Chapter a. True b. True c. False d. True

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1 Chapter 1 Lesson a. b. c. d. e 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 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 y- coordinates 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 a. 9 b. 1 c. 7 d. 4 e. 6 f by 18, 2 by 9, 3 by a. ( ) 2 = 80.5% b.! 83.67% a. Diagrams vary, but one way to describe it is as a whole. b. Diagrams vary; one-half, one-fourth, one and one-half c. Diagrams vary, 33 1 % d. answers vary: 10%, one tenth a. (-3, 1) and (-2, 3) b. (2, -1) and (3, -2) Answer Key 1

2 Lesson 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 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 a. b. c. d e. f. g. h x 5 x 2 2x x a 7 ab a+b b a. 8 b. 29 c. 8 d. 22 e. 80 f. 12 g. 0 h. 3 i. 5 j. 30 k. 0 l a. 5 b. 12 c a. 430 b a. 8, etc. b. 0.8; no c. 80% by 4; P = inches % MT! 45 sq. un., CA! 51 sq. un a. i. ii. iii. iv b. answers vary 2 Algebra Connections

3 Lesson b. answers vary c. The 100 th figure has 10,200 tiles Figure a. 15, 21; add increasing amounts. b. 1 16, 1 32 ; multiply by 1 2. c. 81, 243; multiply by 3. d. -2, -7; subtract increasing amounts. e. 53, 58; subtract 2, then add 5 repeatedly a. b. c. d b d c a e is closer a. 15 by 18 = 270 ft 2 b. 28 by 23 = 644 ft 2 c = 120 ft 2 Answer Key 3

4 Lesson 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 a. If the base is 10, the height must be 18 and the total area is (20)(24) = 480 ft 2. If the height of the living room is 10, the base must be 18 and the total area is (28)(16) = 448 ft 2. 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 ft 2. d. answers vary a. b. c. d e. f. g. h a. 74.5%, No b % c. It lowered her score a lot. Yes He owns 9 spy novels and 17 science-fiction novels 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 a. 9 b. 34 c. 27 d. 20 e. 85 f. 8 g. 5 h. 1 i. 54 j. 25 k l. 26 Lesson bulls-eyes a. Adele $320, Amanda $160, Alisa $160 b. 9 units by 15 units c. 32 laps for Hector d. 984 and 2568 students e. 6 and Todd: 37 years old, Jamal: 27 years old , 21, = 27, 3 4 = 81, 3 5 = 243, 3 6 = 729, 3 7 = 2187, 3 8 = 6561, 3 9 = a. The pattern 3, 9, 7, 1 repeats. b. 3 5, 3 9, and a. 2, 5 b. 2, -4 c. 3, 5 d. 3, cm. 4 Algebra Connections

5 Lesson a. 31 tomato paste, 13 noodles b. It cost $21.70 for tomato paste and $6.50 for the noodles, total cost was $ million $ , 51, 32 inches birds cm, 32 cm, and 28 cm a. A 5 unit square with a 4 unit square inside it. b. 9 sq. units b. It grows by two tiles each time. c. Methods may vary, but the 100 th figure will have 201 tiles 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.! 450; 250; 50 Answer Key 5

6 Chapter 2 Lesson 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 x x 2 + 8x a. perimeter = 120 units, area = 647 sq. units b. perimeter = 70 units, area = 192 sq. units 2-9. perimeter: 52, area: a. 11 b. 14 c.!30 d m and 51 m Lesson unit tile: 4, 5-piece: 12, x-tile: 2x + 2, x 2 -tile: 4x, y-tile: 2y + 2, y 2 -tile: 4y, xy-tile: 2x + 2y a. 4x + 2y + 6 b. 2x + 4 c. 2x + 4y + 2 d. 4x + 2y a. 34 units b. 10 units c. 40 units d. 34 units 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 = 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 a. 7 b. 14 c. 2 d a. 10 feet by 15 feet b. bathroom = 63 sq. ft, kitchen = 105 sq. ft, living room = 150 sq. ft c. 318 sq. ft A( 4, 3), B(2, 1), C( 2, 0), and D( 3, 3) x 2 + 4x + 3xy + 6y + y 2 6 Algebra Connections

7 Lesson Ideas should include subtraction, opposite, and negative They are all correct. 2 4 is a. 4x 2 + 6x + 13 b. 5y 2 + 8x + 19 c. 9x 2 + x + 44 d. 5y 2 + 6xy a. 14 b. 6.5 c. 74 d. 12 e a. a square b. 9 units c. 81 square units d. 36 units beige, 107 red, and 246 navy blue a. 45 b. 10 c. 2 d. 254 e.!1 4 f. 2 Lesson Each situation deals with equal opposites so that the net effect is zero. They can be represented with equations, such as $15 $15 = $0, = 0 yards, and (-x) + x = 0, if x represents the level of the sand above sea level a. One possible expression is 2x! x + 1! 3. b. answers vary answers vary a.!2x! 2 + x! (!2) =!x b. x + 1! x! 2! (x! 3) =!x a. x! 4 b.!2y x x Parts (a) and (b) are possible, but part (c) is not a. 3 b. 18 c: 24 d: 30 e: 6 f: 8 g: 12 h: a. 2x! 3! (x + 1) = x! 4 b. y + 3! y! 1 = 2 c.!x! (x + 2) =!2x! x a. b. c. d Answer Key

8 Lesson answers vary 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 a. left expression is greater b. left expression is greater c. right expression is greater a. both are equal b. right expression is greater a. 4x 2 + 3x! 3 b. 5x 2! 23x c. 4x + 3y d. 2y 2 + 3xy ! 1 6! a. The steeper line is B b.! 3 years c.! $65,000 d. Company B a. 832 b. 416 c. 208 d a b. 36 c. 1.5 d. 14 Lesson a. left b. right c. left d. right e. neither (they are equal) f. right b. No; it depends on the value of x a. 2x! 1 b. 4 c. x 2! y! a. 1=D, 2=A, 3=C, 4=B b. 6, 4, 24 sq. units c. the area " 36 d. Possible solutions: 10 by 3.6, 15 by 2.4, 12 by 3 e. curve a. 17 b. 9 c. 45 d. 10 e. 22 f a. $ = $2.19 b. $ gallons = $32.85 c. $8.76 $2.19 = 4 gallons a. 6 b = 3 4! 8 1 = 6 8 Algebra Connections

9 Lesson a. neither (they are equal) b. cannot be determined (depends on the value of x) c. left a. Not enough information b. The right side is greater c. The left expression is greater d. They are equal 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 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 a. 4y + 2x + 2 ; 4(10) + 2(6) + 2 = 54 units b. y 2 + 2x + 1 = (10) 2 + 2(6) + 1 = 113 square units a. 4 b. 1 c. 7 d. 2 e. 5 f. 6 g. 3 Lesson a. The right side is greater b. Cannot be determined, because it depends on the value of x a. x < 5 b. x > 5 c. x = a. x = 3 b. x =! a. x = 3 b. x = 6 c. x = 2 d. x = 4 e. x = 2 f. no solution a. The right side is greater b. The left and right expressions are equal , a. 2y! 2x + 3 b. 2x 2 + 3x + 6 c. 0 d. x! y (-3, 2), (-6, 4), (9, 6) and various others a.! b. 18 c. 20 Answer Key 9

10 Lesson a. x = 3 b. no solution a. 3 b. 0 c. all numbers a. 2 b. no solution c. 3 d possible solution: 2! 2x! (4! x) = 2! 3! (x! 2) weeks Sample solutions: ( 4, 2), ( 6, 3), (4, 2) a. x = 2 y = 5 b. x = 2 y =!4 c. x = 3 y = 5 d. x =!3 y =! a. 60 b. 32 c d. 6 Lesson a cm and 25.6 cm b. $105, $ c.! 44 minutes, 102 transparencies d. 300 tamales e.! 20 cm,! 1.4 cm P = 88 m, A = 369 sq. m x +1! 2! (!x + 3) = 6! (1!1), x = a. 3 b a. 4x 2 + 5x! 4y! 3 b.!6x 2! 2x + 8 c.!3y x! 5y d The 24-ounce bag is best. It costs roughly 5 per ounce while the 36-ounce bag costs about 8 per ounce a.! $40 b.! $55 c.! $75 10 Algebra Connections

11 Lesson a. 42 tubes in 5 years, 16.8 tubes in 2 years, 11.9 years b. x! 7.83, y = a. 200 blue candies b. $1495, $14950, $7475 c. $ , 32 games a. 3x! 1! (1) = x! 3! (5), x =!3 b.!11 =! a. 2 b a. 35g, 70g b. cookies: 200, 400; sugar: 35, 70, 7, 3.5, answers vary Lesson a. Corresponding vertical sides represent the number of origami cranes, while the bottom pair represent minutes. b.! minutes c. He s trying to find how many cranes he can produce in 250 minutes, which is 7 25 = 175 cranes answers vary eggs a. Car A b. A (2, 120); B (4, 120) c. 60; a. 3 b. 1 c. 1.5 d a. 2x 2 +12x! 8 b. 8y 2!10y +1 c. 30xy + 4y! 4x d. 0.8x Answer Key 11

12 Chapter 3 Lesson a. x (in) y (out) b. x (in) y (out) c. x (in) y (out) A C Easy Hard L N Dark Light heptagon D F Hot Cold quadrilateral Q S Up Down W Y Left Right decagon Rule. y is two letters after x Rule: y is the opposite of x Rule: y has two more sides than x d. x (in) y (out) e. x (in) y (out) f. x (in) y (out) or Rule: Double x and add 1 Rule: Halve x, and add 1 Rule: Square x 12 Algebra Connections

