42.
a. x-values between $10 and $100 are appropriate. b. The x ! y pairs should be solutions to the equation y = 0.15x , where y is the amount of tip and x is the bill. c. ! $5.50; ! $8.00
3-43.
b. (4,-2)
3-44.
See table below. IN (x) –4
c. possible answers: (2,-2) or (4,0)
OUT (y)
4
–3
–2
–1
0
1
2
3
4
4.5
5
5.5
6
6.5
7
7.5
8
b. No; when x = 10 , y = 11 . 3-45.
Rule: y = x + 3
3-47.
c. The point would lie at (5, 19), because the rule is y = 3x + 4 or because it is possible to predict on the graph.
3-48.
See table below. IN (x) OUT (y)
2 2
10 6
6 4
7 4.5
-3 -0.5
28 15
a. Divide x by two and then add one 3-49.
Answer Key
a. 2
-10 -4 b.
x 2
+ 1 or
100 51 1 2
x x 2
+1
x +1
b. 3.5
17
Lesson 3.1.6 3-51.
Each graph has something wrong to make the question unanswerable. Part (a) has no scale, so the coordinates are not identifiable. Part (b) has no arrows, so it is unclear if the function even exists when x = 5 . Finally, the axes are not labeled in part (c), so it is impossible to know which axis is k and which is t.
3-54.
a. (a) and (b) are both lines, while (c) is a parabola. Part (a) goes down as x increases and (b) goes up as x increases. b. (a) (0, 1); (b) (0, 2); (c) (0, -4) c. (a) (1, 0), (b) (– 4, 0), (c) (2, 0) and (–2, 0)
3-55.
See table below. x –3 –2 –1 y 11 6 3
0 2
1 3
2 6
3 11
3-56. See table below. x -3 -2 -1 0 1 y 6 5 4 3 2
2 1
3 0
3-57.
Figure 0 will have 1 square, Figure 4 will have a 5-by-5 square with 8 tiles attached in a row on the right, and Figure 100 will have a 101-by-101 square with 200 tiles attached in a row on the right.
3-58.
2y + 4x + 2 , 2(7) + 4(3) + 2 = 23 units
3-59. a. 1
b. 2
Lesson 3.1.7 3-60.
a. y = 4x + 2 b. For this problem, x should be more than 1. Sample points in table: (2, 10), (3, 14), and (4, 18). c. The line should pass through the points (2, 10), (3, 14), and (4, 18). d. Yes, they should be connected, because the shape (and thus, the perimeter) also exists for values of x between integers, such as x = 1.5 and x = 3.01 .
3-61.
a. (2, 8), (3, 17), (5, 35), and (6, 44) b. i: The point (8, 2) should be at (2, 8); ii: The origin should be (0, 0), not (2, 5); iii: Both axes are not scaled evenly and need to be labeled; iv: The x-axis is spaced unevenly.
3-62.
a. y = x 2 ! 1 b. Because only positive values of x and y make sense for this situation. c. a parabola ` d. ! 4.6 units
3-64.
a. 84 minutes b. ! 109.4 minutes
3-65.
b. (–3, –2)
3-66.
c. The parabola in part (a) points upward, while the parabola in part (b) points downward d. ! ± 2.6 e. ! ± 3.2 x=0
3-67. 3-68.
18
3 12
7 4
4
1 2
c. a parabola d. (–3, –2) should be (–3, 5)
–28 7
56 –4
3
–7
10 –8
–15
2
5 7
Algebra Connections
Lesson 3.2.1 b. x = 4
3-69.
a. 4x ! 5 = 2x + 3
3-70.
a. No, 100 does not make the equation true, x = 10 3 b. Yes, 4 makes the equation true
3-71.
a. 2
b. -2
c. - 3
d. no solution
3-73.
a. -2
b. –1
c. 1
d. 0.5
3-74.
See table below. IN (x) -4 OUT (y) 9
-3 7
-2 5
-1 3
0 1
1 -1
2 -3
3 -5
4 -7
b. A slanted line that decreases as x increases. 3-75.
a. 2x + 6
b. 4x + 6
c. 4x + 4y
3-76.
a. $68.75
b. 20 pounds
3-77. 2
d. 6y ! 2x + 4
Lesson 3.2.2 3-79.
a. 2x + 3 ! x ! 1 = x + 2 b. The equation must always be true, regardless of the value of x. c. 2x ! 3 ! x + 4 = x + 2 ; This equation is never true, no matter the value of x.
3-80.
a. all numbers d. all numbers
3-83.
a. y = 2x + 2
3-84.
a. 2
3-85.
a. yes
3-86.
a. See table below. x -1 0 1 y -3 -1 1
Answer Key
b. –1 b. no
b. x = !4 e. no solution
c. no solution f. x = 2
b. (5, 12) c. all numbers c. no
d. 1.5
d. yes b. y = 2x ! 1 2 3
3 5
19
Lesson 3.2.3 3-87.
303 tiles
3-88.
a. y
3-89.
x = 13.5 ; Since x represents a figure number, it logically should be an integer. However, it can be argued that the pattern is continuous, and thus the “C’s” grow continuously.
3-90.
a. 4
3-91.
x = 2.5
3-92.
b. 45 = 6x + 3 , Figure 7
c. Replace x with 7 in the rule
b. 8
c. any number
d. 0.15
a. 42
b. 3
c. 12
d. 3
3-93.
a. no solution
b. 3.5
c. any number
3-94.
27 and 12 cm
3-95.
!(x ! 3) represents the opposite of x ! 3 , which is !x + 3 .
3-96.
See table below. IN (x)
-1
0
1
2
3
-5 0 3 b. a parabola opening downward
4
3
0
-5
OUT (y)
20
-3
-2
d. 0
Algebra Connections
Lesson 3.2.4 3-97.
a. y = 4x + 5 b. x: number of years after planting, y: height of the tree in feet. c. 85 feet d. 367 = 4x + 5 ; 90.5 years e. No. The numbers are too large to solve the problem efficiently with tiles.
3-98.
a. 10
b. no solution
3-99.
a. y = 0.15x
b. 9 = 0.15x , so x = $60
3-100.
a.
b.
1 0.5
c. -12
2
5
3-101.
a. 4.5 pounds
3-102.
a. any number
3-103.
See table below.
d.
5
– 48
–2.5 –2 – 4.5
7
2.5
c. $60 + $9 = $69
c.
10
2
d. 0.2
8
–6 2
b. 40 dozen b. m = 9
d. p = !10
c. x = 4
IN (x)
2
10
6
7
-3
±5
-10
1 2
x
OUT (y)
-1
95
31
44
4
20
95
!4 43
x2 ! 5
a. Multiply x by itself and subtract 5 b. x 2 ! 5 c. a parabola 3-104.
a. three less than five times x b. twice the sum of x and y c. three decreased by the sum of x and five
Answer Key
21
Chapter 4 Lesson 4.1.1 4-3.
The parabola should pass through the points (0, 0) and (2, 0) and have vertex (1, –1).
4-4.
She should have received two sports cars and ten pieces of furniture.
4-5.
1 ! 3 ! (!2x) = !x ! 3 ! (x + 2)
a. x = !0.75
4-6.
a. 2x 2 + 11x ! 4y ! 3
c. !y 2 + 33y ! 14
4-7.
a. y = x 2 + 1
b. !2x 2 ! 2x ! 8
d. 0
Lesson 4.1.2 4-8.
b. Figure 0 has 2 tiles. d. 100th figure has 402 tiles
c. The pattern is growing by 4 tiles.
4-9.
Both patterns grow by 4 tiles.
4-10.
a. They each have the same growth factor, but start with a different number of tiles. b. #1: y = 4x + 2 , #2: y = 4x + 1 , #3: y = 4x + 3 c. The growth of 4 corresponds to the 4x in each equation. d. You would need to know the number of tiles in figure 0.
4-11.
b. y = 2x + 3 c. They both have 3 tiles in figure 0, but they grow by different amounts. d. The growth factor is the coefficient of the ‘x’ term.
4-13.
x + 1 ! 2 ! (!2x + 3) = x + 5 ! (2 ! 1)
4-14.
a. 3
4-16.
See table below. IN (x) 2 OUT (y)
b. –1
9
a. x = 4
4-15. ! 83 words
10
6
7
–3
5
–10
1000
x
25
17
19
–1
15
–15
2005
2x + 5
a. Multiply x by two and add five b. y = 2x + 5 4-17. 22
Each sister should receive 1 poster, 72 books and 8 CD’s. Algebra Connections
Lesson 4.1.3 4-18.
a. #1: y = 4x + 2 , #2: y = 4x + 1 , #3: y = 4x + 3 , #4: y = 2x + 3 c. In the pattern, the growth is represented by the added tiles. In the rule, the growth shows up as the coefficient of x. In the graph, the growth can be shown as the vertical change between the points for each figure. d. Some possible responses include: “If the graph goes in the same direction, they all add the same amount.” “The pattern in the table (growth) is the number multiplied by x in the rule.” “The first number for figure 0 is always the last number in the rule.” “If we have the rule, we have a shortcut to graph by starting with figure 0 (end #) and going up by the growth number.”
4-19.
a. Figure 4 has 10 tiles. b. A and B both increase by 2, while C increases by 4. c. Since the lines are parallel, tile patterns A and B must grow by the same amount for each figure. Since the lines start at different values on the y-axis, Figure 0 for tile patterns A and B must have a different number of tiles (2 and 8, respectively). d. Since lines A and C have a different “steepness,” tile patterns A and C grow by different amounts for each figure. However, since they start at the same value on the y-axis, Figure 0 for both patterns must have the same number of tiles.
4-21.
c. y = 2x + 3
4-23.
a. 2
4-25.
OUT: -2.5, -2, -1.5, -1, -0.5, 0, 0.5
b. 0
4-22. a. OUT: 5, 0, -3, -4, -3, 0, 5, A parabola opening upward. 4-24. 35 cm long and 15 cm wide.
Lesson 4.1.4 4-26.
a. They all have something added and/or multiplied by x. b. how each figure changes c. The number of tiles in the 0th figure.
4-27.
y = 3x + 1
4-28. a. y = 5x ! 2 b. i: y = 2x + 5 , ii: y = !3x + 7
4-29.
Answers vary
4-30. Answers vary
4-33.
If x is my number, then 5 ! 2x = !1 ; x = 3 .
4-34.
IN: 5, -5, 4, 1.5, 50; OUT: -11
4-35.
a. Answers vary
4-36.
Figure 0 has 7; figure 1 has 9; the growth factor is 2.
Answer Key
4-32. a. all numbers
b. x =
8 3
a. Multiply x by three and subtract two
b. Each figure has 4 more tiles than the figure before it.
23
Lesson 4.1.5 4-37.
a. y = 4x + 4
4-39.
a. 4
4-40.
a. 1 2
4-41.
c. y = !2x + 18
b. y = 9x ! 5
1
b. no solution
c. 3.5
b.
c.
10
2
–2
2.5
–5
-2 -8
d.
5 –5
–7
See table below. x -4 -3 y -14 -11
d. 1
–1
8
–6
-1 -5
0 -2
1 1
2 4
3 7
–112 –14 –6
4 10
b. Yes. These values for the variables make the rule true 4-42.
Answers vary. Typical statements include: “Figure 4 has 32 tiles,” “Each figure has 7 more tiles than the previous figure,” “Figure 0 has 4 tiles,” and “The rule is y = 7x + 4 .”
4-43.
a. !5x ! 1
4-44.
Answers vary
4-45.
a. any number
4-46.
a. This should be a horizontal line with positive y that ends at x = 10. b. This should start with positive y and have 0 slope, then turn to a negative slope, then again to 0 slope at the x- axis. c. This should start horizontally along the line x = 0, then rise quickly to a higher horizontal line, then descend to a lower positive horizontal line, ending at x = 4 .
4-47.
b. Yes, -29 = 3 – 32.
4-48.
24
Number of Letters 10 2 15 7 1 500 42
b. 11x ! 1
b. 0
c. 5x ! 1
c. –6
d. !4x ! 1
d. no solution
Cost of Stamps $3.40 $0.68 $5.10 $2.38 $0.34 $170 $14.28 Algebra Connections
Lesson 4.1.6 4-49.
a. y = 2x + 2
b. y = 4x + 3
c. y = 8x + 5
d. y = !2x + 8
e. y = 4x + 8
f. y = !4x + 12
4-50.
a. (0, 3)
b. (0, 0)
c. (0, 8)
d. (0, –1)
4-51.
a. y = 4x + 3
4-52.
The pattern shrinks by two tiles between figures. Figure 0 has 13 tiles.
4-55.
a. A and C have 5 tiles, B has 15 tiles.
c. y = 2
b. y = !3x + 5
b. C grows fastest at about 13 tiles per figure. It’s steepest. c. Figure 2; the lines intersect at (2, 20). d. y = !10x + 30 4-56.
a. They each have 5 tiles.
b. The second
4-57.
a. 4
b. 34
4-58.
9 43 tsp
Lesson 4.1.7 4-59.
a. y = 3x + 2
4-60.
Figure 19
4-61.
y = 4x ! 8
4-62.
a. x: (1.5, 0), y: (0, 3) b. x and y: (0, 0) c. no x-intercept, y: (0, 3) d. x: (6, 0), y: (0, 3)
4-64.
a. 5
4-65.
Figure 6; 40 = 6x + 4
4-66.
Josie is correct. 10 inches long
Answer Key
b. y = !3x + 14
b. –8
c.
7 6
d. -18
25
Lesson 4.2.1 4-67.
Evie is the traveling toward Fairbanks so she is the circle graph. Joyla is moving away from Fairbanks so she is the triangle graph; Between 90 – 100 hours; 1100 miles; Evie’s line is steeper; Extend the line to 1100 miles and look at the time, about 230 hours.
4-68.
b. (2, 2)
4-69.
a. As San Francisco warms up in the spring, Rio is cooling. b. The cost of machine-made tortillas decreases as more are produced, whereas the handmade tortillas’ cost stays the same c. Boston’s pollution increases more quickly than Denver’s as the population increases. d. The hare catches up to the tortoise and then passes it even after starting later than the tortoise.
4-71.
15 ÷ 8 · 20 = 37.50 minutes
4-72.
a. 4896 sq. units, b. 91 + 13x sq. units
4-73.
a. Leslie passes Gale
c. 900 sq. units
b. 2:00 PM
d. 336 sq. units
c. 18 miles e. at approximately 6:40 PM
d. Gale stopped to rest, flat tire, etc. 4-74.
a. y = 3x ! 1
b. y = 5x + 2
4-75.
a. the opposite of y increased by 8 c. the square of the sum of x and 3
b. 48 less than two times x d. !6x 2 e. x ·x + 5
Lesson 4.2.2 4-76.
They will have the same amount of money in 5 weeks
4-78.
a. y = 50 + 30x b. y = 75 + 25x c. When x = 5 is substituted into both equation, y = 200 .
4-79.
a. No common points appear in his tables. However, using the pattern in the tables, it is reasonable to predict that the lines intersect when x is between 1 and 2. b. (1.5, 0)
4-80.
a. Science: y = 20 + 10x ; Math: y = 50 + 5x
4-81.
a. x 2 + 5x + 6
4-83.
a. (10)(17) = 72 + 30 + 48 + 20 = 170
4-84.
1 x + 74 3
26
b. 4x + 10
b. 6 weeks
c. $80
4-82. y-int: (0, 1), x-int: ( 12 ,!0) b. 54
= 90 ; 48 days later (Thursday, June 19)
Algebra Connections
Lesson 4.2.3 4-85.
7 years
4-86.
a. 3x + 5 = x + 19
4-87.
a. Richmond: y = 1900 ! 15x , Post Falls: y = 1160 + 22x
b. x = 7 years
c. 26 pounds
b. 1900 ! 15x = 1160 + 22x ; x = 20 years
c. 1600 students
4-88.
Ficus: y = 6 + 1.5x , Oak: y = 2x ; x = 12 years; 24 feet
4-89.
The solution of the equation formed is the x-value of the point at which the two rules have equal y-values.
4-90.
8 dumplings, 12 eggrolls, 4 sweet buns and 4 drinks
4-91.
He is correct, because !2(2) + 4 = 0 and 2 ! 2 = 0 .
4-92. 4-93.
a. (2,1) b. (2,1) a. (-2,-7) b. ( 12 , -4)
4-94.
a. If x represents weeks and y represents total money, y = 20 + 6x and y = 172 ! 4x . b. They will have the same amount between weeks 15 and 16.
Lesson 4.2.4 4-95.
a. $1105 b. Town B increases by 6000 in one year. c. They will have the same in 52 weeks d. Answers vary, but the story should include the point of intersection at (4.5, 25). e. $3.50, 24 more
4-96.
She is not correct
4-98.
a. parallel lines
4-99.
a. y = 12, 200 + 300x and y = 21, 000 ! 250x ; y = total population and x = number of years b. They will have the same population in 16 years.
4-100.
a. y = 27 + 43x
4-102.
This parabola should point downward and pass through (0, 0) and (–2, 0). The vertex is at (–1, 2).
4-103.
1296
4-105.
If r is Regina’s favorite number, then 3r = 12 + 2r and r = 12 .
4-106.
a. parallel lines
Answer Key
4-97. a. (5,3) b. no solution
b. (2,-6)
c. parallel lines do not intersect
b. After the 22nd day.
4-104. a. (0, 3)
b. no solution
4-101. –7
5 ) b. (0, ! 12
c. (0, 12)
c. Parallel lines do not intersect. 27
Chapter 5 Lesson 5.1.1 5-1.
b. x + 4 and x + 2
c. x 2 + 6x + 8 = (x + 2)(x + 4)
5-2.
a. (x + 1)(x + 2)
b. 3(2x + 5)
d. (x + 1)(3x + 1)
e. (2x + 6)(x + 2) or (2x + 4)(x + 3)
f. 2y(y + 3)
g. (y + x)(y + 2)
i. (x + y + 1)(x + y + 2)
j. (2y + 1)(x + 2)
5-4.
77 + 56 + 33 + 24 = 190
5-5.
y = 7x + 5
5-6.
(2x + 4)(x + 2) = 2x 2 + 8x + 8
5-7.
Answers vary.
5-8.
No, he is not correct. Parallel lines do not have a point of intersection.
5-9.
62,000 votes
c . (2x + 3)(x + 2) h. (x + 1)(y + 1)
Lesson 5.1.2 5-10.
a. (2x + 3)(x + 2) = 2x 2 + 7x + 6 b. (y + x + 2)(x + 4) = xy + x 2 + 6x + 4y + 8
5-11.
a. 8x 2
b. 2x 2 + 7x + 3
d. 4x 2 + 4x + 1
e. 2x 2 + xy
c. 2x 2 + 10x
f. 2x 2 + 2xy + 9x + 5y + 10
g. 6x + 10
h. 2xy + y 2 + 3y
5-13.
a. 12x 2 + 10x
c. 12xy + 9y
d. 70xy + 77y 2
5-15.
a. (x + 1)(x + 3) = x 2 + 4x + 3
b. (2x + 1)(x + 2) = 2x 2 + 5x + 2
5-16.
a. 238
b. 112
5-17.
a. Multiply by 6
b. x = 15
c. x = 4
5-18.
a. m = 3
b. (0, –2)
c. y = 3x ! 2
5-19.
10,000 + 1,500x = 18,000 + 1,300x, x = 40 months
5-20.
x-intercepts are (1.5, 0) and (–1, 0), y-intercept is (0, –3).
28
b. 24x + 6
Algebra Connections
Lesson 5.1.3 5-21.
a. !18x + 12
b. 4x 2 ! 2x
c. 50t ! 15t 2
d. !32 + 24k ! 4y
5-22.
(4x + 5)(2x + 3) = 8x 2 + 22x + 15
5-23.
The area inside each smaller rectangle of the generic rectangle corresponds to the tiles in the same portions of the rectangle in problem 5-22.
5-24.
a. (3)(y + 5) = 3y + 15
b. (x)(2x) = 2x 2
c. (x + 5)(2x ! 3) = 2x 2 + 7x ! 15
d. (4y ! 7)(6y ! 1) = 24y 2 ! 46y + 7
a. 3x 2 + 17x + 10
b. 10y 2 ! 11y ! 35
c. 18xy ! 33x
d. 15w 2 ! wp ! 2 p 2 ! 20w + 8 p
a. y(x + 3) = xy + 3y
b. (x + 12)(x + 5) = x 2 + 17x + 60
5-25.
5-26.
c. (x ! 2)(2x ! 3y + 5) = 2x 2 ! 3xy + x + 6y ! 10 d. Multiple answers possible 5-27.
a. 2x 2 + 17x + 30
b. 3m 2 ! 4m ! 15
c. 12x 2 ! 59x ! 5
d. 6 ! 7y ! 5y 2
5-28.
a. 6
b. 16
c. 1.5
d. 8
5-29.
a. –4, 0
b. –16, 6
c. 2, 14
d. 20, –1
5-30.
a. y = !2x + 1
5-31.
30 ounces
5-32.
These expressions are equivalent because of the Commutative Properties of Addition and Multiplication.
Answer Key
b. x: (0.5, 0), y: (0, 1)
c. y = !2x
29
Lesson 5.1.4 5-33.
5-34.
a. 12x 2 + 8x ! 55
b. !30x 2 + 6x
c. !y 2 + 4y + 12
d. 48 ! 16m 2
This can be built with 3 rows, each containing 2 x-tiles and 1 unit tile a. 4 b. 3(2·4 + 1) = 8·4 – 5
5-35.
A and C are correct.
5-36.
x = !2
5-37.
a. x = 9
b. x = 0
c. y = 6
d. x = !3.5
e. x = 2
f. x = ! 10 3
5-38.
a. –1
5-39.
The equation in part (b) has no solution. Possible reason: There are the same number of x-terms on each side of the equation, so if you try to solve, you end up with an equation such as 11 = 14 , which is impossible.
5-40.
Rena is correct.
5-41.
y = !2x + 13
5-42.
a. After 3 hours
5-43.
(–2, –1)
5-44.
No, on average he will make about 375 baskets.
30
b. 5
c. –2.5
d. 8
b. 8
Algebra Connections
Lesson 5.1.5 5-45.
a. 5 feet, the y-intercept is the constant b. 4 feet per year, the rate is the coefficient of the x.
5-46.
b. y = 3x + 5 c. y-intercept is (0, 5); growth factor is 3.
5-47.
See table below. a. b. c. d. e. f.
5-48.
5-50.
y = mx + b form y= x!7 y= x+2 y = !5x ! 2 y=x y=0 y = !3x ! 4
Growth factor 1 1 –5 1 0 –3
y-intercept (0, –7) (0, 2) (0, –2) (0, 0) (0, 0) (0, –4)
a. y = 5
b. x = 5 ! 52 y
d. x can be any number
e. x =
g. y = x ! 3
h. q =
a. 4x 2 + 17x + 15
b. !6x 2 ! 20x ! 16
c. !3xy + 3y 2 + 8x ! 8y
d. 3xy + 5y 2 ! 22y ! 12x + 8
y+5 2
5-51.
a. x =
5-52.
y = 3 + 2x
5-53.
b. no solution
b. w =
p!9 !3
!1 y + 2 3 !37 3
c. m = 2n + 5
c. y = 6x ! 11 f. p =
m 2
+4
d. y = !3x
c. there is no solution to the system of equations because the two lines do not intersect 5-54.
Answer Key
100 ÷ 4 = 25, 3,000 ÷ 25 = 120 gallons
31
Lesson 5.1.6 5-55.
b. y = 12 x ! 2
a. x = 3
f. y = !x + 3 g. x =
y! 4 !3
c. x = 3
d. y = ! 12 x + 4
e. x = –2
h. x = 2
i. w = 3 ! 2v
j. no solution
c. y = 0
b. x = 2
5-57.
a. x = 5
5-58.
(x + 3)(x ! 5) = x 2 ! 2x ! 15
5-59. 5-60.
See graph at right. The numbers are 55 and 60.
5-61.
a. 6x + 10 = 10x , after 2.5 hours
5-62.
! 1222
d. x = 38 y 4
–3
b. 25 problems
3
x
Lesson 5.2.1 b. 12
c. 4.5
d.
12 7
5-63.
a. 18
5-64.
a. 34 people
5-65.
a. Ratios vary b. No c. When ratios are of corresponding parts, the ratios are equal 68 = x d. The ratios used will vary. One example: 100 . Therefore, 527
b. Yes
c. 238,085 votes
x = 358.36 , so Mr. Mears would probably receive roughly 358 votes from Carina’s neighborhood. 2 17
=
x 85
5-66.
a.
5-67.
19.9 minutes
5-68.
9.6 weeks; 179.2 pounds
5-69.
answers vary
5-70.
a. x 2 + 9x + 20
5-71.
a. 3x 2 + 5x + 13 d. 5x
32
, x = 10 minutes
b.
8 29
x = 1490 , x ! 411 students
b. 2y 2 + 6y b. 5x 2 + 16x + 5 e. 4x 2 ! 11x ! 2
c. !x 2 + 4x + 6
Algebra Connections
Lesson 5.2.2 5-72.
a. 30
b. 16
5-73.
See table below: # of Cans 50 1,240 b.
50 1240
=
0.77 x
c. 32
d. .3
Weight (kg) 0.77 x , x ! 19.1 kg
b. $74.95
c. ! 31.9 kg
5-74.
a. $8
c. 10 days (rounded up to nearest whole day)
5-75.
15 units
5-76.
74 184
5-77.
a. x = !5
5-78.
! 574 times at bat
5-79.
a.
5-80.
a. (3, 5)
5-81.
a. Yes, because the sum of two even numbers is always an even number. If one even number is written as 2m and the other as 2n , then the sum is 2m + 2n = 2(m + n) , which is even.
= 4x , x ! 1.6 inches
1 32
b. y = 2x ! 3
= 8x , x = 256 miles
c. no solution
b.
g 118
=
1 32
d. y = !3x + 5
, g ! 3.7 gallons
b. Yes, because the product of two even numbers is always even. If one even number is written as 2m and the other as 2n , then the product is (2m)(2n) = 2(2mn) , which is even. 5-82.
a. 2x(x + 5) = 2x 2 + 10x
b. (2x + 5)(x + 3) = 2x 2 + 13x + 15
Lesson 5.2.3 a. x 2 + 10x + 16
b. 2m 2 + 70m + 600
c. xy + 10x
d. 6x 2 + 17x + 12
5-86.
a. 0.08 feet or about 1 inch
b. ! 42 feet
5-87.
a. 11.2
b. 7.5
c. 3.75
d. –0.25
5-88.
(2, -5)
5-89.
a. y = 4x ! 3
b. y = x + 3
c. y = ! 23 x + 6
d. y = ! 23 x + 2
5-85.
Answer Key
33
Chapter 6 Lesson 6.1.1 6-1.
a. There are 23 students in the class. b. There are three fewer boys than girls in the class. c. There are 13 girls in the class.
6-2.
a. 3p + 8
6-3.
Answers vary, but typical responses: a) the value of some quarters and nickels is $5, and b) The area of a rectangle is 30.
6-4.
a. Mountain View has 100 more students because 100 is subtracted to get the number of students at Ferguson. b. The total number of students served by these two high schools is 5980. c. Chapter 2 is the longest, Chapter 3 is the shortest.
b. 3p + 8 = 176
d. p + p + 12 +
p 2
c. 56; 3(56) + 8 = 176
= 182 , p = 68 pages in Chapter 1.
6-5.
a. 2m ! 10
6-7.
side #1 = side #2 = 8 cm and side #3 = 2 · 8 – 1 = 15cm
6-8.
a. x ! 342
6-9.
a = !3, !!b =
6-10.
no, when x = 12, y = 102 so it would have 102 tiles.
6-11.
21.1 minutes
6-12.
a. y = !2x +
34
b. 5c + 2 p = 9.50
c. 4(a + p + b) = 84
c. 3c + 400
b. 2w !1 2
2 3
b. Yes
c. growth factor: –2, y-intercept: (0,
2 3
)
Algebra Connections
Lesson 6.1.2 6-13.
C, A, D, B
6-14
a. Side #2 = x, Side#3 = 2x ! 1 b. x + x + (2x ! 1) = 31 c. x = 8, so Side #1 = Side #2 = 8 and Side #3 = 2 · 8 – 1 = 15; Yes.
6-15.
a. If m = the number of months they have saved money, then 15, 000 + 1000m = 12, 000 + 1300m , and m = 10 months b. If s = the number of sheep, then (21 – s) = the number of chickens. Then 4s + 2(21 ! s) = 56 , and s = 7 sheep. 21 – 7 = 14 chickens c. If p represents the number of pencils Mr. Williams usually orders, then 2 p + 12 = 60 ; p = 24 pencils d. If g represents the number of CDs that George buys, then 15.95g = 13.95g + 8 ; g = 4 CDs e. If s represents the number of slices on an extra-large pizza, then 4s + 3 = 51 ; s = 12 slices
6-16.
a. no solution
6-19.
Lakeisha, Samantha, Carly, Barbara, and Kendra
6-20.
She combined terms from opposite sides of the equation. Instead, line 4 should read 2x = 14. Then x = 7 is the solution.
6-21.
This statement is false because the Distributive Property states that a(b + c) = ab + ac .
Answer Key
b. x = 13
6-17
(-1, 3)
6-18. 31
35
Lesson 6.1.3 6-22.
a. 17 cans and 4 bottles b. Let b = number of bottles, then 4b + 1 = number of cans; 10(4b + 1) + 12b = 218 c. c = 17 cans; yes d. Let b = number of bottles and c = number of cans, then c = 4b + 1 and 10c + 12b = 218 e. 17 = 4(4) + 1 and 10(17) + 12(4) = 218
6-23.
a. 8t + 16c = 400 , t = 5 + c b. t = 50 ! 2c , t = 5 + c c. t = 20, c = 15
6-24.
2y + 8x = 10 becomes y = ! 4x + 5 ; (–2, –13).
6-25.
a. (6, 1)
6-26.
a. t ! 4 ; 2(t ! 4)
6-27.
If Nina has n nickels, then 5n + 9 + 5(2n) = 84 , and n = 5 nickels.
6-28.
8 x
b. (–2, 8) b. 150 ! c
c. 14.95c + 39.99v
3 ; x = 29 1 , so 29 falcons is a good estimate. = 11 3
6-29.
2. Associative Property, 3. Combining like terms, 4. Additive Property of Equality, 5. Combining like terms
6-30.
No; 2 is a prime number and it is even.