13 3-3. a. x (in) y (out) b. x (in) y (out) c. x (in) y (out) or or 7 50 Double x, then subtract 2 Multiply x by 3 Square x and add 1 d. x (in) y (out) e. x (in) y (out) f. x (in) y (out) Subtract 6 from x Multiply x by x + 1 Multiply x by 5 and add a. left. -4, right 7 (left is greater) b. left. -5, right. -1 (right is greater) 3-5. a. -45 b. 6 c. 62 d a. It subtracts 6 from the input value b $50, $250 and $ a. Car B, line is steeper. b. In 2 hours, both cars were 80 miles away from the game. c. 160 miles Answer Key 13

14 Lesson a. 13 ft. b. 33 ft. c. 5 ft. d. about 200 ft a. Line b. Yes c. 11 d. No a. 0 th year, 5 ft; 1 st year, 9 ft; 2 nd year, 13 ft; 6 th year, 29 ft b. y = 4x + 5 c. 205 ft a. x + 4 = 3x! 2, x = 3 b. x + 2 =!x! 1, x =! a. 31 b a. 2x 2! x + 5 b.!2y 2 + 4x! 8 c. 2x y! 3y 2 + 4xy!14! x d a. -13 b. 4 c. 36 d. 4 e. 16 f. 3 g. They are different a. $22.50 b. $ Algebra Connections

15 Lesson 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 a. The points are in a line. b. The points should be co-linear c. The points would follow the pattern 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 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 Using the rule, the 100 th figure will have 6(100) + 3 = 603 tiles. To use a graph, change the viewing window to include x = 100 and y = x +1! 2! (!x + 3) = 6! (!1+1), x = a. -5 b. 10 c. 15 d. 2 e. -10 f. 3, a. 2 b. no solution a. 26 b. 4 c. 2 d See table below. IN (x) x OUT (y) x a. Multiply x by negative 3 b. -3x a. 2x + 1! (!x! 2) = 2! (!2x);!!x =!1 b. 0 = a. 4 b a. 3x 2 + 2x + 8, 5x 2 + 2, 8x 2 + 2x +10 b. 7x 2! 7x!17 c. 3 4 x x = x Answer Key 15

16 Lesson IN (x) OUT (y) IN (x) OUT (y) 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 IN (x) ! OUT (y) b. (0, 0) The line should pass through the points (0, 2) and (1, 3) IN (x) x OUT (y) x+1 a. Multiply x by two and add one b. y = 2x a. 2 b a. 37 b. 5.5 c. 3 d a.! 71.4 or 72 tomatoes b.! 81 people 16 Algebra Connections

17 Lesson 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;! $ b. (4,-2) c. possible answers: (2,-2) or (4,0) See table below. IN (x) OUT (y) b. No; when x = 10, y = Rule: y = x 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 See table below. IN (x) x OUT (y) a. Divide x by two and then add one b. x 2 +1or 1 2 x a. 2 b. 3.5 x 2 +1 Answer Key 17

18 Lesson 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 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) See table below. x y See table below. x y 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 y + 4x + 2, 2(7) + 4(3) + 2 = 23 units a. 1 b. 2 Lesson 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 = 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 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 a. 84 minutes b.! minutes b. ( 3, 2) c. a parabola d. ( 3, 2) should be ( 3, 5) c. The parabola in part (a) points upward, while the parabola in part (b) points downward d.! ± 2.6 e.! ± x = Algebra Connections

19 Lesson a. 4x! 5 = 2x + 3 b. x = a. No, 100 does not make the equation true, x = 10 3 b. Yes, 4 makes the equation true a. 2 b. -2 c. - 3 d. no solution a. -2 b. 1 c. 1 d See table below. IN (x) OUT (y) b. A slanted line that decreases as x increases a. 2x + 6 b. 4x + 6 c. 4x + 4y d. 6y! 2x a. $68.75 b. 20 pounds Lesson 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 a. all numbers b. x =!4 c. no solution d. all numbers e. no solution f. x = a. y = 2x + 2 b. (5, 12) a. 2 b. 1 c. all numbers d a. yes b. no c. no d. yes a. See table below. b. y = 2x!1 x y Answer Key 19

20 Lesson tiles a. y b. 45 = 6x + 3, Figure 7 c. Replace x with 7 in the rule 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 a. 4 b. 8 c. any number d x = a. 42 b. 3 c. 12 d a. no solution b. 3.5 c. any number d and 12 cm 3-95.!(x! 3) represents the opposite of x! 3, which is!x See table below. IN (x) OUT (y) b. a parabola opening downward 20 Algebra Connections

21 Lesson 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 a. 10 b. no solution c. -12 d a. y = 0.15x b. 9 = 0.15x, so x = $60 c. $60 + $9 = $ a. b. c. d a. 4.5 pounds b. 40 dozen a. any number b. m = 9 c. x = 4 d. p =! See table below. IN (x) ± x OUT (y) !4 3 4 x 2! 5 a. Multiply x by itself and subtract 5 b. x 2! 5 c. a parabola 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

22 Chapter 4 Lesson The parabola should pass through the points (0, 0) and (2, 0) and have vertex (1, 1) She should have received two sports cars and ten pieces of furniture ! 3! (!2x) =!x! 3! (x + 2) a. x =! a. 2x 2 +11x! 4y! 3 b.!2x 2! 2x! 8 c.!y y!14 d a. y = x Lesson b. Figure 0 has 2 tiles. c. The pattern is growing by 4 tiles. d. 100 th figure has 402 tiles 4-9. Both patterns grow by 4 tiles 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 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 x + 1! 2! (!2x + 3) = x + 5! (2! 1) a. x = a. 3 b ! 83 words See table below. IN (x) x OUT (y) x + 5 a. Multiply x by two and add five b. y = 2x Each sister should receive 1 poster, 72 books and 8 CD s. 22 Algebra Connections

23 Lesson 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 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 c. y = 2x a. OUT: 5, 0, -3, -4, -3, 0, 5, A parabola opening upward a. 2 b cm long and 15 cm wide OUT: -2.5, -2, -1.5, -1, -0.5, 0, 0.5 Lesson a. They all have something added and/or multiplied by x. b. how each figure changes c. The number of tiles in the 0 th figure y = 3x a. y = 5x! 2 b. i: y = 2x + 5, ii: y =!3x Answers vary Answers vary a. all numbers b. x = If x is my number, then 5! 2x =!1; x = IN: 5, -5, 4, 1.5, 50; OUT: -11 a. Multiply x by three and subtract two a. Answers vary b. Each figure has 4 more tiles than the figure before it Figure 0 has 7; figure 1 has 9; the growth factor is 2. Answer Key 23

24 Lesson a. y = 4x + 4 b. y = 9x! 5 c. y =!2x a. 4 b. no solution c. 3.5 d a. b. c. d See table below. x y b. Yes. These values for the variables make the rule true 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 a.!5x!1 b. 11x! 1 c. 5x! 1 d.!4x! Answers vary a. any number b. 0 c. 6 d. no solution 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 = b. Yes, -29 = Number of Letters Cost of Stamps 10 $ $ $ $ $ $ $ Algebra Connections

25 Lesson a. y = 2x + 2 b. y = 4x + 3 c. y = 8x + 5 d. y =!2x + 8 e. y = 4x + 8 f. y =!4x a. (0, 3) b. (0, 0) c. (0, 8) d. (0, 1) a. y = 4x + 3 b. y =!3x + 5 c. y = The pattern shrinks by two tiles between figures. Figure 0 has 13 tiles a. A and C have 5 tiles, B has 15 tiles. 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 a. They each have 5 tiles. b. The second a. 4 b tsp Lesson a. y = 3x + 2 b. y =!3x Figure y = 4x! 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) a. 5 b. 8 c. 7 6 d Figure 6; 40 = 6x Josie is correct. 10 inches long Answer Key 25

26 Lesson 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 hours; 1100 miles; Evie s line is steeper; Extend the line to 1100 miles and look at the time, about 230 hours b. (2, 2) 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 = minutes a sq. units, b x sq. units c. 900 sq. units d. 336 sq. units a. Leslie passes Gale b. 2:00 PM c. 18 miles d. Gale stopped to rest, flat tire, etc. e. at approximately 6:40 PM a. y = 3x! 1 b. y = 5x a. the opposite of y increased by 8 b. 48 less than two times x c. the square of the sum of x and 3 d.!6x 2 e. x x + 5 Lesson They will have the same amount of money in 5 weeks a. y = x b. y = x c. When x = 5 is substituted into both equation, y = 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) a. Science: y = x ; Math: y = x b. 6 weeks c. $ a. x 2 + 5x + 6 b. 4x y-int: (0, 1), x-int: ( 1 2,!0) a. (10)(17) = = 170 b x + 74 = 90 ; 48 days later (Thursday, June 19) 26 Algebra Connections

27 Lesson years a. 3x + 5 = x +19 b. x = 7 years c. 26 pounds a. Richmond: y = 1900!15x, Post Falls: y = x b. 1900!15x = x ; x = 20 years c students Ficus: y = x, Oak: y = 2x ; x = 12 years; 24 feet The solution of the equation formed is the x-value of the point at which the two rules have equal y-values dumplings, 12 eggrolls, 4 sweet buns and 4 drinks He is correct, because!2(2) + 4 = 0 and 2! 2 = a. (2,1) b. (2,1) a. (-2,-7) b. ( 1 2, -4) a. If x represents weeks and y represents total money, y = x and y = 172! 4x. b. They will have the same amount between weeks 15 and 16. Lesson 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 She is not correct a. (5,3) b. (2,-6) a. parallel lines b. no solution c. parallel lines do not intersect a. y = 12, x and y = 21,000! 250x ; y = total population and x = number of years b. They will have the same population in 16 years a. y = x b. After the 22 nd day This parabola should point downward and pass through (0, 0) and ( 2, 0). The vertex is at ( 1, 2) a. (0, 3) b. (0,! 12 5 ) c. (0, 12) If r is Regina s favorite number, then 3r = r and r = a. parallel lines b. no solution c. Parallel lines do not intersect. Answer Key 27