6-31.
a. H: !x = 8 , C: x = !8 ; Yes it is true because the equality is maintained when you take the opposite of both sides of the equation b. H: 3x + y = !11 , C: 6x + 2y = !22 ; Yes it is true because the equality is maintained when both sides of an equation are multiplied by the same number c. H: “Tomas runs at a constant rate of 4 meters every five seconds”, C: “he will run 50 meters in 1 minute”; No, this statement is false.
36
Algebra Connections
Lesson 6.2.1 6-32.
(–11, 4)
6-33.
a. Yes; the two quantities are equal b. Yes; again, we can switch these values because the top equation indicates that they are equal c. x = !11 , y = 4
6-34.
a. x = 4 , y = 12 b. x = 3 , y = !1 c. no solution d. b = !3 , c = !8
6-35.
Yes, she is correct. To test, substitute the values for x and y into both equations to see if they are correct solutions.
6-36.
There are 28 red and 56 green marbles.
6-37.
a. 0
6-38.
a. #2 b. 4 touchdowns and 9 field goals.
6-39.
a. The graphed line should be y = !2x ! 3 . b. Yes; (–3, 3) and (–2, 1) both make this rule true.
6-40.
Katy is correct because the “6x – 1” should be substituted for y because they are equal.
6-41.
upside-down parabola with x-intercepts (-2, 0) and (5, 0) and y-intercept (0, 10)
6-42.
No. When -2 is substituted into the equation, the equation is false.
Answer Key
b.
16 17
c. 10
d. 2
37
Lesson 6.2.2 6-44.
a. 2y + 1x = 40 b. Answers vary and could include: 10 yodelers and 20 xylophones, 15 yodelers and 10 xylophones c. Answers could include (20,10), (10,15), (4,18), (30,5), (12,14), (14,13) etc. d. No. e. Yes
6-45.
a. y = 2x c. (8,16)
6-46.
a. a line; 2y + x = 40 or y = ! 12 x + 20 b. When each point on the line is substituted into the equation, it makes the equation true c. The line y = 2x should be graphed. d. (8, 16); It makes both equations true e. Answers vary. Common methods: as a point (x, y), as a statement (such as “x = … and y = …”), or as a sentence (such as, “The club had 16 yodelers and 8 xylophones.”)
6-47.
a. x = 6, y = 3 b. See table below. c. The line should have a y-intercept of (0, 9) and slope of –1. x y -2 11 -1 10 0 9 1 8 2 7 3 6 d. (6, 3)
b. (1,2), (2,4), (3,6), (4,8), etc. d. it makes both equations true
e. Yes
6-48.
They would be written in the form (a, b, c), or x = #, y = #, and z = #.
6-50.
Yes, each point makes the equation true.
6-51.
a. no solution
6-52.
a. h = 2c ! 3
6-53.
three dice and two jacks
6-54.
Yes. Adding equal values to both sides of an equality preserves the equality.
6-55.
3y(y ! 4) = 3y 2 ! 12y
38
b. x = 5, y = 2 b. 3h + 1.5c = 201
c. 28 corndogs were sold.
b. (3y ! 4)(y + 5) = 3y 2 ! 7y ! 20 Algebra Connections
Lesson 6.2.3 6-56.
(2, -2)
6-57
b. Equal amounts are being added to both sides. Therefore, both sides remain equal. c. 7x = 14 , x = 2 , the y-terms were eliminated when simplified. d. y = !2 , yes f. x = 1 , y = 4
6-58.
a. If b represents the number of bass and t represents the number of trout, then 3b + t = 30 , 5b ! t = 42 b. Yes – one variable (the variable representing trout) will be eliminated when the equations are combined. c. Pat caught 9 bass and 3 trout
6-59.
x = ! 12 , y = 2
6-60
a. (3, 4)
b. (11, 2)
c. (-1, 1)
6-61.
a. (-5, 1)
b. (3, 1)
c. no solution
6-62.
a. infinite solutions
b. lines coincide
6-63.
a. Let p represent the number of pizza slices and b represent the number of burritos sold. Then 2.50 p + 3b = 358 and p = b ! 20 b. 31 pizza slices were sold.
6-64.
$24.58
6-65.
y = 3x + 3
6-66.
a. x 2 ! 3x ! 10 c. !3xy + 3y 2 + 8x ! 8y
Answer Key
b. y 2 + 5xy + 6x 2 d. x 2 ! 9y 2
39
Lesson 6.2.4 6-67.
a. !3x = !15 , x = 5 , y = !3
6-68.
a. No variable is eliminated. c. (3, 4) d. No – Multiplying the top equation by 2 created a zero with the y-terms.
6-69.
a. m = !4 , n = 5 c. x = !1 , y = 6
b. a = 2 , b = 9 d. infinite solutions
6-70.
Answers vary, but one possible strategy is to multiply the top equation by 4 and the bottom equation by 3. Once strategies are presented, solve the system with the class. Solution: (1, 2).
6-71.
a. (3, 1)
6-72.
These lines coincide.
6-73.
17, 18, and 19
6-74.
a. H: y = 23 x ! 5 , C: the point (6, –1) is a solution; yes this is true because
b. (0, 4)
c. (10, 2)
d. (-4, 5)
!1 = 23 (6) ! 5 . b. H: Figure 2 has 13 tiles and Figure 4 has 15 tiles, C: the pattern grows by 2 tiles each figure; No, this is not correct. The tile pattern grows by one tile each figure. c. H: (3x + 1)(x ! 2) = 4 , C: 3x 2 ! 5x ! 2 = 4 ; yes, this is correct, as can be shown with a generic rectangle. 6-75.
They are both correct. The lines coincide.
6-76.
y = 2x + 5 , 105 tiles
40
Algebra Connections
Lesson 6.2.5 6-77.
a. If c is the number of capped bottles and b is the number of broken bottles, then c + b = 15 and 4c ! 2b = 6 . b. Erica has capped 6 bottles and broken 9.
6-78.
a. substitution e. equal values
b. elimination f. elimination
c. equal values g. elimination
d. substitution h. substitution
6-79.
a. (2, 1) e. (2, 5)
b. (1, -2) f. no solution
c. (-0.5, 0.5) g. (0, -2)
d. infinite solutions h. (10, 7)
6-81.
a. (0, 13 )
b. (-6, 2)
c. No solution
d. (11, -5)
6-82.
2n = p and n + p = 168 , 56 nectarines are needed.
6-83.
a. Yes, because these expressions are equal. b. 5(3y) + y = 32 , y = 2 , x = 3.5 8 = 18 , x=
6-84.
x 8
6-85.
y = 3x
6-86.
It can be concluded that line l is parallel to (or coincides with) line n because all three lines must have the same slope.
Answer Key
32 9
41
Lesson 6.3.1 6-87. Antenna: Leg:
Ant 2 mm 4 mm
Beetle 6 mm 10 mm
b. This rule is correct. 6-89.
Grass Hopper 20 mm 31 mm
c. When x = 1 mm, y = 2.5 mm (2x ! 3)(y + 3x ! 5) = 2xy + 6x 2 ! 19x ! 3y + 15
6-88.
Answers vary
6-90.
a. TeleTalk, 40¢ b. TeleTalk: y = 8x, Americall: y = 30 + 5x, CellTime: y = 60 + 3x c. 10 min, 15 min d. between 10 and 15 minutes
6-91.
a. $18
6-92.
a. none
6-93.
He should get no solution. Lines A and B are parallel, while B and C coincide. That means that A and C are also parallel.
6-94.
Stevie is 6, Joan is 11, and Julio is 14.50. The possible contexts are varied. For example, this could be the price of CD’s by famous artists.
b. She sold 19 brownies. b. one ( t = 3 )
Stevie 3 10 7.50
6-95.
Joan 5 19 14
Number of years at company Salary per hour
c. one ( m = 0 )
Julio 8.50 22.50 17.50
1 $7.00
Total 16.50 51.50 39.00
3 $8.50
d. infinite
31.50? Too low Too high Too high
6 7 $10.75 $11.50
a. $15.25 b. $6.25 · 20 hours = $125 6-96.
(a), (b), and (c) all create equivalent equations. Part (d) is not legal because unless x = 1, -x + 1 " 0.
6-97.
76 roses were sold.
6-98.
x 2 + 5 = x 2 + 2x + 1 , x = 2 and y = 9
6-99.
a. all numbers
42
) b. ( 13 , !3 2
c. (1, 2)
d. (8, 7)
Algebra Connections
6-100.
a. Line b. Answers will vary. Possible solutions: (0, 2), (1, 5), (2, 8), … c. y = 3x + 2 , Yes, the points are the same.
6-101.
y = 2x + 6 , 206 tiles
6-102.
(-1, 0) and (2, 0)
6-103.
Mr. Greer incorrectly distributed. The correct solution is x = 2.
6-104.
n + d = 30 and 0.05n + 0.10d = 2.60 , so n = 8. There are 8 nickels.
6-105.
a, b, and d are correct.
6-106.
y = !5x + 3
6-107.
a. x = 2.2
6-108.
Answers vary, but the answer should have the same number of x-terms on both sides of the equation and the constants on each side should not be equal.
6-109.
It can be concluded that y = –2, because 2(0) ! 3(!2) = 6 .
6-110.
C
Answer Key
b. x = 8
c. x = –10.5
d. x = 0
43
Chapter 7 Lesson 7.1.1 7-2.
Customer A should order y = 4x ! 3 instead; Customer B should order y = !3x + 2 instead; Customer C’s order is correct; Customer D’s table is not linear, so the customer should revise his or her order; Customer E’s order is correct; Customer F’s order is not linear either, since a line must have constant growth everywhere.
7-4.
a. See diagrams at right. b. y = 5x + 1 ; 51 tiles
Figure 0
b. 1
Figure 4
7-5.
a. 3
c. 2
7-6.
a. 250 ÷ 3 ! 5 " 416.7 pounds of food
7-7.
x: (12, 0) and y: (0, 6)
7-8.
a. y = 8 ! 6x
7-9.
Let t = number of teddy bears and d = number of dogs. Then t + d = 356 and d = 2t + 17 , so t = 113 and d = 243 . Thus, he has 113 teddy bears.
b. x = ! 16 y +
8 6
b. 700 ÷ 45 ! 15.6 hours
c. x = 3
Lesson 7.1.2 7-12.
Answers can vary depending on data and the trend line that is chosen.
7-13.
a. 16
b. 2
7-14.
a. 5 e. 38
b. 21 c. the positive value of x ! c f. the positive value of y ! f
7-15.
a. –3
b.
7-16.
a. (3, –0.25)
b. infinite solutions because the lines coincide
7-17.
x = 15 , y = 9
7-18.
a. b. c. d.
44
c. undefined
d. 0 d. 36
1 2
The baby is growing 1.5 inches per month; it is the slope (m). It was 23 inches long when it was born; it is the y-intercept (b). 27.5 inches 11 months after it was born, so in December Algebra Connections
Lesson 7.1.3 7-19.
a. Both have 3 tiles in Figure 0. However, A grows by 2 tiles and B grows by 6 tiles. B is steeper. c. Typical response. It tells how much a pattern grows each figure. The steeper the line, the greater or faster the growth. d. A. y = 2x + 3 , B. y = 6x + 3
7-20.
27 ÷ 3 = 9
7-21.
a. 2
7-22.
a. 6 ÷ 4 = 3 ÷ 2 = 1.5 ; Both give the same result. b. 4.5; The ratio must be equal. c. 1.5; This height is equal to the slope.
7-23.
If !y = 0 , then the line is horizontal; if !x = 0 , then the line is vertical.
7-24.
a. Essie is correct; !y = "3 means that the vertical change is 3 and that the line is pointing downward. = ! 43 b. yes; !3 c. y = ! 43 x + 9 4
7-25.
Possible response. It is a parabola because it has an x 2 -term. (This conclusion is acceptable at this point.)
7-26.
r = 3y and r + y = 124 ; r = 93 and y = 31
7-27.
any horizontal line that is not y = 0
7-28.
a. 5.5
7-29.
Let x represent the number of hours she swims. Then x + 2x = 4.5 and x = 1.5 . Thus, she will swim for 1.5 hours and play volleyball for 3 hours.
7-30.
a. 5x 2 ! 32x ! 21
Answer Key
b. 4
c.
b. 42
1 2
or 0.5
c. ! 25
d. divide !y by !x
e. y = 12 x + 2
d. 2
b. !24x 2 + 18x
45
Lesson 7.1.4 7-31.
b. Downward; !y is –2.
a. B is the steepest; C is the least steep.
c. A. m = 1, B. m = 3, C. m = 12 , D. m = ! 25 d. Possible responses. The value of slope indicates how steep the graph is. The larger the slope, the steeper the line. If the slope is positive, the line travels upward from left to right, while a negative slope indicates that the line travels downward from left to right. e. In this case, she is correct. The steepest line (B) has the largest value for slope (3). Note that for negative slope, the steeper the line, the more negative (smaller) the slope. 7-32.
a. A is the steepest; B is steeper than C. b. A. !x = 1, !y = 2; B. !x = 2, !y = 3; C. !x = 3, !y = 2; D. !x = 5, !y = –1 c. A. m = 2, B. m = 23 , C. m = 23 , D. m = ! 15 d. It would point downward from left to right because its slope is negative. e. It would point downward from left to right because its slope is negative. It would be steeper.
7-33.
A. m = ! 12 , B. m = –2, C. m = 23 , D. m = 0
7-34.
a. The line should have slope m = 6 .
7-35.
a. The slope of each is
7-37.
a. line a. y = 2x ! 2 , line b. y = 2x + 3 b. It would lie between lines a and b, because its y-intercept is at (0, 1). c. It would travel downward but would have the same y-intercept as the line from part (b).
7-38.
a. (–1, –7) b. ( 12 , 2) c. These coincide, so there are an infinite number of points of intersection.
7-39.
a. 12
7-40.
a. 6(13x ! 21) = 78x ! 126 c. 4(4x 2 ! 6x + 1) = 16x 2 ! 24x + 4
7-41.
55 hops
7-42.
y=
46
4 3
b. 0
1 2
.
b. no
c. –8
b. m =
3 5
c. !y =
c. m = ! 23
d. m = 0
1 2
d. no solution b. (x + 3)(x ! 5) = x 2 ! 2x ! 15 d. (3x ! 2)(x + 4) = 3x 2 + 10x ! 8
x!4
Algebra Connections
Lesson 7.1.5 7-43.
a. This is not enough information. An infinite number of lines pass through (2, 5). b. This is enough information. y = !3x . c. This is enough information. y = 2x + 4 . d. This is not enough information. An infinite number of lines, all parallel to each other, have slope of 4. e. This is enough information. y = ! 43 x + 5 . f. You need the slope and a point or two points to determine a unique line.
7-44.
a.
1 2
; parallel lines must have the same growth factor.
b. They are equal. d. y = 12 x ! 6
c. It should have a slope of –2 and pass through (0, –5). 7-45.
a. She subtracted the x- and y-coordinates. b. ! 23 c. 42 = 2 d. ! 63 = ! 12 d e. The student did not make !x negative and forgot to divide !y by !x .
7-46.
The steepest line is vertical. Its slope is undefined since its !x is 0.
7-47.
No; she would be on the same page as Sam on page 310, but there are only 300 pages in the book. Sam only needs 29 minutes and 45 seconds to finish the book, while Jimmica needs 30 minutes.
7-48.
a. See graph at right.
7-49.
a. m = !2
b. m = 0.5
c. undefined
7-50.
a. 4
b. no solution
c. 0.5
7-51.
a. y = 2x – 4
b. x =
7-52.
x-intercepts. (–2, 0) and (4, 0); y-intercept. (0, –24)
y
b. (3, –2)
y! 4 !2
3
x –2
d. 4
= ! 12 y + 2 c. x = 1
d. x = –7
Lesson 7.2.1 7-55.
A person cannot be different distances from the motion detector at the same time.
7-56.
$2,982,000
7-58.
a. There is no solution, so the lines do not intersect. same slope. y = 23 x ! 10 3
7-59.
y = 3x ! 2 , y = x
7-60.
Answers vary. Possible solutions. 5x(3x ! 1) , 5(3x 2 ! x) , x(15x ! 5) , etc.
7-61.
a. ! 43
Answer Key
7-57. a. 4
b. (0, –5)
b. 16
c. It would get steeper. b. Yes; both lines have the
c. y = ! 43 x ! 5 47
Lesson 7.2.2 7-62.
a. Leslie. y = 2x , Kristin. y = 25 x + 8 , Evie. y = 54 x + 6 b. After 8 seconds, both racers were 16 meters from the starting line. c. Leslie won in 10 seconds, Kristin took 30 seconds and Evie took 11.2 seconds. d. Her slope is 25 . Her slope represents her rate of travel. e. She traveled 2 meters every 3 seconds and got a 1-meter head start.
7-63.
a. The customer started walking slowly toward the motion detector from 4 feet away, stopped for about 4 seconds, and then walked away from the machine at a faster rate b. The customer started from 3.5 feet in front of the machine and walked quickly away, stopped for about 4 seconds, and then started to walk slowly toward the machine c. The customer started walking quickly toward the machine from about 10 feet away, after 3 seconds, stopped for 4 seconds, and then started walking slowly toward the machine.
7-64.
a. The slope represents the number of feet a tree grows per year, and the y-intercept represents the height when the measuring began (perhaps when it was planted), m = 2 feet per year, b = (0, 4) b. The slope represents the amount of money being spent from a bank account each month, while the y-intercept represents the beginning balance, m = !10 dollars per month, b = (0, 170) c. The slope represents the distance that can be traveled on a gallon of gas, m = 22 miles per gallon, b = (0, 0).
7-65.
a. Elizabeth won the race, finishing in 5 seconds. b. Barbara: y = 23 x + 3 , Elizabeth: y = 4x c. 5 meters every 2 seconds, or 52 meters per second d. 2 seconds after the start of the race, when each is 6 meters from the starting line
7-66.
a. 0 pounds b. The graph should show a line with positive slope. Units labeled on the axes should be in pounds (vertical axis) and feet (horizontal axis). c. 43 pounds d. " 9 feet
7-67.
x-intercept. (2, 0), y-intercept. (0, –10)
7-68. 7-70.
a. 1 b. 3 c. 2 7-69. y = 3x ! 2 Let t = number of t-shirts and s = number of shorts. Then 12t + 8s = 780 and t + s = 77 ; t = 41 and s = 36 ; they sold 41 t-shirts.
48
Algebra Connections
Lesson 7.2.3 7-71.
Rider A: y =
D Leslie A
3 x, 2 1 x + 10 , 2
C B
Distance from Starting Line (meters)
Rider B: y = Rider C: y = x + 4 ,
Rider D: y = x + 1 , 5 2
Elizabeth: y = 14 x + 10 , Leslie: y = 2x ! 4 ; Rider D wins the race. Solution graph is shown at right.
Elizabeth
Time (seconds)
7-72.
a. Rider D; Elizabeth b. Rider D. 5 meters every 2 seconds; Elizabeth. 1 meter every 4 seconds c. 8 seconds after the start of the race, Rider A, Rider C, Elizabeth, and Leslie were all 12 meters from the starting line.
7-73.
(–2, –15)
7-74.
a. (4, 0) and (0, –2)
7-75.
a. –1 b. ! 12 c. 23 d. ! 15 e. The line travels downward from left to right, so m = !1 .
7-76.
a. 4
7-77.
a. !12x
7-78.
C
Answer Key
b. 3
b. (8, 0) and (0, 4)
c. 1 b. 8x + 7
d. 2 c. x 2 ! 3
d. 5x + 7 ! 3y
49
Lesson 7.3.1 7-79.
While the graph will only approximate the solution, a table helps show that the chick weighed 51.6 grams when it was born and will weigh 140 grams after 17 days
7-80.
a. It grows 5.2 grams per day, so the slope of the line is 5.2. b. slope. m = 5.2 ; should pass through (9, 98.4) c. Units on the axes should be in grams (vertical axis) and days (horizontal axis). d. While the y-intercept is (0, 51.6), answers close to 50 grams are acceptable. e. when the chick is 17 days old
7-81.
a. The chick weighed 98.4 grams after 9 days. b. On Day 0, the chick weighed 51.6 grams. c. Colleen’s chick weighed 51.6 grams when hatched. The chick will weigh 140 grams when it is 17 days old. For this problem, a table can produce exact results, while the graph is approximate.
7-82.
a. The variable m is the slope, or the chick’s daily rate of growth, 5.2 grams per day; b is the y-intercept, or the chick’s weight when hatched, which is unknown. b. The slope is 5.2, so y = 5.2x + b . The y-intercept is still unknown. c. Possible response. A point on a line is a solution to the equation, so when substituting the x- and y-values into the equation, the equation is still true. d. Substituting x = 9 and y = 98.4 into the equation, the equation becomes 98.4 = 5.2(9) + b . After solving, b = 51.6 and y = 5.2x + 51.6 . e. The equation is exact. f. Starting with 140 = 5.2x + 51.6 , the solution becomes x = 17 days.
7-83.
a. y = !3x ! 5
7-84.
Mt. Everest was 8750.05 meters tall in the year 0; y = 0.05x + 8750.05 .
7-85.
a. y = 1.5x + 0.5 b. Answers vary, but solutions should lie on y = 1.5x + 0.5 .
7-86.
b. y = 0.5x ! 14
–11
–15 –1
11 10
1
–3
5 2
13 2 3 13 6 6
7-87.
a. ( 7, 11)
7-88.
a. 14
7-89.
a. (3.5, 0) and (0, –2.33)
7-90.
a. 3; (0, 5)
b. ! 54 ; (0, 0)
e. ! 43 ; (0, –1)
f. ! 15 ; (0, 6)
50
–14 2
–7 –5
b. (–8, –10) b.
5 2
c. –3 b. y =
d. 11 13 3
" 4.33
c. 0; (0, 3)
d. 4; (0, 7)
Algebra Connections
Lesson 7.3.2
Lesson 7.3.3 7-102.
a. m = 3 b. Substitute the point into the equation y = 3x + b in order to solve for b; y = 3x + 10 . c. No; both will give the same equation. d. Substitute both points into the equation to verify that both are solutions.
7-104.
b. Although there are many possible solutions, the points for 1961 and 1989 are good choices. This will create a line. y = 9.82x ! 19234.82 . c. See part (b) above. e. Answers will vary. In the case stated in part (b), the y-intercept is (0, –19234.82); Dizzyland did not exist in the year 0. f. Answers will vary. Approximately the year 2060.
7-105.
a. y = 15 x + 4 b. y = !3x + 376
7-106.
y=
If d = number of Democrats, then d + (d + 30) + 1 = 435 , and d = 202 Democrats. Thus, there were 202 + 30 = 232 Republicans.
7-108.
a. There are no x-intercepts; yintercept. (0, 10). b. (3, 1)
7-109.
a. –8 c. 3
b. –56 d. 6
7-99.
(–5, 0) and (3, 0)
7-110.
a. m = ! 56
b. m = 0
+ 5x ! 3 a. 2 b. 15x ! 33x c. 5x 2 ! 27x + 10 d. 300x ! 50
d. undefined,
7-100.
c. m = ! 56 e. m = – 4
f. m =
7-91.
b. ! 23 c. Answers vary. d. They are opposite reciprocals. e. ! 53 .
7-92.
a. y = 52 x ! 8
7-93.
line L. y = ! 16 x + 6 ;
b. y = 23 x + 1
line M. y = 23 x + 1 ; (6, 5) 7-94.
Lines A and E are perpendicular, so the slope of line E is ! 56 .
7-96.
a. 0.75 meters b. Dean. y = 43 x + 5 , Carlos. c. after 20 seconds y=x
7-97.
a. The slope represents the cost per mile for a ride in a taxi, m = $3.50 per mile; b. The slope represents the gallons per minute of water draining from a tub, m = ! 4 gallons per minute.
7-98.
7-101.
Answer Key
2x 2
B
9 4
x+9
7-107. ( 11 , –3) 3
1 4
g. m = 4 h. m = ! 65 ; Lines (b) and (d) are perpendicular; lines (e) and (f) are perpendicular; lines (a) and (c) are parallel. 51
Lesson 7.3.4 7-111.
Logo A: y = 2 , y = x + 2 , y = 12 x + 2 , y = ! 23 x + 6 , y = ! 12 x + 6 , y = 23 x ! 6 , y = ! 12 x + 10 ; Logo B: y = 3 , y = 2x , y = 43 x , y = !2x + 6 , y = ! 43 x + 6 , y = 43 x ! 3 , y = ! 43 x + 9
7-112.
a. (2y + 4)(3 + 5x) = 6y + 20x + 10xy + 12 b. (5 + 2x)(2x ! 3) = 4x 2 + 4x ! 15
7-113.
If b = number of brownies sold and c = the number of cookies sold, then 3b + 2.50c = 218 and b = c + 3 ; she sold 41 brownies.
7-114.
a. y = ! 23 x + 1
7-115.
a. The slope represents the change in height of a candle per minute, m = 0 cm per minute b. The slope represents the gallons per month of water being removed from a well, m = –900 gallons per month.
c. y = 23 x
b. y = –11
7-116. –4
16 –1
4
–4
3
7-117.
52
a. 15x 2
–33 –4
–8
b. 8x
–7 –3
11 8
c. 6x 2
–1
7 6
d. 7x
Algebra Connections
Chapter 8 Lesson 8.1.1 b. 18x 2 + 9x ! 2
8-1.
a. (x + 4)(y + x + 2) = xy + x 2 + 6x + 4y + 8
8-2.
a. (2x + 3)(x + 2) b. (2x + 1)(3x + 2) c. no solution d. (2x + y)(y + 3) ; Conclusion. Not every expression can be factored.
8-3.
a. (3x + 1)(2x + 5) = 6x 2 + 17x + 5 b. (5x ! 2)(y + 3) = 5xy + 15x ! 2y ! 6 c. (4x ! 3)(3x + 4) = 12x 2 + 7x ! 12
8-4.
The product of each diagonal is equal. 6x 2 ! 5 = 30x 2 and 2x !15x = 30x 2 .
8-5.
Diagonals: part (a) are both 30x 2 , part (b) are both !30xy , part (c) are both !144x 2 . Typical response: “The product of one diagonal always equals the product of the other diagonal.”
8-6.
(2x ! 3)(x + 2y ! 4) = 2x 2 + 4xy ! 11x ! 6y + 12
8-7.
a. 12x 2 + 17x ! 5 b. 4x 2 ! 28x + 49
8-8.
–80 10
12 –8
–3
2
8-9.
–4 –7
7
0 7
6x 2
–81
0 9
–9 0
2x
3x 5x
!7x 2 –7x x –6x
a. m = 2 , (0,!! 12 ) b. m = !3 , (0,!!7) c. m = ! 23 , (0,!8) d. m = 0 , (0,!!2)
8-10.
a. (0, –8); It is the constant in the equation. b. (–2, 0) and (4, 0); The product of the x-intercepts equals the constant term. c. (1, –9); Its x-coordinate is midway between the x-intercepts.
8-11.
a. !1
Answer Key
b. ! 7.24
c. ! " 4.24
53
Lesson 8.1.2 8-12.
a. (5x ! 2)(2x ! 7) = 10x 2 ! 39x + 14
8-13.
a. (2x + 3)(x + 1) b. One corner should contain 4x , while the other should contain 6x ; (3x + 4)(x + 2) . c. Their sum is 7x, and their product is 12x 2 . d. The product 12x 2 should be placed at the top of the diamond problem, 7x at the bottom, and terms 3x and 4x should be in the middle. e. (2x + 3)(x + 2)
8-14.
a. One corner contains 6x 2 , and the opposite corner contains 12. b. The product of the x 2 and units terms (in this case, 72x 2 ) goes on top, while the x-term ( 17x ) goes on bottom. d. (2x + 3)(3x + 4)
8-15.
a. (x + 3)(x + 6) b. (4x ! 3)(x + 5) c. (2x ! 3)(2x ! 1) d. not factorable because there are no integers that multiply to get !9x 2 (the diagonal of the generic rectangle) and add to get 5x.
8-16.
a. (x ! 6)(x + 2) d. (x + 4)(3x ! 2)
8-17.
a. x-intercepts (–1, 0) and (3, 0), y-intercept. (0, –3) b. x-intercept (2, 0), no y-intercept c. x-intercepts (–3, 0), (–1, 0), and (1, 0), y-intercept (0, 2) d. x-intercept (8, 0), y-intercept (0, –20)
8-18.
a. (0, –9); It is the constant in the equation. b. (3, 0) and (–3, 0)
8-19.
a. (6, 9)
b. (0, 2)
8-20.
a. x = ! 10 23
b. all numbers
8-21.
y=
54
1 4
b. !35x " ! 4x = 10x 2 "14 = 140x 2
b. (2x + 1)2
c. (x ! 5)(2x + 1)
c. c = 0
x + 400
Algebra Connections
Lesson 8.1.3 8-22.
Lesson 8.1.4 8-33. a. (3x ! 2)2 b. (9m + 1)(9m ! 1) c. (x ! 4)(x ! 7) d. (3n + 3)(n + 2) or (n + 1)(3n + 6) .
a. (x + 3)2 b. (2x + 3)(x + 1) c. not factorable d. (3m + 7)(m ! 2)
8-23.
a. (3x ! 2)(3x + 2) b. 4x(3x ! 4) c. (4k ! 3)(2k ! 1) d. 20(2 ! 5m)
8-24.
(2x ! 6)(2x + 1) or (x ! 3)(4x + 2)
8-25.
See table below
8-34. a. Yes, because there are two different arrangements of tiles that build a rectangle. b. Because there is a common factor of 3 in each of the terms of the original expression and in one of the two binomials in either of the two partially factored forms. c. (i) and (iii) both have common factors, so they could have more than one factored form.
Multiply
x!2
2x + 1
x+7
x 2 + 5x ! 14
2x 2 + 15x + 7
3x + 1
3x 2 ! 5x ! 2
6x 2 + 5x + 1
8-27.
s + 2s + s + 3 = 51 ; 12, 24, and 15 cm
8-28.
a. 9 units c. 10 units
8-29.
a. (k ! 2)(k ! 10) b. (2x + 7)(3x ! 2) c. (x ! 4)2 d. (3m + 1)(3m ! 1)
b. 15 units d. 121 un 2
8-35. a. 5 b. 5(2x 2 + 5x ! 3) c. Yes; 5(2x ! 1)(x + 3) . 8-36. a. 5(x + 4)(x ! 1) b. 3x(x + 3)(x ! 5) c. 2(x + 5)(x ! 5) d. y(x ! 5)(x + 2) 8-37. a. (2x + 5)(x ! 1) b. (x ! 3)(x + 2) c. (3x + 1)(x + 4) d. It is not factorable because no integers have a product of 14 and a sum of 5. 8-38.