28 Chapter 5 Lesson 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) c. (2x + 3)(x + 2) d. (x + 1)(3x + 1) e. (2x + 6)(x + 2) or (2x + 4)(x + 3) f. 2y(y + 3) g. (y + x)(y + 2) h. (x + 1)(y + 1) i. (x + y + 1)(x + y + 2) j. (2y + 1)(x + 2) = y = 7x (2x + 4)(x + 2) = 2x 2 + 8x Answers vary No, he is not correct. Parallel lines do not have a point of intersection ,000 votes Lesson a. (2x + 3)(x + 2) = 2x 2 + 7x + 6 b. (y + x + 2)(x + 4) = xy + x 2 + 6x + 4y a. 8x 2 b. 2x 2 + 7x + 3 c. 2x x d. 4x 2 + 4x + 1 e. 2x 2 + xy f. 2x 2 + 2xy + 9x + 5y + 10 g. 6x + 10 h. 2xy + y 2 + 3y a. 12x x b. 24x + 6 c. 12xy + 9y d. 70xy + 77y a. (x + 1)(x + 3) = x 2 + 4x + 3 b. (2x + 1)(x + 2) = 2x 2 + 5x a. 238 b a. Multiply by 6 b. x = 15 c. x = a. m = 3 b. (0, 2) c. y = 3x! , ,500x = 18, ,300x, x = 40 months x-intercepts are (1.5, 0) and ( 1, 0), y-intercept is (0, 3). 28 Algebra Connections

29 Lesson a.!18x + 12 b. 4x 2! 2x c. 50t! 15t 2 d.! k! 4y (4x + 5)(2x + 3) = 8x x The area inside each smaller rectangle of the generic rectangle corresponds to the tiles in the same portions of the rectangle in problem 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 a. 3x x + 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 x + 60 c. (x! 2)(2x! 3y + 5) = 2x 2! 3xy + x + 6y! 10 d. Multiple answers possible a. 2x x + 30 b. 3m 2! 4m! 15 c. 12x 2! 59x! 5 d. 6! 7y! 5y a. 6 b. 16 c. 1.5 d a. 4, 0 b. 16, 6 c. 2, 14 d. 20, a. y =!2x + 1 b. x: (0.5, 0), y: (0, 1) c. y =!2x ounces These expressions are equivalent because of the Commutative Properties of Addition and Multiplication. Answer Key 29

30 Lesson a. 12x 2 + 8x! 55 b.!30x 2 + 6x c.!y 2 + 4y + 12 d. 48! 16m This can be built with 3 rows, each containing 2 x-tiles and 1 unit tile a. 4 b. 3( ) = A and C are correct x =! a. x = 9 b. x = 0 c. y = 6 d. x =!3.5 e. x = 2 f. x =! a. 1 b. 5 c. 2.5 d 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 Rena is correct y =!2x a. After 3 hours b ( 2, 1) No, on average he will make about 375 baskets. 30 Algebra Connections

31 Lesson a. 5 feet, the y-intercept is the constant b. 4 feet per year, the rate is the coefficient of the x b. y = 3x + 5 c. y-intercept is (0, 5); growth factor is See table below. y = mx + b form Growth factor y-intercept a. y = x! 7 1 (0, 7) b. y = x (0, 2) c. y =!5x! 2 5 (0, 2) d. y = x 1 (0, 0) e. y = 0 0 (0, 0) f. y =!3x! 4 3 (0, 4) a. y = 5 b. x = 5! 5 y 2 c. y = 6x! 11 d. x can be any number e. x =!1 3 y + 2 f. p = m g. y = x! 3 h. q =! a. 4x x + 15 b.!6x 2! 20x! 16 c.!3xy + 3y 2 + 8x! 8y d. 3xy + 5y 2! 22y! 12x a. x = y y = 3 + 2x b. w = p!9!3 c. m = 2n + 5 d. y =!3x b. no solution c. there is no solution to the system of equations because the two lines do not intersect = 25, 3, = 120 gallons Answer Key 31

32 Lesson a. x = 3 b. y = 1 2 x! 2 c. x = 3 d. y =! 1 x + 4 e. x = 2 2 f. y =!x + 3 g. x = y!4!3 h. x = 2 i. w = 3! 2v j. no solution a. x = 5 b. x = 2 c. y = 0 d. x = (x + 3)(x! 5) = x 2! 2x! 15 y See graph at right The numbers are 55 and a. 6x + 10 = 10x, after 2.5 hours b. 25 problems 3 3 x 5-62.! 1222 Lesson a. 18 b. 12 c. 4.5 d a. 34 people b. Yes c. 238,085 votes a. Ratios vary b. No c. When ratios are of corresponding parts, the ratios are equal d. The ratios used will vary. One example: = x 527. Therefore, x = , so Mr. Mears would probably receive roughly 358 votes from Carina s neighborhood a. 17 = x 85, x = 10 minutes b = x, x! 411 students minutes weeks; pounds answers vary a. x 2 + 9x + 20 b. 2y 2 + 6y a. 3x 2 + 5x +13 b. 5x 2 +16x + 5 c.!x 2 + 4x + 6 d. 5x e. 4x 2!11x! 2 32 Algebra Connections

33 Lesson a. 30 b. 16 c. 32 d See table below: # of Cans Weight (kg) ,240 x b = 0.77 x, x! 19.1 kg c.! 31.9 kg a. $8 b. $74.95 c. 10 days (rounded up to nearest whole day) units = 4 x, x! 1.6 inches a. x =!5 b. y = 2x! 3 c. no solution d. y =!3x ! 574 times at bat a = 8 x, x = 256 miles b. g 118 = 32 1, g! 3.7 gallons a. (3, 5) 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. 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 a. 2x(x + 5) = 2x x b. (2x + 5)(x + 3) = 2x x + 15 Lesson a. x x + 16 b. 2m m c. xy + 10x d. 6x x a feet or about 1 inch b.! 42 feet a b. 7.5 c d (2, -5) a. y = 4x! 3 b. y = x + 3 c. y =! 3 2 x + 6 d. y =! 2 3 x + 2 Answer Key 33

34 Chapter 6 Lesson 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 a. 3p + 8 b. 3p + 8 = 176 c. 56; 3(56) + 8 = Answers vary, but typical responses: a) the value of some quarters and nickels is $5, and b) The area of a rectangle is 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 c. Chapter 2 is the longest, Chapter 3 is the shortest. d. p + p p 2 = 182, p = 68 pages in Chapter a. 2m! 10 b. 5c + 2 p = 9.50 c. 4(a + p + b) = side #1 = side #2 = 8 cm and side #3 = = 15cm 6-8. a. x! 342 b. 2w c. 3c a =!3,!!b =! no, when x = 12, y = 102 so it would have 102 tiles minutes a. y =!2x b. Yes c. growth factor: 2, y-intercept: (0, 2 3 ) 34 Algebra Connections

35 Lesson 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 = = 15; Yes a. If m = the number of months they have saved money, then 15, m = 12, m, 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 = 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 a. no solution b. x = (-1, 3) Lakeisha, Samantha, Carly, Barbara, and Kendra She combined terms from opposite sides of the equation. Instead, line 4 should read 2x = 14. Then x = 7 is the solution This statement is false because the Distributive Property states that a(b + c) = ab + ac. Answer Key 35

36 Lesson 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) = a. 8t + 16c = 400, t = 5 + c b. t = 50! 2c, t = 5 + c c. t = 20, c = y + 8x = 10 becomes y =! 4x + 5 ; ( 2, 13) a. (6, 1) b. ( 2, 8) a. t! 4 ; 2(t! 4) b. 150! c c c v If Nina has n nickels, then 5n (2n) = 84, and n = 5 nickels x = 11 3 ; x = 29 1, so 29 falcons is a good estimate Associative Property, 3. Combining like terms, 4. Additive Property of Equality, 5. Combining like terms No; 2 is a prime number and it is even 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

37 Lesson ( 11, 4) 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 = a. x = 4, y = 12 b. x = 3, y =!1 c. no solution d. b =!3, c =! Yes, she is correct. To test, substitute the values for x and y into both equations to see if they are correct solutions There are 28 red and 56 green marbles a. 0 b c. 10 d a. #2 b. 4 touchdowns and 9 field goals a. The graphed line should be y =!2x! 3. b. Yes; ( 3, 3) and ( 2, 1) both make this rule true Katy is correct because the 6x 1 should be substituted for y because they are equal upside-down parabola with x-intercepts (-2, 0) and (5, 0) and y-intercept (0, 10) No. When -2 is substituted into the equation, the equation is false. Answer Key 37

38 Lesson 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 a. y = 2x b. (1,2), (2,4), (3,6), (4,8), etc. c. (8,16) d. it makes both equations true a. a line; 2y + x = 40 or y =! 1 2 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. ) 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 d. (6, 3) e. Yes They would be written in the form (a, b, c), or x = #, y = #, and z = # Yes, each point makes the equation true a. no solution b. x = 5, y = a. h = 2c! 3 b. 3h + 1.5c = 201 c. 28 corndogs were sold three dice and two jacks Yes. Adding equal values to both sides of an equality preserves the equality y(y! 4) = 3y 2! 12y b. (3y! 4)(y + 5) = 3y 2! 7y! Algebra Connections