8-30.
y=
(2, 5)
x!3
8-39. a. in 7 weeks b. Joman will score more with 1170 points, while Jhalil will have 970.
8-31.
y = !x + 8
8-32.
a. 5 c. 5 or –6
b. –6 d. ! 14
e. 8
f. ! 14 or 8
8-40. a. Michelle is correct. b. (– 4, 0) 8-41. 45, 46, 47; x + (x + 1) + (x + 2) = 138 8-42. a. 2
Answer Key
3 4
b. 3
c. 1 55
Lesson 8.2.1 8-45.
a. 2
b. –3
c. " –6.1
8-46.
y = !3x + 25
8-47.
y = 3x ! 5 ; m = 3 and b = 5
8-48.
There is only one line of symmetry. horizontal through the middle.
8-49.
a. x-intercepts (–2, 0) and (0, 0), y-intercept (0, 0) b. x-intercepts (–3, 0) and (5, 0), y-intercept (0, 3) c. x-intercepts (–1, 0) and (1, 0), y-intercept (0, –1) d. x-intercept (9, 0), y-intercept (0, 6)
8-50.
a. 6x 2 + x ! 12 b. 25x 2 ! 20x + 4
Lesson 8.2.2 8-51.
Jen
a. Longest: Maggie, Highest: Jen b. Jen. (0, 0) and (8, 0), Maggie. (3, 0) and (14, 0), Imp. (2, 0) and (12, 0), Al. (10, 0) and (16, 0); the x-intercepts tell where the balloon was launched and where it landed. c. Jen. (4, 32), Maggie. (8.5, 30.25), Imp. (7, 25), and Al. (13, 27); maximum height.
Maggie
30
Al
Imp
25 20 15 10 5
2
4
6
8
10
12
Solution to part (a)
14
16
8-52.
You should be able to connect rule ! table, table " graph, graph " situation, and table " situation.
8-53.
a. One way to write the rule is y = (x + 1)(x + 2) + 2 .
8-54.
vertex. (4, –9), x-intercepts. (1, 0) and (7, 0), y-intercept. (0, 7)
8-55.
a. 3 –7 6 –2 c. It tells us that a = 0 .
8-56.
a. x-intercepts (2, 0), (– 4, 0), and (3, 0), y-intercept. (0, 18) b. x-intercepts (3, 0) and (8, 0), y-intercept. (0, –3) c. x-intercept (1, 0) and y-intercept (0, – 4)
8-57.
a. (–3, 0)
b. ! 12
8-58.
a. no solution
b. (7, 2)
56
b. Yes
b. …it does not change the value of the number. d. 0 for all e. …the result is always 0.
Algebra Connections
Lesson 8.2.3 8-59.
a. No; the y-intercept is not enough information. b. No; the parabola could vary in width and direction. c. Yes; solution shown at right.
8-60.
a. y = 0 for all x-intercepts and x = 0 for all y-intercepts. b. (0, –12) c. 0 = 2x 2 + 5x ! 12 d. Not yet, because it has an x 2 term.
8-61.
a. At least one of the two numbers must be zero. b. At least one of the three numbers must be zero. c. Typical response: “If the product of two or more numbers is zero, then you know that one of the numbers must be zero.”
8-62.
a. 0 = (2x ! 3)(x + 4) b. 2x ! 3 = 0 or x + 4 = 0 , so x =
3 2
or x = ! 4 .
c. The roots are at ( 23 , !0) and (– 4, 0). d. The solution graph is shown at right. 8-63.
This parabola should have roots (–3, 0) and (2, 0) and y-intercept (0, –6).
8-64.
roots: (–1, 0) and (–2, 0), y-intercept: (0, 4)
8-65.
a. One is a product and the other is a sum. b. first: x = !2 or x = 1 ; second: x = ! 12
8-66.
a. x = 2 or x = !8 c. x = !10 or x = 2.5
8-67.
a. The line x = 0 is the y-axis, so this system is actually finding where the line 5x ! 2y = 4 crosses the y-axis. b. (0, –2)
8-68.
a. 4; Since the vertex lies on the line of symmetry, it must lie halfway between the x-intercepts. b. (4, –2)
8-69.
a. 2(x ! 2)(x + 1)
8-70.
a. The symbol “#” represents “greater than or equal to” and the symbol “>” represents “greater than.” b. 5 > 3 c. x ! 9 d. –2 is less than 7.
Answer Key
b. x = 3 or x = 1 d. x = 7
b. 4(x ! 3)2
57
Lesson 8.2.4 8-71.
The parabola should have y-intercept (0, 2) and roots (–1, 0) and (–2, 0).
8-72.
a. x = –2 or x = – 4 d. x = 0 or x = –6
8-73.
a. y = (x + 3)(x ! 2) = x 2 + x ! 6 b. y = (x + 5)(x ! 1) = x 2 + 4x ! 5
8-74.
By symmetry, (12, 0) is also a root. Thus, the quadratic must be of the form y = a(x ! 2)(x ! 12) . Since the parabola points down, a must be negative. Testing a point shows that y = !(x ! 2)(x ! 12) = !x 2 + 14x ! 24 is correct.
8-76.
The result must be the original expression because multiplying and factoring are opposite processes; 65x 2 + 212x ! 133 .
8-77.
a. x = 3 or x = ! 23 b. x = 2 or x = 5 c. x = !3 or x = 2 d. x = 12 or x = ! 12
b. x = 1 or x = 43 e. x = 5 or x = –1.5
c. x = –5 or x = 23 f. x = 2 or x = –6
8-78.
8-79.
a. true
8-80.
a. –1
8-81.
y = ! 43 x
b. false
c. true b. " 1.6
d. true
e. false
f. false
c. –3
a. ! 43 b. Yes; it makes the equation true and lies on the graph of the line.
58
Algebra Connections
Lesson 8.2.5 8-82.
(1) " b, (2) " e, (3) " a, (4) " g, (5) " d, (6) " i
8-83.
Letter A. The client should order the parabola y = (x ! 1)(x + 6) . Letter B. The parabola y = (x ! 5)2 should be recommended. Letter C. The parabola y = !(x + 3)(x ! 2) should be recommended. b. y = !3(x ! 10)(x ! 16)
8-84.
a. y = !2x(x ! 8) = !2x 2 + 16x
8-85.
a. y = x 2 + 2x ! 8 d. y = !x 2 ! 4x + 5
8-86.
m = 12 , (0, 4)
8-88.
a. x = 4 or x = –10
b. x = –8 or x = 1.5
8-89.
a. 4
c. –8
8-90.
a. (1, –1)
b. y = x 2 ! 6x + 9
c. y = x 2 ! 7x
8-87. a. " –1.4 and " 0.3
b. –10
b. (–2,
b. The quadratic is not factorable.
d. 1.5; They are the same. 1 2
)
Lesson 8.3.1 8-91.
a. The quadratic is not factorable. c. The intercepts are " –1.5 and 4.5.
b. There are two roots (x-intercepts).
8-92.
a. a = 1 , b = !3 , c = !7
b.
8-93.
a. Graphing and factoring with the Zero Product Property
8-94.
a. x = –2 or ! 13
b. x = 7 or –2.5
8-96.
a. x = 6 or 7
b. x =
8-97.
x = 6 or 7; yes
8-98.
no a. The parabola should be tangent to the x-axis. b. Answers vary, but the parabola should not cross the x-axis.
8-99.
y = 12 x + 9
8-101.
A and D
8-102.
a. false g. true
Answer Key
8-100.
b. true h. false
2 3
or – 4
3± 37 2
" 4.5 and –1.5; yes
c. x = –0.5 or –0.75 c. x = 0 or 5
d. no solution
d. x = 3 or –5
line. (a) and (c); parabola. (b) and (d)
c. true
d. true
e. true
f. false 59
Lesson 8.3.2 8-103.
Lesson 8.3.3
a. (3x ! 2)(2x + 5) = 0 ,
x=
2 3
or !
b. a = 6 , b = 11 , c = !10 , x =
8-114.
a. The Zero Product Property only works when a product equals zero. b. x 2 ! 3x ! 4 = 0 c. x = 4 or x = –1; no
8-115.
a. x = 0 seconds and x = 2.4 seconds, so it is in the air for 2.4 seconds. b. at x = 1.2 secs c. 2.88 feet d. The sketch should have roots (0, 0) and (2.4, 0) and vertex (1.2, –2.88).
8-116.
If n = # nickels and q = # of quarters, 0.05n + 0.25q = 1.90 , n = 2q + 3 , and n = 13, so Daria has 13 nickels.
8-117.
a. x = ± 0.08 b. x = c. no solution d. x " 1.4 or –17.4
8-118.
While the expressions may vary, each should be equivalent to y = x 2 + 4x + 3 .
8-119.
a. x = 2 c. x = –2
8-120.
Line L has slope 4, while line M has slope 3. Therefore, line L is steeper.
8-121.
D
a. x = 5.5 or x = !5.5 b. x = 2 or x = ! 12 c. x = !3 or x = 14
d. x =
5 6
8-105.
a. " 315 and –315 feet; The bases of the arch are 315 feet from the center. b. " 630 feet c. 630 feet; y-intercept
8-106.
a. x = 5 c. x = –1 or 1 3
b. x = 5 3
1 3
or –6
d. x = ± 43
8-107.
x=
8-108.
a. y = (x + 3)(x ! 1) = x 2 + 2x ! 3 b. y = (x ! 2)(x + 2) = x 2 ! 4
or –6; yes
8-109.
If x = width, x(2x + 5) = 403 ; width = 13 cm.
8-110.
(b) and (c) are solutions.
8-111.
a. She solved for x when y = 0 . b. The y-intercept is (0, –5), and a shortcut is to solve for y when x = 0; y = 53 x ! 5 . c. x. (8, 0), y. (0, –12)
60
a. x = !3 or x = !9 b. x = !17.6 or x = !0.36 c. x = ! 43 or x = 12 d. x = 4 e. no solution f. x = 2.5 or x = !0.9
2 3
or ! 52 c. Yes 8-104.
8-112.
5 2
2 9
or – 4
b. x = 15 d. all numbers
Algebra Connections
Chapter 9 Lesson 9.1.1 9-2.
a. Inequalities have multiple solutions, but equalities only have one solution. b. infinite c. The result of !1 " x " 4 does not extend infinitely. It has two endpoints. The result of x 2 ! 4 has two endpoints (2 and –2) and extends infinitely in both directions.
9-3.
a. x ! 1
9-4.
a. infinite b. 4 c. It is the solution to the equality 2x ! 5 = 3 ; it is the “starting point” for the solution. d. x ! 4 , 4 x
9-5.
a. Solve the corresponding equality for the variable. 3 ! 2x = 1 , x = 1 . b. There should be an unfilled circle at x = 1 . c. If the point is not a solution, then the solutions lie on the other side of the boundary; x > 1 ; -4 -3 -2 -1 0 1 2 4 x
b. x < 1
c. x ! 3
d. ! 4 < x " 2
3
9-6.
x < 2,
9-7.
Let n = # of North American countries. Then n + (2n) + (2n + 7) = 122 and n = 23.
9-8.
a. p > !1
b. k < 2
9-9.
a. k = 1.5 or –2
b. m = 3 or –3
c. w = 2 or –6
d. n " 2.12 or –0.79
9-10.
a. always true e. always true
b. sometimes true f. never true
c. never true
d. sometimes true
9-11.
1; Any non-zero number divided by itself is 1.
9-12.
a. (5, 0) and (8, 0); Robbie backed up 5m and the rocket landed 8m away from him. b. 3 m
9-13.
a.
Answer Key
5± 13 2
-4 -3 -2 -1 0 1 2 3 4
" 0.7 or 4.3
x
c. 1 ! k or k ! 1
b. !1 ±
7 " –3.6 or 1.6
61
Lesson 9.1.2 b. m ! 2 d. x ! 3 f. no possible solution
9-14.
a. x < 5 c. p > !3 e. all numbers
9-15.
Let t = number of Turks and k = number of Kurds. Then t + k = 66, 000, 000 and t = 4k . There are 13,200,000 Kurds and 52,800,000 Turks.
9-16.
a. Since trees bloom when they are taller than 150 cm, an inequality is fitting. b. 6 + 9x > 150 , x > 16 , so the trees should be more than 16 years old. c. 6 + 9x ! 240 , x ! 26 , so the trees should be less than 26 years old. Thus, the trees should be between 16 and 26 years of age or 26 years old, which can be written. 16 < x $ 26.
9-17.
a. k < 2 b. p ! 15 c. n > 12 d. t ! 0
9-18.
The graph should be a line with x-intercept (1.5, 0) and y-intercept (0, 3).
9-19.
y = ! 53 x !
9-20.
3x 2 + x ! 10 a. 3x ! 5 ; The product divided by one factor gives the other factor. b. x + 2
9-21.
x=
9-22.
a. (2x ! 5)2 b. not possible c. 3x(x ! 4) d. 5(x ! 4)(2x + 1)
62
5 3
8 5
and x = ! 52
Algebra Connections
Lesson 9.2.1 9-23.
a. Yes; the point is on the line and makes the equation true. b. The point (2, –1) is a solution because it is on the line and makes the equation true. The point (0, 0) is not a solution because it is not on the line and does not make the equation true. c. The points on the line make the equation true.
9-24.
b. Any point above the line is a solution. Verify any questioned points by substituting them into the inequality. Yes; they make the inequality statement true. infinite Points below the line do not satisfy the inequality.
9-25.
a. y ! "2x + 3
b. The boundary should be a dashed line.
9-26.
9-28. a:
b: 4
x
2
x
c:
-3
x
0
x
d:
9-29.
1200 + 300x ! 2700 , so x ! 5 . Algeria can order an advertisement up to 5 inches high.
9-30.
Let x = number correct for Part 1 and y = number correct for Part 2. Then x + y = 33 and 3x + 2y = 85 , and x = 19 and y = 14 . Rowan answered 19 problems correctly on Part 1.
9-31.
a. y = ! 27 x ! 2 b. Yes; verify by substituting the coordinates into the equation and testing. c. y = 27 x ! 22
9-32.
B
9-33.
D
Answer Key
63
Lesson 9.2.2 9-34.
a. Just one. If the point does not make the equation true, then the side that does not contain the point is the solution. b. Zero conveniently eliminates the variable terms in the inequality. c. No; (–3, 2) lies on the line. She needs to test a point off the line.
9-35.
a. y > 12 x b. Honduras, Zambia, Madagascar, and Uganda will receive aid. Rwanda is on the boundary, but the boundary is not included, so it is not a solution.
9-36.
9-37.
9-38.
y < ! 13 x ! 1
9-39.
A
9-40.
A
9-41.
a. $16; Subtract and then make the result positive. b. 42° c. 360 students d. The difference does not depend on which is greater. It represents how much greater the larger amount is than the smaller amount.
9-42.
a. x =
9-43.
No; Bernie would pass Barnaby after 40 seconds, when each was 90 meters from the starting line. Since the race was only 70 meters, that would occur after the race was over.
9-44.
x = 0.6 or x = –2
64
1 3
b. x = 16
c. x = ± 5
d. x > 5
Algebra Connections
Lesson 9.2.3 9-45.
a. The absolute-value relation always returns the positive value of the input. b. You need it to make something positive, like finding a distance or difference, such as those in problem 9-41 in homework.