39 Lesson (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 = 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 x =! 1 2, y = a. (3, 4) b. (11, 2) c. (-1, 1) a. (-5, 1) b. (3, 1) c. no solution a. infinite solutions b. lines coincide 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 $ y = 3x a. x 2! 3x! 10 b. y 2 + 5xy + 6x 2 c.!3xy + 3y 2 + 8x! 8y d. x 2! 9y 2 Answer Key 39

40 Lesson a.!3x =!15, x = 5, y =! a. No variable is eliminated. c. (3, 4) d. No Multiplying the top equation by 2 created a zero with the y-terms a. m =!4, n = 5 b. a = 2, b = 9 c. x =!1, y = 6 d. infinite solutions 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) a. (3, 1) b. (0, 4) c. (10, 2) d. (-4, 5) These lines coincide , 18, and a. H: y = 2 x! 5, C: the point (6, 1) is a solution; yes this is true because 3!1 = 2 3 (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 They are both correct. The lines coincide y = 2x + 5, 105 tiles 40 Algebra Connections

41 Lesson 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 a. substitution b. elimination c. equal values d. substitution e. equal values f. elimination g. elimination h. substitution a. (2, 1) b. (1, -2) c. (-0.5, 0.5) d. infinite solutions e. (2, 5) f. no solution g. (0, -2) h. (10, 7) a. (0, 1 ) b. (-6, 2) c. No solution d. (11, -5) n = p and n + p = 168, 56 nectarines are needed a. Yes, because these expressions are equal. b. 5(3y) + y = 32, y = 2, x = 3.5 x = 8 18, x = y = 3x 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 41

42 Lesson Ant Beetle Grass Hopper Antenna: 2 mm 6 mm 20 mm Leg: 4 mm 10 mm 31 mm b. This rule is correct. c. When x = 1 mm, y = 2.5 mm Answers vary (2x! 3)(y + 3x! 5) = 2xy + 6x 2! 19x! 3y a. TeleTalk, 40 b. TeleTalk: y = 8x, Americall: y = x, CellTime: y = x c. 10 min, 15 min d. between 10 and 15 minutes a. $18 b. She sold 19 brownies a. none b. one (t = 3) c. one ( m = 0 ) d. infinite 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 Stevie is 6, Joan is 11, and Julio is The possible contexts are varied. For example, this could be the price of CD s by famous artists. Stevie Joan Julio Total 31.50? Too low Too high Too high Number of years at company Salary per hour $7.00 $8.50 $10.75 $11.50 a. $15.25 b. $ hours = $ (a), (b), and (c) all create equivalent equations. Part (d) is not legal because unless x = 1, -x + 1 " roses were sold x = x 2 + 2x + 1, x = 2 and y = a. all numbers b. ( 1 3,!3 ) c. (1, 2) d. (8, 7) 2 42 Algebra Connections

43 a. Line b. Answers will vary. Possible solutions: (0, 2), (1, 5), (2, 8), c. y = 3x + 2, Yes, the points are the same y = 2x + 6, 206 tiles (-1, 0) and (2, 0) Mr. Greer incorrectly distributed. The correct solution is x = n + d = 30 and 0.05n d = 2.60, so n = 8. There are 8 nickels a, b, and d are correct y =!5x a. x = 2.2 b. x = 8 c. x = 10.5 d. x = 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 It can be concluded that y = 2, because 2(0)! 3(!2) = C Answer Key 43

44 Chapter 7 Lesson 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 a. See diagrams at right. b. y = 5x + 1 ; 51 tiles Figure 0 Figure a. 3 b. 1 c a ! 5 " pounds of food b ! 15.6 hours 7-7. x: (12, 0) and y: (0, 6) 7-8. a. y = 8! 6x b. x =! 1 6 y c. x = 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. Lesson Answers can vary depending on data and the trend line that is chosen a. 16 b. 2 c. undefined d a. 5 b. 21 c. the positive value of x! c d. 36 e. 38 f. the positive value of y! f a. 3 b a. (3, 0.25) b. infinite solutions because the lines coincide x = 15, y = a. The baby is growing 1.5 inches per month; it is the slope (m). b. It was 23 inches long when it was born; it is the y-intercept (b). c inches d. 11 months after it was born, so in December 44 Algebra Connections

45 Lesson 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 = a. 2 b. 4 c. 1 2 or 0.5 d. divide!y by!x e. y = 1 2 x 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 If!y = 0, then the line is horizontal; if!x = 0, then the line is vertical a. Essie is correct;!y = "3 means that the vertical change is 3 and that the line is pointing downward. b. yes;!3 4 =! 3 4 c. y =! 3 4 x Possible response. It is a parabola because it has an x 2 -term. (This conclusion is acceptable at this point.) r = 3y and r + y = 124 ; r = 93 and y = any horizontal line that is not y = a. 5.5 b. 42 c.! 2 5 d 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 a. 5x 2! 32x! 21 b.!24x x Answer Key 45

46 Lesson a. B is the steepest; C is the least steep. b. Downward;!y is 2. c. A. m = 1, B. m = 3, C. m = 1 2, D. m =! 2 5 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 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 = 3 2, C. m = 2 3, D. m =! 1 5 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 A. m =! 1 2, B. m = 2, C. m = 2 3, D. m = a. The line should have slope m = 6. b. m = 3 5 c. m =! 3 2 d. m = a. The slope of each is 1 2. b. no c.!y = 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) a. ( 1, 7) b. ( 1 2, 2) c. These coincide, so there are an infinite number of points of intersection a. 12 b. 0 c. 8 d. no solution a. 6(13x! 21) = 78x! 126 b. (x + 3)(x! 5) = x 2! 2x! hops c. 4(4x 2! 6x + 1) = 16x 2! 24x + 4 d. (3x! 2)(x + 4) = 3x x! y = 4 3 x! 4 46 Algebra Connections

47 Lesson 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 =! 3 4 x + 5. f. You need the slope and a point or two points to determine a unique line a. 1 ; parallel lines must have the same growth factor. 2 b. They are equal. c. It should have a slope of 2 and pass through (0, 5). d. y = 1 2 x! a. She subtracted the x- and y-coordinates. b.! 2 3 c. 4 2 = 2 d.! 3 6 =! 1 2 d e. The student did not make!x negative and forgot to divide!y by!x The steepest line is vertical. Its slope is undefined since its!x is 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 a. See graph at right. b. (3, 2) a. m =!2 b. m = 0.5 c. undefined a. 4 b. no solution c. 0.5 d a. y = 2x 4 b. x = y!4!2 =! 1 y + 2 c. x = 1 d. x = x-intercepts. ( 2, 0) and (4, 0); y-intercept. (0, 24) Lesson A person cannot be different distances from the motion detector at the same time $2,982, a. 4 b. 16 c. It would get steeper a. There is no solution, so the lines do not intersect. b. Yes; both lines have the same slope. y = 2 3 x! y = 3x! 2, y = x Answers vary. Possible solutions. 5x(3x! 1), 5(3x 2! x), x(15x! 5), etc a.! 4 3 b. (0, 5) c. y =! 4 3 x! 5 Answer Key 47 y 2 3 x

48 Lesson a. Leslie. y = 2x, Kristin. y = 2 5 x + 8, Evie. y = 5 4 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 2. Her slope represents her rate of travel. 5 e. She traveled 2 meters every 3 seconds and got a 1-meter head start 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 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) a. Elizabeth won the race, finishing in 5 seconds. b. Barbara: y = 3 x + 3, Elizabeth: y = 4x 2 c. 5 meters every 2 seconds, or 5 2 meters per second d. 2 seconds after the start of the race, when each is 6 meters from the starting line 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. 4 pounds d. " 9 feet x-intercept. (2, 0), y-intercept. (0, 10) a. 1 b. 3 c y = 3x! 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

49 Lesson Rider A: y = 3 2 x, Rider B: y = 1 2 x + 10, Rider C: y = x + 4, Rider D: y = 5 2 x + 1, Elizabeth: y = 1 4 x + 10, Leslie: y = 2x! 4 ; Rider D wins the race. Solution graph is shown at right. Distance from Starting Line D Leslie A C B Elizabeth 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 ( 2, 15) a. (4, 0) and (0, 2) b. (8, 0) and (0, 4) Time (seconds) a. 1 b.! 1 2 c. 3 2 d.! 1 5 e. The line travels downward from left to right, so m =! a. 4 b. 3 c. 1 d a.!12x b. 8x + 7 c. x 2! 3 d. 5x + 7! 3y C Answer Key 49

50 Lesson 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 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 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 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 e. The equation is exact. f. Starting with 140 = 5.2x , the solution becomes x = 17 days a. y =!3x! 5 b. y = 0.5x! Mt. Everest was meters tall in the year 0; y = 0.05x a. y = 1.5x b. Answers vary, but solutions should lie on y = 1.5x a. ( 7, 11) b. ( 8, 10) a. 14 b. 5 2 c. 3 d a. (3.5, 0) and (0, 2.33) b. y = 13 3 " a. 3; (0, 5) b.! 5 ; (0, 0) 4 c. 0; (0, 3) d. 4; (0, 7) e.! 3 4 ; (0, 1) f.! 1 ; (0, 6) 5 50 Algebra Connections

51 Lesson b.! 3 2 c. Answers vary. d. They are opposite reciprocals. e.! a. y = 5 2 x! 8 b. y = 3 2 x line L. y =! 1 6 x + 6 ; line M. y = 2 x + 1; (6, 5) Lines A and E are perpendicular, so the slope of line E is! a meters b. Dean. y = 3 4 x + 5, Carlos. y = x c. after 20 seconds 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 If d = number of Democrats, then d + (d + 30) + 1 = 435, and d = 202 Democrats. Thus, there were = 232 Republicans ( 5, 0) and (3, 0) a. 2x 2 + 5x! 3 b. 15x 2! 33x c. 5x 2! 27x + 10 d. 300x! B Lesson a. m = 3 b. Substitute the point into the equation y = 3x + b in order to solve for b; y = 3x c. No; both will give the same equation. d. Substitute both points into the equation to verify that both are solutions b. Although there are many possible solutions, the points for 1961 and 1989 are good choices. This will create a line. y = 9.82x! c. See part (b) above. e. Answers will vary. In the case stated in part (b), the y-intercept is (0, ); Dizzyland did not exist in the year 0. f. Answers will vary. Approximately the year a. y = 1 x + 4 b. y =!3x y = 9 4 x ( 11 3, 3) a. There are no x-intercepts; y- intercept. (0, 10). b. (3, 1) a. 8 b. 56 c. 3 d a. m =! 5 6 b. m = 0 c. m =! 5 6 d. undefined, e. m = 4 f. m = 1 4 g. m = 4 h. m =! 6 5 ; Lines (b) and (d) are perpendicular; lines (e) and (f) are perpendicular; lines (a) and (c) are parallel. Answer Key 51

52 Lesson Logo A: y = 2, y = x + 2, y = 1 2 x + 2, y =! 3 2 x + 6, y =! 1 2 x + 6, y = 3 2 x! 6, y =! 1 2 x +10 ; Logo B: y = 3, y = 2x, y = 3 4 x, y =!2x + 6, y =! 3 4 x + 6, y = 3 4 x! 3, y =! 3 4 x a. (2y + 4)(3 + 5x) = 6y + 20x + 10xy + 12 b. (5 + 2x)(2x! 3) = 4x 2 + 4x! If b = number of brownies sold and c = the number of cookies sold, then 3b c = 218 and b = c + 3 ; she sold 41 brownies a. y =! 2 3 x + 1 b. y = 11 c. y = 3 2 x 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 a. 15x 2 b. 8x c. 6x 2 d. 7x 52 Algebra Connections

53 Chapter 8 Lesson a. (x + 4)(y + x + 2) = xy + x 2 + 6x + 4y + 8 b. 18x 2 + 9x! 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 a. (3x + 1)(2x + 5) = 6x x + 5 b. (5x! 2)(y + 3) = 5xy + 15x! 2y! 6 c. (4x! 3)(3x + 4) = 12x 2 + 7x! The product of each diagonal is equal. 6x 2! 5 = 30x 2 and 2x!15x = 30x 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 (2x! 3)(x + 2y! 4) = 2x 2 + 4xy! 11x! 6y a. 12x x! 5 b. 4x 2! 28x x x 3x x!7x 2 7x x 6x 8-9. a. m = 2, (0,!! 1 2 ) b. m =!3, (0,!!7) c. m =! 2 3, (0,!8) d. m = 0, (0,!!2) 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 a.!1 b.! 7.24 c.! " 4.24 Answer Key 53

54 Lesson a. (5x! 2)(2x! 7) = 10x 2! 39x + 14 b.!35x "! 4x = 10x 2 "14 = 140x 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) 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) 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 a. (x! 6)(x + 2) b. (2x + 1) 2 c. (x! 5)(2x + 1) d. (x + 4)(3x! 2) 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) a. (0, 9); It is the constant in the equation. b. (3, 0) and ( 3, 0) a. (6, 9) b. (0, 2) a. x =! b. all numbers c. c = y = 1 4 x Algebra Connections

55 Lesson a. (x + 3) 2 b. (2x + 3)(x + 1) c. not factorable d. (3m + 7)(m! 2) a. (3x! 2)(3x + 2) b. 4x(3x! 4) c. (4k! 3)(2k! 1) d. 20(2! 5m) (2x! 6)(2x + 1) or (x! 3)(4x + 2) See table below Multiply x! 2 2x + 1 x + 7 x 2 + 5x! 14 2x x + 7 3x + 1 3x 2! 5x! 2 6x 2 + 5x s + 2s + s + 3 = 51; 12, 24, and 15 cm a. 9 units b. 15 units c. 10 units d. 121 un a. (k! 2)(k! 10) b. (2x + 7)(3x! 2) (2, 5) c. (x! 4) 2 d. (3m + 1)(3m! 1) y =!x a. 5 b. 6 c. 5 or 6 d.! 1 4 e. 8 f.! 1 4 or 8 Lesson 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. 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 a. 5 b. 5(2x 2 + 5x! 3) c. Yes; 5(2x! 1)(x + 3) a. 5(x + 4)(x! 1) b. 3x(x + 3)(x! 5) c. 2(x + 5)(x! 5) d. y(x! 5)(x + 2) 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 y = 3 4 x! a. in 7 weeks b. Joman will score more with 1170 points, while Jhalil will have a. Michelle is correct. b. ( 4, 0) , 46, 47; x + (x + 1) + (x + 2) = a. 2 b. 3 c. 1 Answer Key 55

56 Lesson a. 2 b. 3 c. " y =!3x y = 3x! 5 ; m = 3 and b = There is only one line of symmetry. horizontal through the middle 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) a. 6x 2 + x! 12 b. 25x 2! 20x + 4 Lesson 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 You should be able to connect rule! table, table " graph, graph " situation, and table " situation a. One way to write the rule is y = (x + 1)(x + 2) + 2. b. Yes vertex. (4, 9), x-intercepts. (1, 0) and (7, 0), y-intercept. (0, 7) a b. it does not change the value of the number. c. It tells us that a = 0. d. 0 for all e. the result is always 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) Jen Maggie Imp Al Solution to part (a) a. ( 3, 0) b.! a. no solution b. (7, 2) 56 Algebra Connections

57 Lesson 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 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 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 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 ( 3 2,!0) and ( 4, 0). d. The solution graph is shown at right This parabola should have roots ( 3, 0) and (2, 0) and y-intercept (0, 6) roots: ( 1, 0) and ( 2, 0), y-intercept: (0, 4) a. One is a product and the other is a sum. b. first: x =!2 or x = 1 ; second: x =! a. x = 2 or x =!8 b. x = 3 or x = 1 c. x =!10 or x = 2.5 d. x = 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) a. 4; Since the vertex lies on the line of symmetry, it must lie halfway between the x-intercepts. b. (4, 2) a. 2(x! 2)(x + 1) b. 4(x! 3) 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 57

58 Lesson The parabola should have y-intercept (0, 2) and roots ( 1, 0) and ( 2, 0) a. x = 2 or x = 4 b. x = 1 or x = 4 3 c. x = 5 or x = 3 2 d. x = 0 or x = 6 e. x = 5 or x = 1.5 f. x = 2 or x = a. y = (x + 3)(x! 2) = x 2 + x! 6 b. y = (x + 5)(x! 1) = x 2 + 4x! 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 x! 24 is correct The result must be the original expression because multiplying and factoring are opposite processes; 65x x! a. x = 3 or x =! 2 3 b. x = 2 or x = 5 c. x =!3 or x = 2 d. x = 1 2 or x =! a. true b. false c. true d. true e. false f. false a. 1 b. " 1.6 c y =! 4 3 x a.! 4 3 b. Yes; it makes the equation true and lies on the graph of the line. 58 Algebra Connections

59 Lesson (1) " b, (2) " e, (3) " a, (4) " g, (5) " d, (6) " i 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 a. y =!2x(x! 8) =!2x x b. y =!3(x! 10)(x! 16) a. y = x 2 + 2x! 8 b. y = x 2! 6x + 9 c. y = x 2! 7x d. y =!x 2! 4x m = 1, (0, 4) a. " 1.4 and " 0.3 b. The quadratic is not factorable a. x = 4 or x = 10 b. x = 8 or x = a. 4 b. 10 c. 8 d. 1.5; They are the same a. (1, 1) b. ( 2, 1 2 ) Lesson a. The quadratic is not factorable. b. There are two roots (x-intercepts). c. The intercepts are " 1.5 and a. a = 1, b =!3, c =!7 b. 3± 37 2 " 4.5 and 1.5; yes a. Graphing and factoring with the Zero Product Property a. x = 2 or! 1 3 b. x = 7 or 2.5 c. x = 0.5 or 0.75 d. no solution a. x = 6 or 7 b. x = 2 3 or 4 c. x = 0 or 5 d. x = 3 or x = 6 or 7; yes no a. The parabola should be tangent to the x-axis. b. Answers vary, but the parabola should not cross the x-axis y = 1 x line. (a) and (c); parabola. (b) and (d) A and D a. false b. true c. true d. true e. true f. false g. true h. false Answer Key 59

60 Lesson a. (3x! 2)(2x + 5) = 0, x = 2 3 or! 5 2 b. a = 6, b = 11, c =!10, x = 2 3 or! 5 2 c. Yes a. x = 5.5 or x =!5.5 b. x = 2 or x =! 1 2 c. x =!3 or x = 14 d. x = a. " 315 and 315 feet; The bases of the arch are 315 feet from the center. b. " 630 feet c. 630 feet; y-intercept a. x = 5 b. x = 1 3 c. x = 1 or 5 3 d. x = ± 3 4 or x = 1 or 6; yes a. y = (x + 3)(x! 1) = x 2 + 2x! 3 b. y = (x! 2)(x + 2) = x 2! If x = width, x(2x + 5) = 403 ; width = 13 cm (b) and (c) are solutions 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 = 3 5 x! 5. c. x. (8, 0), y. (0, 12) Lesson a. x =!3 or x =!9 b. x =!17.6 or x =!0.36 c. x =! 4 3 or x = 1 2 d. x = 4 e. no solution f. x = 2.5 or x =! 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 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 feet d. The sketch should have roots (0, 0) and (2.4, 0) and vertex (1.2, 2.88) If n = # nickels and q = # of quarters, 0.05n q = 1.90, n = 2q + 3, and n = 13, so Daria has 13 nickels a. x = ± 0.08 b. x = 2 9 or 4 c. no solution d. x " 1.4 or While the expressions may vary, each should be equivalent to y = x 2 + 4x a. x = 2 b. x = 15 c. x = 2 d. all numbers Line L has slope 4, while line M has slope 3. Therefore, line L is steeper D 60 Algebra Connections