9-46.
a. Add –5 and 1 and then change the result to a positive value. Then subtract 3. The result is 1. b. i. 2, ii. 30, iii. 13, iv. 7 c. Answers vary.
9-47.
either 15 or –15; yes
9-48.
a.
9-49.
She should add 1 first, since the addition is placed inside the absolute value symbol, which acts as a grouping symbol.
b. Answers will vary.
9-50.
x
9-51.
1, 400, 000 ! 50x > 1, 200, 000 , less than 4000 square miles per year
9-52.
a. 1
9-53.
no; 3(7 ! 2) = 15 and 15 > 4
b. 2
c. –11
d. 28
9-54.
9-55.
718 ! 14x = 212 + 32x , x = 11 months
9-56.
b. Answers vary, but using data from games 1 and 3. y = 6x . c. " 90 baskets
Answer Key
65
Lesson 9.3.1 9-57.
a. See graphs at right. c. The point (0, 0) satisfies both inequalities. d. There are multiple ways to justify that the region found in part (b) is the only possible solution. One way is to test points in every other region to demonstrate that they do not make both inequalities true.
9-58.
9-59.
a. y = x 2 + x ! 6 is part of y ! x 2 + x " 6 , while y = 23 x is not part of y > 23 x . b. 5 c. 2. d. Because it lies on one of the boundary graphs. It will not tell you anything about the regions separated by the boundaries.
9-60.
9-61.
y ! x + 3 , y ! 12 x " 1 , y ! 2 , y ! " 23 x + 4 , y ! " 23 x " 1
9-62.
a. 3
b. 1
c. 4
d. 2
9-63. a. c.
-5 -4 -3 -2 -1 0 1 2 3 x -5 -4 -3 -2 -1 0 1 2 3 m
b.
2 3 4 5 6 7 8 9 10 x
d. no solution
9-64.
All equal 1
9-65.
a. (5x ! 2)(x + 3)
9-66.
2a + 3c = 27.75 , 3a + 2c = 32.25 , a = $8.25 , c = $3.75
9-67.
B
66
b. 2(3t ! 1)(t ! 4)
-5 -4 -3 -2 -1 0 1 2 3 x
c. 6(x ! 2)(x + 2)
Algebra Connections
Lesson 9.3.2 9-68.
6
y
3 –6
–3
3
6 x
–3 –6
9-69.
y ! " 12 x + 2000 , y ! 23 x " 2000 , y ! " 15 x ; The search-and-rescue teams should search near the island of Samoa.
9-70.
a. the region between and including the parallel lines b. the region below and including the boundary line y = 23 x c. no solution
9-72.
9-73.
9-74.
a. x ! 6
b. x > 1
c. 2 ! x < 7
d. !3 " x " !1
9-75.
a. false
b. false
c. true
d. false
9-76.
C
9-77.
a. r + 2.50c ! 15 b. r + c ! 25 c. No; the club cannot sell a negative number of items. d. See graph. The solution points represent the possible sales of rulers and compasses that would allow the club to break even or make a profit while falling within the sales limit.
Answer Key
67
9-78.
A. y ! 500 " 1.5x , B. y ! 300 " 0.25x , C. y > 180 ! 23 x , D. y > 450 ! 52 x , E. x < 250 , Special Assignment. x ! 100
Number of Medicine Packages
Lesson 9.3.3 Special Assignment E
500
450
A
400
350
D
300
B
250
200
150
100
C
50
50
100
150
200
250
300
Number of Food Packages 9-79.
a. 210; estimate from the graph or substitute 185 for y into the equation for Country A ( y = ! 23 x + 500 ) and solve for x since that line is the upper boundary when x = 185 . b. 275; It is the point in the region with the largest y-value. One way to determine this is to substitute 100 for x into the equation for Country B and solve for y.
9-80.
200 of each should be requested; graph the line y = x and find its intersection with the line for Country A.
9-81.
a. Yes, they are equivalent. One way to determine this is to change both to y = mx + b form and compare slope and y-intercept. b. Answers vary. Multiply or divide both sides of either equation to find an equivalent equation. For example, 2x + y = 3 and 8x + 4y = 12 are both equivalent equations.
9-82.
3280x + 1500 < 50, 000 , less than 14.8 pounds
9-83.
a. m > 5
9-84.
c. x > 7
d. no solution
b. No, it is not; it lies on both boundaries, but the boundary to y < x is not part of the solution.
9-85.
x = 9 or x = 0.5
9-86.
D
68
b. x ! "6
Algebra Connections
Chapter 10 Lesson 10.1.1 10-1.
1 is special because any non-zero number divided by itself is 1, and anything multiplied by 1 remains the same.
10-2.
a. yes
c. Yes; x ! 3
b. You cannot divide by zero.
d. Answers vary; sample solutions. e. Yes, because
z z
x x
,
x+5 x+5
, and
n2
.
n2
= 1. The fact that anything multiplied by 1 stays the same is called
the Identity Property of Multiplication. 10-3.
a. 1, x ! 0
b.
e. hk , h ! 0
f.
h.
1 , 4 x!1
x!
1 4
or
x , x!0 3 2m!5 x ! , 3m+1
c.
" 6 or " 13
x+5 x!1
, x ! 1 or 2
d. 1, x ! 0
g. 2(n ! 2) , x ! 2
3 2
10-4.
a. Yes; you can tell by substituting any number (other than zero). b. No; you can tell by substituting a number (other than 1). c. They can be simplified like this when the numerator and denominator are single terms and are products of factors. d. (i) is not simplified correctly; (ii) is simplified correctly.
10-5.
a.
10-7.
2 x!5 3x+1
c. 1
d.
a. 1
b. none
c. 2
d. 1
10-8.
Yes, he can.
a. x = 2
b. Divide both sides by 100.
10-9.
a. x < 0
10-10.
a. x ! "4 or 2 ,
10-11.
a.
10-12.
B
Answer Key
x+ 3 x! 3
3 7
b.
x 2
b. x ! " 4 x+ 4 x!2
b. x ! "2 or 3 , b.
2(x+2) (x! 3)2
5 4
69
Lesson 10.1.2 10-13.
3x!1 ; 2 x+ 3
x cannot be – 4 or ! 23 .
3 5 , 7 4
10-14.
x!
or "
d. 10-16.
a.
4 11
10-18.
a. x ! 2 = 4
b. For each, x = 6 .
c. x + 3 = 8 , x = 5
10-19.
a. m = ! 65 , b = (0, –7)
b. m = 23 , b = (0, –5)
c. m = 2 , b = (0, –12)
10-20.
a. ( 13 , –2)
b. (4, –9)
10-22.
a. m = !6
b. x = 5.5
10-23.
after 44 minutes
a.
b.
or " 3
b.
y ! 0,!"2, or 2
e. 2x, x ! 0 and y ! 0 f.
7+ 4 m m ! 2 or " , 3m!2 3 3x+1 x ! 3 ,!4, or , 2 x! 3 2
4 x+ 3 , x!5 x+ 3 2 (y!2) (y+5) , 3y(y+2)
10-15.
3 25
c.
5(x!2) (x+ 4)2
d.
x+2 , 2 x+1
1 9
2(2 x! 3) (x!5)(x+1)
10-21. a.
e.
1 2
c.
5 x!1 2(x! 3)
x! 4 3x+2
=
5 x!1 2 x!6
5 x! 3
b.
f.
3 2 2 5
1 2 x+5
c. 2 d. x = 90
c. k = 4
Lesson 10.1.3 2x 3(2 x!1)
=
2x 6 x! 3
a.
10-25.
a. She can multiply each term in the equation by a number to eliminate the decimals. b. 2x 2 + 3x ! 5 = 0 , (2x + 5)(x ! 1) = 0 , so x = !2.5 or 1.
10-26.
a. 3x ! 5 = 7 , x = 4
b. 3x 2 ! 2x ! 5 = 0 , x = !1 or
c. 1 + x = 10 , x = 9
d. 4x 2 + 8x ! 5 = 0 , x =
10-27.
b.
x! 4 x+ 4
10-24.
1 2
5 3
or x = ! 52
= 24 ; One denominator was eliminated, but there is still a fraction left over. a. x ! 15 4 = 27 43 . c. 24 or any multiple of 24; 24 is the least common multiple of 6 and 8; 4x ! 15 = 96 . d. x = 27 43 b. Multiply by 4; 4x ! 15 = 96 , x =
111 4
10-28.
Equations: 4x ! 2 = 6x , 1 ! 2x = 5 , 5 ! 3x = !7 , 2x + 1 = 5 , x ! 6 = 3 + 2x , 4 ! x = !1 ; solutions: a: x = !1 , b: x = !2 , c: x = 4 , d: x = 2 , e: x = !9 , f: x = 5
10-30.
a.
10-31.
y = ! 13 x + 2
70
5(3x!1) 2(4 x+1)
b. 1
c.
p+9 3 p!2
d.
4 x!2
10-32. y < ! 23 x + 2 10-33. a. 2d ! 3 b. 2d ! 3 = 19 , 11 candies Algebra Connections
Lesson 10.1.4 10-34.
a. possible equation. 10x + 45 = 15 , x = !3 b. Six is the smallest number that can eliminate all fractions. Any multiple of 6 works, as does multiplying first by 3 and then again by 2.
10-35.
a. 3x ! 2x = 8 , x = 8 c. !14x + 7 ! 3x ! 9 = 168 , x = !10
10-36.
a. p % 2 or – 4 because they would cause one (or both) denominators to be zero. b. p = 3 or – 4. However, p = ! 4 is an excluded value, and therefore is extraneous. Hence, the solution is p = 3 .
10-37.
a. x 2 + 4x + 3 = 0 , x = !1 or –3 c. 12m ! 2 = 38 + 10m , m = 20
b. a 2 ! 6a + 9 = 0 , a = 3 d. 2(x ! 3) + 3x = 4(x + 5) , x = 26
10-38.
a. x = 4
c. x = 16 3
10-39.
a.
10-41.
t + s = 27 ,
10-42.
See diagram at right.
b. x = !5 or 2
(x!2)(x+6) (x+ 4)(2 x+ 3)
1t 2
b.
b. 5 ! 2x 2 = 3x , x = 1 or –2.5 d. x + 3 + 2x ! 4 = x + 5 , x = 3
10-40. a. t = 5 seconds
2 x+1 x!5
d. x =
1 2
b. 100 feet
+ 14 s = 11 12 , t = 19 Times papers 10-43. C
Lesson 10.2.1 10-44.
x=2
10-46.
a. Divide by 4; dividing “undoes” multiplication, because they are inverse operations. b. after dividing, x + 3 = 5 , x = 2; yes c. Using the opposite operation eliminates (or “undoes”) the multiplication.
10-47.
a. If you treat (x + 3) as a group, then 4 · (group) = 20. Therefore, x + 3 = 5 b. x + 3 = 5 , x = 2; yes
10-48.
1 " iii, 2 " i, 3 " ii
10-49.
a. 34
10-50.
a. x = 10 or 4
10-51.
a. Both Hank and Frank are correct.
10-52.
a. 4 or – 4
10-53.
(a), (b), and (c) all are equal to 1.
10-54.
a. x > !2
10-55.
a. 3 + 4x = 14 , x = 11 4
10-56.
Let x represent the amount of money the youngest child receives. Then x + 2x + x + 35 = 775 ; $185, $370, and $220.
s
Answer Key
10-45. 4x + 12 = 20 , x = 2
b. 100 or –100
b. x ! 12
b. 15 c. 11
d. 2
e. 25
c. no solution
d. –3 or 7
c. x > 1
d. x ! 15
f. 27 b. two
b. rewriting
71
Lesson 10.2.2 b. x = 3 ± 12
10-57.
a. two solutions
10-58.
Reminder. the scope of this course is limited to real numbers. a. two solutions. ! 4 ± 20 b. no solution c. two solutions. 5 or –2 d. one solution.
1 2
c. ! 6.46 or –0.46
e. no solution b. one solution
f. one solution. –11
10-59.
a. no solution
c. two solutions
10-60.
a. Two solutions because 2x – 5 can equal 9 or –9 b. Looking inside offers a quick solution. c. Since 2x ! 5 = 9 or –9, then x = 7 or x = !2 .
10-61.
a. Answers vary b. Answers vary, but it should contain an absolute-value expression equal to zero.
10-63.
a. x = 3 or –11
10-64. 10-65.
No, because –1 is not greater than –1. 4 m+5 a. 4x+ b. m+ x! 3 4
10-66.
a. (3x ! 1)(3x ! 1) b. 7 ! 7 ! 7 ! 7 d. w ! w ! w ! w ! w ! w ! w ! w ! w ! w
10-67.
a. 4x(x ! 3)
10-68.
t = number of toppings, 1.19(3) + 0.49t = 4.55 , and t = 2
b. x = 14
b. 3(y + 1)2
c. x = 2
d. x = 2
c. m ! m ! m
c. (2m + 1)(m + 3)
d. (3x ! 2)(x + 2)
Lesson 10.2.3 10-69.
a. x = 4 or –6 e. x = !15 or –9
10-70.
If x = the length of a side of the hot tub, then (x + 3)2 ! 169 and !16 " x " 10 . However, since the minimum side length is 4 feet, the possible measurements that Ernie can order are 4 ! x ! 10 feet.
10-71.
See graph at right.
10-72.
When any real number is squared, the result is either positive or zero.
10-73.
a. 7 or 1
b. 4 or 8
c. 3
d. no solution
10-74.
a. x < 2
b. x ! 6
c. x > 4
d. x ! 18
10-75.
a.
72
x! 3 3x!14
b.
2 x!1 x+1
b. x = 15 f. x = 5 or 11
c. x = 53
d. x = !3 or –7
10-76. 5xxxyy Algebra Connections
Lesson 10.2.4 10-77.
x < 2.5 ; a. 2.5; No, because the inequality does not include -2 -1 0 1 2 3 4 5 6 x equality. b. Find the boundary point by solving the corresponding equality. Then test a point on either side of the boundary point.
10-78.
a. x < !1 or x > 5 ; -2 -1 0 1 2 3 4 5 in each possible solution region.
10-79.
a. The solution points are those x-values for which y = x ! 2 is greater than y = 3 . b. !1 " x " 5 ; ; The solution points are the x-values for which -2 -1 0 1 2 3 4 5 6 x
6
x
; After finding dividing points, test a point b. two boundary points.
y = x ! 2 is below (less than) y = 3 , including the points of intersection. c. All numbers; y = x ! 2 is always above the line y = !1 .
10-80.
a. two. x = 1 and x = !3 c. !3 < x < 1 ;
10-81.
Solutions vary; in general, any inequality that states that x ! 2 is less than a negative number will work.
10-83.
(x + 2)(2x + 3) = 2x 2 + 7x + 6
10-85.
y = 23 x + 18
10-87.
a.
10-88.
a. x = 13 b. x = 3
2(x+5) 3
or
b. three. x < !3 , !3 < x < 1 , and x > 1 d. Yes, see graph at right
10-84. a. 2
b. 1
c. 0
d. 1
10-86. See graph at right. 2 x+10 3
b.
7 x!2 x+ 4
c. !5 " x " 5 d. x < ! 23 or x > 2
Lesson 10.2.5 10-89.
x ! "2 and x ! 4 ,
10-90.
a. x < !3 or x > !1
b. ! 4 < x < 3
e. no solution
f. x < 2 !
d. x ! 1
c. all numbers
7 or x > 2 +
7
10-91.