61 Chapter 9 Lesson 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 a. x! 1 b. x < 1 c. x! 3 d.! 4 < x " 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; x 9-6. x < 2, x 9-7. Let n = # of North American countries. Then n + (2n) + (2n + 7) = 122 and n = a. p >!1 b. k < 2 c. 1! k or k! a. k = 1.5 or 2 b. m = 3 or 3 c. w = 2 or 6 d. n " 2.12 or a. always true b. sometimes true c. never true d. sometimes true e. always true f. never true ; Any non-zero number divided by itself is a. (5, 0) and (8, 0); Robbie backed up 5m and the rocket landed 8m away from him. b. 3 m a. 5± 13 2 " 0.7 or 4.3 b.!1 ± 7 " 3.6 or 1.6 Answer Key 61

62 Lesson a. x < 5 b. m! 2 c. p >!3 d. x! 3 e. all numbers f. no possible solution 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 a. Since trees bloom when they are taller than 150 cm, an inequality is fitting. b x > 150, x > 16, so the trees should be more than 16 years old. c x! 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 $ a. k < 2 b. p! 15 c. n > 1 2 d. t! The graph should be a line with x-intercept (1.5, 0) and y-intercept (0, 3) y =! 3 5 x! x 2 + x! 10 a. 3x! 5 ; The product divided by one factor gives the other factor. b. x x = 5 3 and x =! a. (2x! 5) 2 b. not possible c. 3x(x! 4) d. 5(x! 4)(2x + 1) 62 Algebra Connections

63 Lesson 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 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 a. y! "2x + 3 b. The boundary should be a dashed line a: c: 4 x 2 x b: d: - 3 x 0 x x! 2700, so x! 5. Algeria can order an advertisement up to 5 inches high 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 a. y =! 2 7 x! B D b. Yes; verify by substituting the coordinates into the equation and testing. c. y = 7 2 x! 22 Answer Key 63

64 Lesson 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 a. y > 1 2 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 y <! 1 3 x! A A 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 a. x = 1 3 b. x = 16 c. x = ± 5 d. x > 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 x = 0.6 or x = 2 64 Algebra Connections

65 Lesson 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 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 either 15 or 15; yes a. b. Answers will vary She should add 1 first, since the addition is placed inside the absolute value symbol, which acts as a grouping symbol x ,400,000! 50x > 1,200,000, less than 4000 square miles per year a. 1 b. 2 c. 11 d no; 3(7! 2) = 15 and 15 > ! 14x = x, x = 11 months b. Answers vary, but using data from games 1 and 3. y = 6x. c. " 90 baskets Answer Key 65

66 Lesson 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 a. y = x 2 + x! 6 is part of y! x 2 + x " 6, while y = 2 3 x is not part of y > 2 3 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 y! x + 3, y! 1 2 x " 1, y! 2, y! " 2 3 x + 4, y! " 2 3 x " a. 3 b. 1 c. 4 d a. b x x c. d. no solution m x All equal a. (5x! 2)(x + 3) b. 2(3t! 1)(t! 4) c. 6(x! 2)(x + 2) a + 3c = 27.75, 3a + 2c = 32.25, a = $8.25, c = $ B 66 Algebra Connections

67 Lesson y x y! " 1 2 x , y! 2 3 x " 2000, y! " 1 x ; The search-and-rescue teams should 5 search near the island of Samoa a. the region between and including the parallel lines b. the region below and including the boundary line y = 2 3 x c. no solution a. x! 6 b. x > 1 c. 2! x < 7 d.!3 " x "! a. false b. false c. true d. false C a. r c! 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

68 Lesson A. y! 500 " 1.5x, B. y! 300 " 0.25x, C. y > 180! 2 3 x, D. y > 450! 5 2 x, E. x < 250, Special Assignment. x! 100 Number of Medicine Packages D A C Special Assignment E B a. 210; estimate from the graph or substitute 185 for y into the equation for Country A ( y =! 3 x ) and solve for x since that line is the upper boundary when 2 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 of each should be requested; graph the line y = x and find its intersection with the line for Country A 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 x < 50,000, less than 14.8 pounds Number of Food Packages a. m > 5 b. x! "6 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 x = 9 or x = D 68 Algebra Connections

69 Chapter 10 Lesson is special because any non-zero number divided by itself is 1, and anything multiplied by 1 remains the same a. yes b. You cannot divide by zero. c. Yes; x! 3 d. Answers vary; sample solutions. x x, x+5 n2, and x+5. n 2 e. Yes, because z = 1. The fact that anything multiplied by 1 stays the same is called z the Identity Property of Multiplication a. 1, x! 0 b. x x+5, x! 0 c., x! 1 or 2 d. 1, x! 0 3 x!1 e. hk, h! 0 f. 2m!5 3m+1, x! " 6 or " 1 g. 2(n! 2), x! h. 4 x!1, x! 1 4 or 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 a. x+3 x!3 b. 2x!5 3x+1 c. 1 d. x a. 1 b. none c. 2 d Yes, he can. a. x = 2 b. Divide both sides by a. x < 0 b. x! " a. x! "4 or 2, x+4 x!2 b. x! "2 or 3, 2(x+2) (x!3) a. 3 7 b B Answer Key 69

70 Lesson x!1 2x+3 ; x cannot be 4 or! , a. 4 x+3 x+2, x! 5 or " 3 b. x+3 2x+1, x! 1 9 or " 1 2 d. (y!2)2 (y+5) 3y(y+2) c. 7+4m 3m!2, m! 2 3 or " 3 2, y! 0,!"2, or 2 e. 2x, x! 0 and y! 0 f. 3x+1 2x!3, x! 3 2,!4, or a b c. 5(x!2) (x+4) 2 d. 2(2x!3) (x!5)(x+1) e. 5x!1 2(x!3) = 5x!1 2x!6 f. 1 2x a. x! 2 = 4 b. For each, x = 6. c. x + 3 = 8, x = a. m =! 6 5, b = (0, 7) b. m = 3, b = (0, 5) c. m = 2, b = (0, 12) a. ( 1 3, 2) b. (4, 9) a. x!4 3x+2 b. 5 x!3 c a. m =!6 b. x = 5.5 c. k = 4 d. x = after 44 minutes Lesson a. 2x 3(2x!1) = 2x 6x!3 b. x!4 x 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 a. 3x! 5 = 7, x = 4 b. 3x 2! 2x! 5 = 0, x =!1 or 5 3 c. 1 + x = 10, x = 9 d. 4x 2 + 8x! 5 = 0, x = 1 2 or x =! a. x! 15 = 24 ; One denominator was eliminated, but there is still a fraction left over. 4 b. Multiply by 4; 4x! 15 = 96, x = = c. 24 or any multiple of 24; 24 is the least common multiple of 6 and 8; 4x! 15 = 96. d. x = 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 = a. 5(3x!1) 2(4 x+1) b. 1 c. p+9 3p!2 d. 4 x! y =! 1 3 x y <! 2 x a. 2d! 3 b. 2d! 3 = 19, 11 candies 3 70 Algebra Connections

71 Lesson 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 a. 3x! 2x = 8, x = 8 b. 5! 2x 2 = 3x, x = 1 or 2.5 c.!14x + 7! 3x! 9 = 168, x =!10 d. x x! 4 = x + 5, x = 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 = a. x 2 + 4x + 3 = 0, x =!1 or 3 b. a 2! 6a + 9 = 0, a = 3 c. 12m! 2 = m, m = 20 d. 2(x! 3) + 3x = 4(x + 5), x = a. x = 4 b. x =!5 or 2 c. x = 16 3 d. x = a. (x!2)(x+6) (x+4)(2x+3) b. 2x+1 x! a. t = 5 seconds b. 100 feet t + s = 27, 1 2 t s = 11 1, t = 19 Times papers See diagram at right C Lesson x = x + 12 = 20, x = 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 a. If you treat (x + 3) as a group, then 4 (group) = 20. Therefore, x + 3 = 5 b. x + 3 = 5, x = 2; yes " iii, 2 " i, 3 " ii a. 34 b. 15 c. 11 d. 2 e. 25 f. 27 s a. x = 10 or a. Both Hank and Frank are correct. b. two a. 4 or 4 b. 100 or 100 c. no solution d. 3 or (a), (b), and (c) all are equal to a. x >!2 b. x! 12 c. x > 1 d. x! a x = 14, x = 11 4 b. rewriting Let x represent the amount of money the youngest child receives. Then x + 2x + x + 35 = 775 ; $185, $370, and $220. Answer Key 71

72 Lesson a. two solutions b. x = 3 ± 12 c.! 6.46 or 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 e. no solution f. one solution a. no solution b. one solution c. two solutions 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 =! a. Answers vary b. Answers vary, but it should contain an absolute-value expression equal to zero a. x = 3 or 11 b. x = 14 c. x = 2 d. x = No, because 1 is not greater than a. x+4 m+5 b. 4 x!3 m a. (3x! 1)(3x! 1) b. 7! 7! 7! 7 c. m! m! m d. w! w! w! w! w! w! w! w! w! w a. 4x(x! 3) b. 3(y + 1) 2 c. (2m + 1)(m + 3) d. (3x! 2)(x + 2) t = number of toppings, 1.19(3) t = 4.55, and t = 2 Lesson a. x = 4 or 6 b. x = 15 c. x = 53 d. x =!3 or 7 e. x =!15 or 9 f. x = 5 or 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 See graph at right When any real number is squared, the result is either positive or zero a. 7 or 1 b. 4 or 8 c. 3 d. no solution a. x < 2 b. x! 6 c. x > 4 d. x! a. x! 3 3x!14 b. 2x!1 x xxxyy 72 Algebra Connections

73 Lesson x < 2.5 ; a. 2.5; No, because the inequality does not include x equality. b. Find the boundary point by solving the corresponding equality. Then test a point on either side of the boundary point a. x <!1 or x > 5 ; ; After finding dividing points, test a point x in each possible solution region. b. two boundary points a. The solution points are those x-values for which y = x! 2 is greater than y = 3. b.!1 " x " 5 ; x ; The solution points are the x-values for which 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 =! a. two. x = 1 and x =!3 b. three. x <!3,!3 < x < 1, and x > 1 c.!3 < x < 1 ; d. Yes, see graph at right Solutions vary; in general, any inequality that states that x! 2 is less than a negative number will work (x + 2)(2x + 3) = 2x 2 + 7x a. 2 b. 1 c. 0 d y = 3 x See graph at right a. 2(x+5) or 2x+10 b. 7x!2 3 3 x a. x = 13 b. x = 3 c.!5 " x " 5 d. x <! 2 3 or x > 2 Lesson x! "2 and x! 4, a. x <!3 or x >!1 b.! 4 < x < 3 c. all numbers d. x! 1 e. no solution f. x < 2! 7 or x > !16t t " 140, 2.5! t! 3.5 seconds x =!105,!!9,!!1,!or! (y! 2)(y! 2)(y! 2) It has no solution, because an absolute value cannot equal a negative number a. x = 1 b. x > 3 or x <!3 c. 0! x! 4 3 d. x = 3 e.!2 " x " 3 f a. y =! 2 3 x! 1 3 b. Yes; the slope of 6x! 4y = 8 is a. x! 4 b. 7m!1 3m+2 c. (4z!1)2 z+2 d. x!3 x! C Answer Key 73

74 Lesson Both have the same solutions.!2 ± 3 " 3.73 or 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 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 a. (x + 3) 2 = 4, x =!1 or!5 b. (x + 1) 2 = 6, x =!1 ± 6 " 1.45 or 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 (2x + 3) 2 = 16, x = 1 2 or! a. 4 x!3 5x+1 b. 5x+2 5x!2 c. 4(2x+1) x!2 d. 2y 3x Both result in no solution a.!2 < x < 2 b. x! 2.5 c. x = 1 4 f.!5 " x " 3 d. no solution e. x =! See graph below. The portion for x between 5 and 3 (inclusive) should be highlighted m m = 80, m = 8; It will be full after 8 months. 74 Algebra Connections

75 Lesson a. 2 b. 0 c. 1 d 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 ( b 2 )2! c unit tiles need to be added to both sides of the equation a. (w + 14) 2 = 144, w =!2 or 26 b. (x + 2.5) 2 = 2.25, x =!1 or 4 c. (k! 8) 2 = 81, k =!1 or 17 d. (z! 500) 2 = , z = 65 or ! 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 " or a. x+2 3x!2 b. (x!2) 2 (x!11)(x+4) a. x = 5 8 b.!6 < x < 0 c. x! "9 or x! 7 d. no solution e. x =! 4 or 4 f. all numbers See graph at right They will be the same after 20 years, when both are $1800. Lesson 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. x 12, viii. 3x 3, challenge. x a. y 7 b. w 3 c. x 8 d. x 22 e. 13p 2 q 3 f. x6 y 3 g. 10h 25 h. 5m 22 i. 81k 80 j. 8 = g 7 8g!7 k. m12 40 l. pw Answers vary a. incorrect, x 100 b. correct c. incorrect, 27m 6 n a. 1 h 2 b. x7 c. 9k 10 d. n 8 e. 8y 3 f. 28x 3 y a. Haley is correct. You cannot add unlike terms. b. Haley is incorrect. The bases differ a. 1 unit tile b. (x + 1) 2 = While there are multiple ways to write the rule, one possible way is y = x 2 + 5x a. x = 2 b. k! 3.76 or 1.24 c.!2 < x < 10 d. x = 5 Answer Key 75 n

76 Lesson a. x 11 b. x 9 c. m 12 d. x 8 y 8 e. x!3 y 8 or y8 x 3 f. 1 4 x Expression Generalization Why is this true? a. x 25! x 40 = x 65 x m! x n =? x m+n b. x 36 x13 = x23 x m x n =? xm!n If you have m x s and n x s multiplied together, you then have a product of (m + n) x s. 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 Its value is 1, and it can be rewritten as x m!m = x 0. Thus x 0 = a. 1 x = x!1 b. 1 x 2 = x!2 c. 1 x 3 = x! a. 1 k 5 b. 1 c. x3 d. p!2 e. y f. 1 x 6 g. 1 a 2 b h. x b, c, d, f y > 2x! (15, 2) y =!x(x! 20) =!x x ; Its maximum height is 100 feet when x = Perfect square form. (x! 5) 2 = 0 ; x = 5; answers vary a. x!4 3x+2 b. 5 x!3 c Algebra Connections

77 Lesson a. The result is the square root of the base. b. 2, 7 c. It is inductive reasoning (based on patterns) 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 = , so 32 3/2 = = 8 ; 25 = 5 5, so 25 3/2 = = 125 d. 5 x ; fifth root a. (3 1/2 ) 4 = 3 2 = 9 b. 3 7 = 2187 c. (2 5 ) 1/3 = 2 5/ a. 3 b. 8 c. 7 d. 1 e. 6 f. 4 g. 2 h a. 4,600,000,000 years b. 1.5!10 11 meters c seconds a. 100,000,000,000,000,000 b. 1! c. While opinions may vary, since most calculators cannot handle the calculations in standard form, scientific notation is easier a.!15x b. 64 p 6 q 3 c. 3m ,261,392,000 or feet See solution graph at right; ( 2, 6) and (1, 6) x! 20, x! 3.2 hours; Vinita can rent a skateboard for up to 3.2 hours (3 hours and 12 minutes) 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 77

78 Chapter 11 Lesson (3x! 2) 2 = 9x 2! 12x a. 228 shoppers b. 58 people/hr c. at 3:00 PM a b. 99 c. 2 d. & The x-coordinate of the vertex must be at x = 1 because of symmetry of the parabola a + 3c = 56.5, a + 4c = 49.5, a = $15.50, c = $ a. m =! 2 7, b = 2 b. m =! 1 3, b = 6 c. m = 5, b = 1 d. m = 3, b = a. x! 2 b. x >!1 c. x! 9 d. x > a. 2 x 4 b. s11 u 2 c. 81 w 8 d. 1 m a. 3 b. 1 c. 4 d a. x!8 6x!1 2(x!6) b. = 2x!12 x!2 x! a. (0, 3) b. ( 1, 0) and (3, 0) a. x 4 y 3 b. xy c.!6x 6 d. 8x 3 Lesson y = x 2! f (4) = f (x) = x 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 a. 0 b. 3 and 0 c. x! "0.5, , 9, and t y =! 4 3 x Marley is correct; they are perpendicular since the slopes are 2 7 and! miles a. x+2 x!6 b. 4 x!3 x!5 c. x(x!6) d. x+3 x!1 x!7 e. 3x!1 x!5 f. x!3 x a. 1 4 b. 1 c = 1 25 d. 1 x 2 78 Algebra Connections

79 Lesson 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 Typical response. A function is a relation in which each input has only one output 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 No; vertical lines are not functions Some possible machines are an ATM machine, a calculator, a radio, etc , 5,! u = 4, v =! a. 7 b. 1 c. 9 d a. x = 8 or x =!2 b. x = ± 7 c. x = 1 or x =!3 d. x =!5 or x =! a. 25a!22 b 36 b. 5! 3 "1 x "9 y B Answer Key 79

80 Lesson a.! 2.3, 0, 3.3 b. 2, 3 c.! 3.7 d. No 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 = 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 See solutions below. a. D.!3 " x " 3, R.!3 " y " 3 b. D.! " < x < ", R.! " < y < " c. D.!2 " x " 4, R.! 4 " y " 2 d. D.! " < x < ", R. y! 4 e. D. 2! x! 4, R.!3 " y " 2 f. D. 2, 1, and 4, R. 4, 1, 1, and 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 a.! " < x < " b. All y-values greater than There are many possible solutions. See example at right 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 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) Marisol. y = 2x, Mimi. y = 3x! 3, solution: x = 3 hrs, so 6 miles suur No; the slope of AB is 3 5 suur, while the slope of AC is a. x = y b. y = 3 2 x! 9 c. r = d t d. r = C 2! and the slope of BC suur is a. 2x!5 x!6 b. x Algebra Connections