!16t 2 + 96t " 140 , 2.5 ! t ! 3.5 seconds
10-93. 10-94.
(y ! 2)(y ! 2)(y ! 2) It has no solution, because an absolute value cannot equal a negative number.
10-95.
a. x = 1
10-96.
a. y = ! 23 x ! 13
10-97. Answer Key
a. x ! 4
b. x > 3 or x < !3
b.
7m!1 3m+2
c. 0 ! x !
10-92. x = !105,!!9,!!1,!or!95
4 3
d. x =
3 4
e. !2 " x " 3
b. Yes; the slope of 6x ! 4y = 8 is c.
(4 z!1)2 z+2
d.
x! 3 x!2
3 2
f. 3
.
10-98. C 73
Lesson 10.3.1 10-99.
Both have the same solutions. !2 ±
10-100.
a. They have the same answer. This tells us that the equations are equivalent, which can be shown by expanding (x + 2)2 . b. Answers vary.
10-101.
a. Add 3 to both sides of the equation and then write the square as a product. b. You must add tiles to complete a square.
10-102.
a. (x + 3)2 = 4 , x = !1 or !5
10-103.
a. (x ! 3)2 = 10 , x ! "0.16 or 6.16; b. (x + 2)2 = 9 , x = !5 or 1; c. (x ! 1)2 = 5 , x ! 3.24 or –1.24
10-104.
(2x + 3)2 = 16 , x =
10-105.
a.
10-106.
Both result in no solution.
10-107.
a. !2 < x < 2 f. !5 " x " 3
10-108.
See graph below. The portion for x between –5 and 3 (inclusive) should be highlighted.
10-109.
18 + 3m + 22 + 2m = 80 , m = 8; It will be full after 8 months.
74
4 x! 3 5 x+1
1 2
b.
3 " –3.73 or –0.27.
b. (x + 1)2 = 6 , x = !1 ± 6 " 1.45 or –3.45
or ! 27 5 x+2 5 x!2
b. x ! 2.5
c.
c. x =
1 4
4(2 x+1) x!2
d. no solution
d.
2y 3x
e. x = !12
Algebra Connections
Lesson 10.3.2 a. 2
10-111.
b. The constant in the parentheses will be half the coefficient of the x-term in the original quadratic equation. c. If the quadratic is in the form x 2 + bx + c , then ( b2 )2 ! c unit tiles need to be added to both sides of the equation.
10-112.
a. (w + 14)2 = 144 , w = !2 or –26 c. (k ! 8)2 = 81 , k = !1 or 17
10-113.
! 20 7
10-114.
a. (x + 2)2 = 1 , x = !3 or –1 b. (x ! 4)2 = 9 , x = 1 or 7 c. (x ! 12)2 = 15 , x = 12 ± 15 , x " 15.87 or 8.13
10-115.
a.
10-116.
a. x = 58 d. no solution
10-117.
See graph at right.
10-118.
They will be the same after 20 years, when both are $1800.
x+2 3x!2
b. 0
d. 0.
10-110.
b.
c. 1
b. (x + 2.5)2 = 2.25 , x = !1 or – 4 d. (z ! 500)2 = 189225 , z = 65 or 935
(x!2)2 (x!11)(x+ 4)
b. !6 < x < 0 e. x = ! 4 or 4
c. x ! "9 or x ! 7 f. all numbers
Lesson 10.4.1 10-119.
a. i. y ! y ! y ! y ! y ! y ! y , ii. 5(2m)(2m)(2m) , iii. (x ! x ! x)(x ! x ! x) , iv. 4x ! x ! x ! x ! x ! y ! y b. i. x 2 , ii. x 3 , iii. x 5 , iv. k 8 , v. 2k , vi. m 7 , vii. x12 , viii. 3x 3 , challenge.
10-120.
x 35 2
a. y 7
b. w 3
c. x 8
d. x 22
g. 10h 25
h. 5m 22
i. 81k 80
j.
8 g7
10-122. a. incorrect, x100
e. 13p 2 q 3
= 8g !7
k.
m12 n 40
x6 y3
l. pw 7 c. incorrect, 27m 6 n 45
10-121.
Answers vary.
10-123.
a.
10-124. 10-125.
a. Haley is correct. You cannot add unlike terms. bases differ. a. 1 unit tile b. (x + 1)2 = 9
10-126.
While there are multiple ways to write the rule, one possible way is y = x 2 + 5x + 7 .
10-127.
a. x = 2
Answer Key
1 h2
b. x 7
c. 9k10
b. k ! 3.76 or 1.24
b. correct
f.
d. n 8
e. 8y 3
f. 28x 3 y 6
b. Haley is incorrect. The
c. !2 < x < 10
d. x = 5 75
Lesson 10.4.2 10-128.
a. x11
b. x 9 c. m12
e. x !3 y 8 or
d. x 8 y 8
y8 x3
f.
1 10 x 4
10-129. Expression
Generalization
Why is this true?
a.
x 25 ! x 40 = x 65
xm ! xn = ?
x m+n
If you have m x’s and n x’s multiplied together, you then have a product of (m + n) x’s.
b.
x 36 x13
xm = ? xn
x m!n
Many of the x’s on top and bottom make “ones,” leaving (m ! n) x’s.
c.
(x 5 )12 = x 60
(x m )n = ?
x mn
If you have n sets of m x’s, then you have mn x’s.
= x 23
10-130.
Its value is 1, and it can be rewritten as x m! m = x 0 . Thus x 0 = 1.
10-131.
a.
1 x
10-132.
a.
1 k5 1 a2b
g.
= x !1
b. b. 1
1 x2
= x !2
c. x 3
c. d. p !2
1 x3
e. y
= x !3 f.
1 x6
h. x
10-135.
b, c, d, f
10-136.
y > 2x ! 1
10-137.
(15, 2)
10-138.
y = !x(x ! 20) = !x 2 + 20x ; Its maximum height is 100 feet when x = 10 .
10-139.
Perfect square form. (x ! 5)2 = 0 ; x = 5; answers vary.
10-140.
a.
76
x! 4 3x+2
b.
5 x! 3
c. 3
Algebra Connections
Lesson 10.4.3 10-141.
a. The result is the square root of the base. c. It is inductive reasoning (based on patterns).
b. 2, 7
10-142.
a. 4 b. 2, 5; Rewrite each as 3 repeated factors and then take one of them. c. 27 = 3·3·3 , so 27 2/3 is 3·3 = 9 ; 32 = 2 ·2 ·2 ·2 ·2 , so 32 3/2 = 2 ·2 ·2 = 8 ;
25 = 5 ·5 , so 25 3/2 = 5 ·5 ·5 = 125 d.
5
x ; fifth root
10-143.
a. (31/2 )4 = 32 = 9
b. 37 = 2187
10-144.
a. 3 g. 2
c. 7
10-145.
a. 4,600,000,000 years
10-146.
a. 100, 000, 000, 000, 000, 000 b. 1 ! 1017 c. While opinions may vary, since most calculators cannot handle the calculations in standard form, scientific notation is easier.
10-147.
a. !15x
10-148.
1,261,392,000 or 1.261 · 109 feet
10-149.
See solution graph at right; (–2, 6) and (1, –6).
10-150.
8 + 3.75x ! 20 , x ! 3.2 hours; Vinita can rent a skateboard for up to 3.2 hours (3 hours and 12 minutes).
10-151.
a. The person starts walking away from the detector and then turns around and walks slightly faster toward the machine b. The person does not move c. This is not possible because the motion detector cannot simultaneously record two different distances.
Answer Key
b. 8 h. 5
c. (2 5 )1/3 = 2 5/3 d. 1
b. 1.5 !1011 meters
b. 64 p 6 q 3
e. 6
f. 4
c. 0.0000000033 seconds
c. 3m 8
77
Chapter 11 Lesson 11.1.1
Lesson 11.1.2
11-4.
(3x ! 2)2 = 9x 2 ! 12x + 4
11-16.
y = x2 ! 1
11-5.
a. 228 shoppers b. 58 people/hr
11-17.
f (4) = 15
c. at 3:00 PM
a. 0.75 c. 2
b. 99 d. &
11-18.
f (x) =
11-6. 11-7.
The x-coordinate of the vertex must be at x = 1 because of symmetry of the parabola.
11-8.
2a + 3c = 56.5 , a + 4c = 49.5 , a = $15.50 , c = $8.50
11-9.
a. m = ! 27 , b = 2 b. m = ! 13 , b = 6 c. m = 5, b = –1 d. m = 3, b = 0
11-10.
a. x ! 2 c. x ! 9
11-11.
2 x4 81 w8
a. c.
11-12.
11-13.
b. x > !1 d. x > 10 b. s11u 2 d.
a. 3 c. 4 a.
x!8 6 x!1
11-14.
a. (0, 3)
11-15.
a. x 4 y 3 c. !6x 6
1 m3
11-19.
a. 10 b. 1 c. 125 d. no output because you cannot take the square root of a negative number, e. –5 f. 10 or –10 g. no input will yield a negative absolute value h. 6 i. 2
11-20.
a. 0 b. 3 and 0 c. x ! "0.5 , 2.5
11-21.
1, 9, and t 2
11-22.
y = ! 43 x + 12
11-23.
Marley is correct; they are perpendicular since the slopes are 2 and ! 27 . 7
11-24.
45 miles
11-25.
a.
b. 1 d. 2 b.
2(x!6) x!2
=
2 x!12 x!2
b. ( 12 , 0) and (3, 0)
c. e.
b. xy d. 8x 3
11-26.
a. c.
78
x
x+2 x!6 x(x!6) x!1 3x!1 x!5
b.
1 4 1 52
b. 1
=
1 25
d. f.
d.
4 x! 3 x!5 x+ 3 x! 7 x! 3 x+11
1 x2
Algebra Connections
Lesson 11.1.3 11-27.
a. Each input (pushing a button) relates to an output (a can of soda). b. Input: possible buttons to push and money; output: possible types of soda c. No, it is not. You cannot predict the output when Lemon Twister is selected. d. Yes; Based on this information, you can predict that every time the Blast is selected, the output will be a can of Blast. e. Yes; Based on this information, you can predict that every time Slurp is selected, the output will be a Lemon Twister. f. Relations that are functions have a predictable (and unique) output value for each input value. Relations that are not functions have more than one output for at least one input, which makes them unpredictable.
11-28.
Typical response. “A function is a relation in which each input has only one output.”
11-29.
a. No; Button 2 gave two different types of candy b. Yes; each input has only one output c. No; x = 2 has two different outputs d. No; at least one x-value has more than one y-value e. Yes; each x-value has only one y-value f. Yes; each x-value has only one h(x)-value g. Yes; each input has only one output
11-30.
No; vertical lines are not functions.
11-31.
Some possible machines are an ATM machine, a calculator, a radio, etc.
11-32.
1, 5, ! 8.54
11-33.
u = 4 , v = !3
11-34.
a. –7
11-35.
a. x = 8 or x = !2
b. x = ± 7
11-36.
a. 25a !22b 36
b. 5 ! 3"1 x "9 y 5
11-37.
B
Answer Key
b. –1
c. 9
d. 34 c. x = 1 or x = !3
d. x = !5 or x = !1
79
Lesson 11.1.4 11-38.
a. ! –2.3, 0, 3.3
11-39.
a. –1, –2, –6 b. There is no solution because you cannot divide by zero. c. No; the error occurs when the denominator is 0, and 3 is the only value that causes that to happen. d. All numbers except x = 3 a. Yes; each input has exactly one output. b. !2 " x " 4 c. !1 " y " 3 d. No; he is missing all the values between those numbers. The curve is continuous, so our description needs to include all the numbers, not just the integers.
11-40.
11-41.
b. 2, –3
c. ! – 3.7
See solutions below. a. D. !3 " x " 3 , R. !3 " y " 3 c. D. !2 " x " 4 , R. ! 4 " y " 2 e. D. 2 ! x ! 4 , R. !3 " y " 2
11-42.
d. No
b. D. ! " < x < " , R. ! " < y < " d. D. ! " < x < " , R. y ! 4 f. D. –2, –1, and 4, R. – 4, –1, 1, and 2
a. No; we only know that the integers used in the table worked. We do not know about the numbers between the integers or those beyond the table. b. Not quite. If we knew that f (x) was a parabola, then (0, – 4) would be the vertex and then the range would be the set of numbers greater than or equal to – 4. However, since we were not told the rule, that is an assumption. In fact, we cannot even assume that the relation is continuous; it could just consist of the points listed in the table. c. No.
11-43.
a. ! " < x < "
11-44.
There are many possible solutions. See example at right.
11-45.
a. not a function as more than one y-value is assigned for x between –1 and 1inclusive b. appears to be a function c. not a function because there are two different y-values for x = 7 d. function
11-46.
a. x-intercepts (–1, 0) and (1, 0), y-intercepts (0, –1) and (0, 4) b. x-intercept (19, 0), y-intercept (0, –3) c. x-intercepts (–2, 0) and (4, 0), y-intercept (0, 10) d. x-intercepts (–1, 0) and (1, 0), y-intercept (0, –1)
11-47.
Marisol. y = 2x , Mimi. y = 3x ! 3 , solution: x = 3 hrs, so 6 miles
11-48.
suur No; the slope of AB is
11-49.
a. x = 6 + 23 y
80
b. All y-values greater than –2.
3 5
suur , while the slope of AC is
b. y = 23 x ! 9
c. r =
d t
d. r =
C 2!
5 8
suur and the slope of BC is 11-50.
a.
2 x!5 x!6
2 3
.
b. x + 4
Algebra Connections
Lesson 11.1.5 11-53.
a. 4
b. 3
c. 1
11-54.
(1) D: ! " < x < " , R: ! " < y < " ; (2) D: ! 4 " x " 0 , R: 0 ! y ! 2 ; (3) D: all x-values except x = 0 , R: all y-values except y = 1 ; (4) D: x " 3, R: y " 0; (5) D: ! " < x < " , R: y > 0; (6) D: all x-values except x = 0 , R: y > 0
11-55.
They are the same shape, but one is shifted up two units. ; x ! "0.5 or –5
d. 2
e. 6
f. 5
11-56.
3x! 4 2 x+1
11-58.
y = 5x ! 22
11-60.
a. 15x 3 y b. y
c. x 5 d.
11-62.
a. ! 4 " x " 4
b. 0 < x < 3
11-63.
Graphs (a) and (b) have a domain of ! " < x < " , while graphs (a) and (c) have a range of ! " < y < " .
11-64.
a. x = ± 4 d. x =
11-65.
11-57. a. 8
b. 1
11-59. a. 9
b.
8 x3
c.
13 5
d. no solution = 2.6
d. 7
11-61. a. 2
b. 53
c. 1 < x ! 6
d. ! 5 " x < !4
c. x = 4 or x = !2
b. (–5, –17)
!1± 57 4
14 3
c. –2
" 1.64 or –2.14
See graph at right.
Lesson 11.1.6 b. y = ! x + 5 + 4
11-68.
a. y =
11-69.
a. D: the set of non-negative numbers; R: the set of non-negative numbers. b. D: x " –2; R: y " –3. c. The domain and range are each shifted along with the graph.
11-70.
a. 3
11-71.
x. (0, 0) and (4, 0), y. (0, 0), vertex. (2, 4)
11-72.
a. 12
11-73.
All equations are equivalent and have the same solution. x = 4 .