81 Lesson a. 4 b. 3 c. 1 d. 2 e. 6 f (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 > They are the same shape, but one is shifted up two units. 3x! ; x! "0.5 or a. 8 b. 1 c. 2 d. no solution 2x y = 5x! a. 9 b c = 2.6 d a. 15x 3 y b. y c. x 5 d. 8 x a. 2 b a.! 4 " x " 4 b. 0 < x < 3 c. 1 < x! 6 d.! 5 " x <! Graphs (a) and (b) have a domain of! " < x < ", while graphs (a) and (c) have a range of! " < y < " a. x = ± 4 b. ( 5, 17) c. x = 4 or x =!2 d. x = See graph at right. Lesson !1± 57 4 " 1.64 or a. y = x! b. y =! x 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 a. 3 b. 1 c x. (0, 0) and (4, 0), y. (0, 0), vertex. (2, 4) a. 12 b. 59 c. 7 d. 9 e. 13 f All equations are equivalent and have the same solution. x = a. 12x 2 + 5x! 2 b. 3m 2 + m! 2 c.!5k k! a. x! 12 b.!10 < x < 10 c. x < 0 d. x <!5, x > 1 Answer Key 81

82 Lesson 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 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 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) The solutions are ( 1 2, 2) and (!1,! 1). The graph shows y = 1 with no roots x nor y-intercept and y = 2x + 1 with x-intercept (! 1,!0) and y-intercept (0, 1) No solution; you cannot divide by zero Yes, they will intersect; top line. y =! 1 4 x + 10, bottom line. y = 1 3 x + 3; they will cross at (12, 7) top line. x-intercept (40, 0) and y-intercept (0, 10); bottom line. x-intercept. ( 9, 0) and y-intercept (0, 3) 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 3 2 ) The line should pass through ( 2, 5) and (0, 2); y =!3 2 x (a) and (b) are functions because each only has one output for each input a. D.! < x <!, R. 1! y! 3 b. D.! < x <!, R. y! 0 c. D. x! "2, R.! < y <! 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 = Algebra Connections

83 Lesson Solutions vary. C is impossible a. one solution. (1, 1) b. no solutions c. two solutions. (1, 2) and (3, 2) d. one solution. ( 1, 0) e. no solution f. two solutions. (1, 4) and ( 2, 7) a. Using a standard window, it appears that there is only one point of intersection. b. There are actually two points. ( 4 3,! 28 9 ) and ( 3 2,! 3) a. 1 5 or 3 b. 1 2 or 3 c. 3 d. 7 or a. (3, 4) b. D:! " < x < ", R. y! He sold 9 watermelons (6, 20) and ( 1, 6) A B Lesson The only relation with no match is y = x 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 = The graphs in parts (a) and (d) are not functions because they have two y-values for at least one x-value x! "5 because the denominator cannot be y =! 2 5 x ( 2, 10) a. 4x2 y 4 b. 18x 2 y c. xy 2 Answer Key 83

84 Chapter 12 Lesson a. (x + 7)(x! 7) b. (x + 6)(x! 4) c. (x! 5) 2 d. (3x + 2) 2 e. (5x + 1)(x! 1) f. (2x + 5)(2x! 5) g. (x! 3) 2 h. (x + 6)(x! 6) i. (7x + 1)(x! 3) j. (2x + 5) 2 k. not factorable l. (3x! 1)(3x + 1) (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) 12-3.!b!abx!b 2 b abx b 2 ax a 2 x 2 abx ax a 2 x 2 abx ax +b ax b a. x+3 x!3 b. 1 x+7 c. 1 d. x+6 x! a. x = 1.5y + 5 b. x = 24 c. x = 22 27! 0.81 d. x = a. 1 b. 20 x c. 5 t 3 d. x9 y a. 3 b. 2 c. does not exist d. 0 e. 1 f a.!1 " x " 3!4± 28 b. x = " 0.65 or! c. all numbers d. x =!2.5 or Typical response: Add (or subtract) the numerators, but keep the denominator the same. a. i. 5 x+2, ii b Algebra Connections

85 Lesson , x+2 6, a. 1 x!3 b. 5 x+1 c. 1 2(2x!5) = 1 4 x!10 d a. x!2 x!5 b. 2x!3 x a. Because if x = 4, then the denominator is zero. Since dividing by zero makes the expression undefined, x # 4. b. a. x #! 1 3 and x # 5; b. x # 3 or 3 c. Answers vary. Example: 1 (x+6)(3x!1) a. 9m+27 m+3 = 9 b. a+3 a! a. all numbers b.!5 < x < 4 c. no solution d. x = a. 5m 2 + 9m! 2 b.!x 2 + 4x + 12, c. 25x 2! 10xy + y 2 d. 6x 2! 15xy + 12x a. not a function, D.!3 " x " 3 and R.!3 " y " 3 b. a function, D.!2 " x " 3, R.!2 " x " 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 a. 15 b. 4 c. 3 d.!m 2 Lesson a b. 2 9 c. 3 d. x+3 x 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. b. 5x 2!5x+10 (x+4)(x! 7)(3x+1) c. x+2 x a. 2x!7 x!9 b. x!3 2x+1 c. 5 d. x+2 2x+3 e. 2x+5 x 2!2x!15 f. 3x+1 4 x a. 2 3x+1 b. x!7 x!3 c. x!2 2x+12 d. 1 x a. 1 b. 4 c. 2 d a. x = 3 b. 0! x! 6 c. x = 1 or 5 d. x < 2 or x > x =!3 or a. 1 b. 3 c ( 3, 8) and (1, 12) Answer Key 85

86 Lesson a of the lawn b of the lawn c = 1 6 of the lawn per hour, then it will take 6 hours to d. 6 hours; if together they mow 1 6 mow the entire lawn. e. Equations vary; possible equations where t represents the time it takes for both teams to mow the entire lawn: = 1 t and t 10 + t 15 = = 1 t ; t = 7.5 minutes t = 1 ; t = 24 minutes streamers x = 4 ± = 1 x, x = 6 5! hrs = 1 1! hrs = 5 1 hour and 12 minutes 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) = $ They all are equivalent to 12x !x 2 + 2x a. x = 1 b. x <!1 or x > 7 c. x = 7 Lesson If x = number of 7 th graders and y = number of 8 th graders, 0.4x + 0.9y = 720 and x + y = Then x = 360 and y = 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 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 x + 12 = 4.75(x + 3), 9 5 pounds a. 3 b. 2 c.! 3.24 d. There is no solution The graph is a V shape with vertex at (3, 1) and x-intercepts (4, 0) and (2, 0) a. x!4 x+5 b. x 2 +4 (x+3)(x+2) (3, 14) and (0, 2) c. 6x+9 x+1 d. 5 x a. (x! 7) 2 = 25, x = 2 or 12 b. (x + 3) 2 = 18, x =!3 ± 3 2 c. (x! 99) 2 = 4, x = 97 or 101 d. (x! 4.5) 2 = 36, x = 1.5 or Algebra Connections

87 Lesson a and 2.5 b.!5 ± 31 c. no solution d. ± a. 5 ± 3 b.!1 ± (2 + x) = 0.08(2) (x), 1 11 gallon The graph starts at (3, 1); D. x! 3, R. y! a. 1 b. 2 c. 2 d a. (x! 9)(x + 9) b. (x + 6) 2 c. (2x + 1)(2x! 3) d. (4x! 5)(4x + 5) (1, 3) Answers vary. Typical responses: x5 x 3, x3 x!5, and (x!1 ) 2 Lesson Teresito is correct because neither point makes both equations true See solution graph at right; D. x! 3, R. y! a. not possible because of dividing by 0 b.!27 c. 8 d. 0 e Answers vary, but likely answers are 6(m! 2), 2(3m! 6), 3(2m! 4), and 1(6m! 12) (1, 12) and ( 5, 42) Answer Key 87

88 Lesson 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 a. 94,557 tiles b. Figure a. x =!12 b. 4! 12 < b < c.!33 " x " 27 d. n = 1 5 or Roots (! 0.2, 0) and (! 3.1, 0), vertex (! 1.7,! "6.3) ; solution graph is shown at right a b. 7 c. undefined d. undefined e. undefined f (x!1) g. 3(x + 2)! 2 = 3x + 4 h. 2 (x!1)+5 = x2!2x+1 for x "!4 x a. 3x 2 +x!3 2x 3 +9x 2!5x b. 3x!5 2x A a. D. x! 2, R. y! 4 b. Yes; each input has only one output a. 44,941 tiles b. Figure hours, or 1 hour and 52.5 minutes y =! 3 x! After 365 days (1 year), each will have $267, B 2 Lesson 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) ! "3.83, " 1,!and dashed curve, shade above and inside the curve No a. yes b. no; most inputs have two outputs c. no; x =!1 has two outputs a. 3 ± 21 b. 2 ± a. 0 b. 2 c. 2 d a. Answers vary but should be close to y =!54x b. about the year a. The line passes through ( 1, 7) and (3, 8). b. f (x) = 0.25x c Algebra Connections

89 Lesson 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 a. See graph to the right. c. No, because these represent the lengths that yield zero area b. 7.5 by 15 meters c. Between 112 and 115 square meters is acceptable. Actual maximum area is square meters Area of Pen (in m 2 ) a. 30! 2x b. y = x(30! 2x), which simplifies to y =!2x x c. upside-down parabola d. (0, 0) and (15, 0) e. (7.5,112.5) f square meters, 7.5 m by 15 m; The equation yields an exact answer. Length of the Sides Attached to Barn (in meters) (!2.5,! 36.75) ; Due to the parabola s symmetry, the vertex must have an x-intercept that is halfway between the x-intercepts a. y = 2!or!!2 b. x = 8 ( 3 is excluded) (6, 8) and (5, 3) D. x < 2 and x > 2, R. y < 0 and y > 0 ; Solution graph shown at right x + 10(6) = 8.5(x + 6), 18 pounds If t represents the time, then t t = 252 and t = 4.5 hours ( 2, 5) and (6, 21) y =! 1 3 x See graph at right a. 0 b. 2 c. 1 d. 2 Answer Key 89