11-74.
a. 12x 2 + 5x ! 2 b. 3m 2 + m ! 2
c. !5k 2 + 26k ! 24
11-75.
a. x ! 12
c. x < 0
Answer Key
x!3+2
b. 1
c. 2
b. 59
c. 7
b. !10 < x < 10
d. 9
e. –13
f. –5
d. x < !5 , x > 1 81
Lesson 11.2.1 11-76.
a. x: (–2, 0) and (5, 0), y: (0, –10) b. x: (1, 0), y: (0, 2) c. (–3, 8) and (4, –6) d. They both represent a point at which two paths “cross,” but intercept specifically represents a curve crossing an axis, while intersect refers to any point where the graphs of two equations cross.
11-77.
a. Intercept since the candle will stay lit until the weight is 0 grams b. Since we have two different growing patterns and we want to know when they charge the same amount, we are looking for an intersection. c. While it sounds like an intersection problem (two people are involved), the question asks for the amount of money when time = 0. Thus, we are looking for intercepts.
11-78.
a. Equal Values Method, Substitution Method, and Elimination Method b. Using the Substitution Method, x 2 ! x ! 12 = 0 , and x = 4 or x = !3 . The points are (4, –6) and (–3, 8).
11-79.
The solutions are ( 12 , 2) and (!1, ! 1) . The graph shows y =
1 x
with no roots
nor y-intercept and y = 2x + 1 with x-intercept (! 12 ,!0) and y-intercept (0, 1) . 11-81.
No solution; you cannot divide by zero.
11-82.
Yes, they will intersect; top line. y = ! 14 x + 10 , bottom line. y = 13 x + 3 ; they will cross at (12, 7).
11-83.
top line. x-intercept (40, 0) and y-intercept (0, 10); bottom line. x-intercept. (–9, 0) and y-intercept (0, 3)
11-84.
Both (a) and (d) are equivalent. One way to test is to check that the solution to 4(3x ! 1) + 3x = 9x + 5 makes the equation true (the solution is 23 ).
11-85.
The line should pass through (–2, 5) and (0, 2); y =
11-86.
(a) and (b) are functions because each only has one output for each input.
11-87.
a. D. –! < x < ! , R. 1 ! y ! 3 c. D. x ! "2 , R. –! < y < !
11-88.
All graphs have lines of symmetry. Graph (a) has multiple vertical lines of symmetry, one at each maximum and minimum; graph (b) has one line of symmetry at x = 1 ; graph (c) has one line of symmetry at y = 1 .
82
!3 x + 2 . 2
b. D. –! < x < ! , R. y ! 0
Algebra Connections
Lesson 11.2.2 11-89.
Solutions vary. C is impossible
11-90.
a. one solution. (1, 1) d. one solution. (–1, 0)
11-91.
a. Using a standard window, it appears that there is only one point of intersection. ) and ( 23 , ! 3) . b. There are actually two points. ( 43 , ! 28 9
11-92.
a.
11-93.
a. (3, – 4)
11-94.
He sold 9 watermelons.
11-95.
(6, 20) and (–1, 6)
11-96.
A
11-97.
B
1 5
or –3
b.
1 2
b. no solutions e. no solution
or –3
c. two solutions. (1, 2) and (3, 2) f. two solutions. (1, – 4) and (–2, –7)
c. 3
d. –7 or 2
b. D: ! " < x < " , R. y ! – 4
Lesson 11.3.1 11-98.
The only relation with no match is y = x .
11-99.
a. D. !2 " x " 2 , R. !4 " y " 4 b. D. ! " < x < " , R. y = 3 c. D. 0 ! x ! 4 , R. 1 ! y ! 5 d. D. x > 0, R. all y-values except y = 0
11-100.
The graphs in parts (a) and (d) are not functions because they have two y-values for at least one x-value.
11-101.
x ! "5 because the denominator cannot be 0
11-102.
y = ! 25 x + 4
11-103.
(–2, –10)
11-105.
a.
Answer Key
4 x2 y4
b. 18x 2 y
c. xy 2
83
Chapter 12 Lesson 12.1.1 12-1.
a. (x + 7)(x ! 7) b. (x + 6)(x ! 4) e. (5x + 1)(x ! 1) f. (2x + 5)(2x ! 5) i. (7x + 1)(x ! 3) j. (2x + 5)2
12-2.
(a) and (f) are both difference of squares, (c) and (d) are perfect square trinomials, and (b) and (e) are neither. Factored forms: a. (5x ! 1)(5x + 1) b. (x ! 9)(x + 4) c. (x + 4)2 d. (3x ! 2)2 e. not factorable f. (3x ! 10)(3x + 10)
c. (x ! 5)2 g. (x ! 3)2
d. (3x + 2)2 h. (x + 6)(x ! 6) k. not factorable l. (3x ! 1)(3x + 1)
12-3. !b !abx
!b 2
b
abx
b2
a2 x 2
abx
ax
a2 x 2
abx
ax
+b
ax
b
ax
x+ 3 x! 3
b.
1 x+ 7
12-5.
a.
12-6.
a. x = 1.5y + 5
12-7.
a. 1
b.
12-8.
a. 3
b. 2
12-9.
a. !1 " x " 3
20 x
c. 1 b. x = 24 c.
d.
x+6 x!5
c. x = 5 t3
22 27
! 0.81
d. x = 1
d. x 9 y 3
c. does not exist
d. 0
e. 1
f. 1
b. x = !4 ±2 28 " 0.65 or ! 4.65 c. all numbers d. x = !2.5 or 5.5 12-10.
84
Typical response: Add (or subtract) the numerators, but keep the denominator the same. 5 11 a. i. 11 , ii. x+2 b. 15 6
Algebra Connections
Lesson 12.1.2 x+2 6
5 11
,
12-13.
a.
x!2 x!5
12-14.
a. Because if x = 4, then the denominator is zero. Since dividing by zero makes the expression undefined, x # 4. 1 b. a. x # ! 13 and x # 5; b. x # 3 or –3 c. Answers vary. Example: (x+6)(3x!1)
12-16.
a.
12-17.
a. all numbers
12-18.
a. 5m 2 + 9m ! 2 b. !x 2 + 4x + 12 , c. 25x 2 ! 10xy + y 2 d. 6x 2 ! 15xy + 12x
12-19.
a. not a function, D. !3 " x " 3 and R. !3 " y " 3 b. a function, D. !2 " x " 3 , R. !2 " x " 2
12-20.
See graph at right: x-intercept. (–2, 0), y-intercept. (0, –2); there is no value for f (1) , which creates a break in the graph.
12-21.
a. –15
9m+27 m+ 3
,
11 15
12-12. a.
1 x! 3
=9
b.
5 x+1
b.
2 x! 3 x+ 3
=
12-11.
b.
1 2(2 x!5)
1 4 x!10
d. 1
a+ 3 a!1
b. !5 < x < 4
b. – 4
c.
d. x =
c. no solution
c. 3
d. !m 2
c. 3
d.
1 3
Lesson 12.1.3 19 24
x+ 3 x+5
a.
12-23.
a. The first term needs to be multiplied on top and bottom by x ! 7 , while the second expression needs to be multiplied on top and bottom by 3x + 1 . x+2 x 2 !5 x+10 b. (x+54)(x! c. x+ 4 7)(3x+1)
12-24.
a.
2 x! 7 x!9
12-25.
a.
2 3x+1
12-26.
a. 1
b. 4
c. 2
d. 5
12-27.
a. x = 3
b. 0 ! x ! 6
c. x = 1 or 5
d. x < 2 or x > 4
12-28.
x = !3 or –11
12-29.
a. 1
b. 3
12-30. Answer Key
b.
2 9
12-22.
b.
x! 3 2 x+1
c. 5 b.
x! 7 x! 3
d.
x+2 2 x+ 3
c.
x!2 2 x+12
e.
2 x+5 x 2 !2 x!15
d.
f.
3x+1 4 x+1
1 x+2
c. 2
(–3, 8) and (1, –12) 85
Lesson 12.2.1 12-32.
1 10
a.
of the lawn
b.
1 15
of the lawn
c.
1 10
1 + 15 =
1 6
d. 6 hours; if together they mow 16 of the lawn per hour, then it will take 6 hours to mow the entire lawn. e. Equations vary; possible equations where t represents the time it takes for both 1 + 1 =1 teams to mow the entire lawn: 10 and 10t + 15t = 1 . 15 t
+
1 10
12-35.
125 streamers
12-37.
1 2
1 30
= 1t ; t = 7.5 minutes
12-33.
12-34.
1 8
+ 1t =
1 6
; t = 24 minutes
12-36. x = 4 ± 5
+ 13 = 1x , x = 65 ! hrs = 1 15 ! hrs = 1 hour and 12 minutes
12-38.
If d = number of dimes and q = number of quarters, then q = 2d ! 6 and d + q = 147 . Then d = 51 and q = 96 , so Jessica has 51(0.10) + 96(0.25) = $29.10 .
12-39.
They all are equivalent to 12x 6 . 12-40. !x 2 + 2x + 3
12-41.
a. x = 1
b. x < !1 or x > 7
c. x = 7
Lesson 12.2.2 12-42.
If x = number of 7th graders and y = number of 8th graders, 0.4x + 0.9y = 720 and x + y = 1000 . Then x = 360 and y = 640 .
12-43.
Let c = number of ounces of Choco-nuts and m = number of ounces of Munchies. Then 1c + 2m = 5.4 and c + m = 4 ; c = 2.6 ounces and m = 1.4 ounces.
12-44.
Let p = amount of powder blue and s = amount of spring blue. Then, 0.02 p + 0.1s = 0.04(1) and p + s = 1 , so p = 0.75 gallons and s = 0.25 gallons.
12-46.
6x + 12 = 4.75(x + 3) ,
12-47.
a. 3
12-48.
The graph is a “V” shape with vertex at (3, –1) and x-intercepts (4, 0) and (2, 0).
12-49.
a.
12-50.
(3, 14) and (0, 2)
12-51.
a. (x ! 7)2 = 25 , x = 2 or 12 c. (x ! 99)2 = 4 , x = 97 or 101
86
b. 2
x! 4 x+5
9 5
pounds
c. ! 3.24
b.
x2 + 4 (x+ 3)(x+2)
d. There is no solution
c.
6 x+9 x+1
d.
5 x+2
b. (x + 3)2 = 18 , x = !3 ± 3 2 d. (x ! 4.5)2 = 36 , x = –1.5 or 10.5 Algebra Connections
Lesson 12.3.1 12-53.
a. 0.75 and –2.5
b. !5 ±
12-54.
a. 5 ±
b. !1 ± 19
12-55.
0.12(2 + x) = 0.08(2) + 1.00(x) ,
12-56.
The graph starts at (3, 1); D. x ! 3 , R. y ! 1 .
12-57.
a. –1
12-58.
a. (x ! 9)(x + 9)
12-59.
(1, 3)
12-60.
Answers vary. Typical responses:
3
b. 2
c. –2
1 11
31
c. no solution
d. ± 2.5
gallon
d. –1
b. (x + 6)2
c. (2x + 1)(2x ! 3)
x5 x3
d. (4x ! 5)(4x + 5)
, x 3 x !5 , and (x !1 )2
Lesson 12.4.1 12-66.
Teresito is correct because neither point makes both equations true.
12-67.
16
12-68.
See solution graph at right; D. x ! 3 , R. y ! 0 .
12-69.
a. not possible because of dividing by 0 b. !27 c. 8 d. 0 e. –1
12-70.
Answers vary, but likely answers are 6(m ! 2) , 2(3m ! 6) , 3(2m ! 4) , and 1(6m ! 12) .
12-71.
(1, 12) and (–5, 42)
Answer Key
87
Lesson 12.4.2 12-72.
A. y = x 2 + 6x + 6 , B. y = x 2 + 4x + 6 , C. y = x 2 + 4x + 5 A. 292,678, B. 291,602, C. 291,601 Figure 742, Figure 699, Figure 778
12-73.
a. 94,557 tiles
b. Figure 411
12-74.
a. x = !12 c. !33 " x " 27
b. 4 ! 12 < b < 4 + 12 d. n = 15 or 2
12-75.
Roots (! 0.2, 0) and (! 3.1, 0) , vertex (! 1.7, ! "6.3) ; solution graph is shown at right.
12-76.
3 a. 3 11
b. 7
d. undefined
e. undefined f. 3 97
3(x + 2) ! 2 =
g.
c. undefined
3x + 4
h.
(x!1)2 (x!1)+5
=
x 2 !2 x+1 x+ 4
for x " !4
12-77.
a.
3x 2 + x! 3 2 x 3 +9 x 2 !5 x
12-78.
A
12-79.
12-80.
a. 44,941 tiles
12-82.
y = ! 23 x ! 1 12-83. After 365 days (1 year), each will have $267,884. 12-84. B
b.
3x!5 2 x+ 3
a. D. x ! 2 , R. y ! 4 b. Yes; each input has only one output. b. Figure 80
12-81. 1 87 hours, or 1 hour and 52.5 minutes
Lesson 12.4.3 12-85.
Information that should be included. it has symmetry about the line x = !1 , it has minimums at (–3, 2) and (1, 2) and has a local maximum at (–1, 6).
12-86.
223
12-88.
dashed curve, shade above and inside the curve
12-90.
a. yes
12-91.
a. 3 ± 21
12-93.
a. Answers vary but should be close to y = !54x + 1617 . b. about the year 2030
12-94.
a. The line passes through (–1, 7) and (3, 8).
88
12-87. ! "3.83, " 1,!and 1.83 12-89. No
b. no; most inputs have two outputs c. no; x = !1 has two outputs b. 2 ±
7
12-92. a. 0
b. 2
c. 2
d. 1
b. f (x) = 0.25x + 7.25
c. – 4
Algebra Connections
Lesson 12.4.4 a. If x represents the length of the sides of the pen that are attached to the barn, then possible points are. (1, 28), (2, 52), (3, 72), (4, 88), (5, 100), (6, 108), (7, 112), (8, 112), (9, 108), (10, 100), (11, 88), (12, 72), (13, 52), and (14, 28) b. Of the pens with integer lengths listed in part (a), the largest pens would be when x is 7 or 8 meters.
12-97.
a. See graph to the right. c. No, because these represent the lengths that yield zero area.
12-98.
b. 7.5 by 15 meters c. Between 112 and 115 square meters is acceptable. Actual maximum area is 112.5 square meters
12-99.
a. 30 ! 2x Length of the Sides Attached to b. y = x(30 ! 2x) , which simplifies to y = !2x 2 + 30x Barn (in meters) c. upside-down parabola d. (0, 0) and (15, 0) e. (7.5,112.5) f. 112.5 square meters, 7.5 m by 15 m; The equation yields an exact answer.
12-101.
(!2.5, ! 36.75) ; Due to the parabola’s symmetry, the vertex must have an x-intercept that is halfway between the x-intercepts.
12-102.
a. y = 2!or!!2
12-103.
(6, 8) and (5, 3)
12-104.
D. x < 2 and x > 2 , R. y < 0 and y > 0 ; Solution graph shown at right.
12-105.
8x + 10(6) = 8.5(x + 6) , 18 pounds
12-106.
If t represents the time, then 38 + 16t + 52 + 20t = 252 and t = 4.5 hours
12-107.
(–2, 5) and (6, 21)
12-108.
y = ! 13 x
12-109.
See graph at right.
12-110.
a. 0
Answer Key
b. 2
Area of Pen (in m2)
12-96.
b. x = 8 (–3 is excluded)
c. 1
d. 2
89
