Download Answers for the lesson “Find Square Roots and Compare Real

Answers for the lesson “Find Square Roots
and Compare Real Numbers”
LESSON
2.1
}
rational number, Ï144 : real
number, rational number, integer,
}
whole number; 211.5, 2Ï 121 ,
Skill Practice
1. real numbers
2. If it is not a perfect square, the
square root is irrational.
3. 2
4. 27
5. 23
6. 61
7. 14
8. 611
9. 650
10. 216
11. 215
12. 19
13. 613
14. 240
15. 3
16. 24
17. 22
18. 12
19. 29
20. 6
21. 14
22. 28
23. B
}
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
24. Ï 49 : real number, rational
number, integer, whole number,
8: real number, rational number,
}
integer, whole number, 2Ï4 : real
number, rational number, integer,
23: real number,}rational
number,
}
integer; 23, 2Ï4 , Ï49 , 8
}
25. 2Ï 12 : real number, irrational
number, 23.7: real}number,
rational number, Ï9 : real number,
rational number, integer, whole
number, 2.9: real number,
rational
}
}
number; 23.7, 2Ï12 , 2.9, Ï 9
26. 211.5: real number, rational
}
number, 2Ï 121 : real number,
rational number, integer, 210:
real number, rational number,
}
25
210, Ï 144 , }
2
}
27. Ï 8 : real number, irrational
2
number, 2}5: real number,
rational number, 21: real number,
rational number, integer, 0.6:
real
}
number, rational number, Ï 6 : real
number, irrational number; 21,
2
}
}
2}5, 0.6, Ï6 , Ï 8
8
28. 2}: real number, rational
3
}
number, 2Ï5 : real number,
irrational number, 2.6: real
number, rational number, 21.5:
real
number, rational number,
}
Ï 5 : real number, irrational
8
}
}
number; 2}3, 2Ï5 , 21.5, Ï 5 , 2.6
29. 28.3: real number, rational
}
number, 2Ï80 : real number,
17
irrational number, 2}
: real
2
number, rational number, 28.25:
real}
number, rational number,
2Ï 100 : real number,}
rational}
number, integer; 2Ï100 , 2Ï 80 ,
17
2}
, 28.3, 28.25
2
25
integer, }
: real number,
2
Algebra 1
Answer Transparencies for Checking Homework
26
30. If a number is a whole number,
then it is a real number; true.
41. B
}
31. If a number is a real number, then
it is an irrational number; false.
Sample answer: 3 is a real number
and a rational number.
42. For 0 , x , 1, x , Ï x ; for
1 }
1
example, if x 5 }4 , Ïx 5 }2 , and
1
1
}
} , }. For x . 1, x . Ï x ; for
2
4
}
example, if x 5 9, Ïx 5 3,
and 9 . 3.
32. If a number is a perfect square,
then it is not a whole number;
false. Sample answer: 9 is a
perfect square and a whole
number.
1
43. 4, 64, 324, 1024, 2500; take } of
2
an even perfect square, and then
square it.
44. 31; the square root of 1000 is
33. If a number is an irrational
between 31 and 32, so there are
31 perfect squares less than 1000.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
number, then it is not a whole
number; true.
34. 6
35. 2
36. 28
Problem Solving
37. 242
38. 240
39. 63
45. 60 in.
40.
Set
Whole
numbers
Integers
Rational
numbers
Real
numbers
Additive
identity?
46. 40 yd
Additive Multiplicative Multiplicative
inverses? identity?
inverses?
Yes
No (except 0)
Yes
No (except 1)
Yes
Yes
Yes
No (except 1)
Yes
Yes
Yes
Yes (except 0)
Yes
Yes
Yes
Yes (except 0)
Algebra 1
Answer Transparencies for Checking Homework
27
47. 35 ft; 24 ft; 48 ft; 30 ft; they are
all rational numbers.
48. No; each side is a little more than
38 feet. 38 feet times 4 is 152
feet, which is more than 150 feet.
49. 2.2 feet
50. a. 290 K
b. 342 m/sec
51. a. 144 tiles
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
b. 16 ft. Sample answer: If the
homeowner can buy 144 tiles
that are each 256 square
inches, then the total area is
(144 tiles) + (256 square
inches per tile) 5 36,864
square inches. Divide 36,864
square inches by 144 square
inches to find the number of
square feet, 256 square feet. If
the area of the square is 256
square feet, take the square
root of 256 to find the side
length, 16 feet.
52. a.
Quotient
Quotient
Quotient of area
of side
of
length
of
pyramids
of
bases
bases
}}
Khafre
Menafaure
3.9
2.0
}}
Khufu
Menafaure
4.6
2.1
Khufu
Khafre
1.2
1.1
}
The quotient of the areas is the
square of the quotient of the side
lengths.
}
b. q 5 Ï r
53. * 5
Ï
}
A
6
}
Mixed Review of Problem
Solving
1. a. 1104 points
b. 354 points
2. a. T 5 15 1 (20.0065)e
b. 256.58C
Algebra 1
Answer Transparencies for Checking Homework
28
3. 3 in.;
7. Sample answer: A stock increases
3
in value by $8.90 one week,
decreases by $2.34 the next week
and decreases $1.15 the following
week. The expression
8.90 1 (22.34) 1 (21.15)
3
}}
represents the stock’s average
change over 3 weeks; $1.80.
4. a. 40 cm; 35 cm
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
b. The yellow bin.
Sample answer: For each bin,
divide the cost by the surface
area; the yellow bin costs less
per cubic centimeter.
5. a. c 5 23.29x; 2$13.16
b. No; it depends on the amount
of change for each item.
6. a. C 5 4x 1 5600
b. $6000
c. Yes; the cost will be $6200.
Algebra 1
Answer Transparencies for Checking Homework
29
LESSON
2.2
Answers for the lesson “Solve One-Step
Equations”
Skill Practice
49. 54 5 12x; 4.5 in.
1. inverse operations
2. Division property of equality; to
solve 14x 5 35, you need to
divide each side by 14.
3. 3
4. 27
5. 5
6. 6
7. 23
8. 28
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
9. 7
10. 12
11. 17
12. 13
13. 4
14. 22
15. C
16. A
17. 4
18. 213
19. 6
20. 212
21. 215
22. 211
23. 15
24. 28
25. 48
26. 249
27. 22
28. 242
29. The student multiplied x by 100 to
produce a number with a decimal
part identical to the decimal part
of x. When the student subtracted,
the result was a whole number.
7
30. a. }
9
31. 3.5
2
b. }
11
32. 24.2
34. 28.3
35. 22.05 36. 26
5
37. }
8
38. 211.5 39. 0.06
40. 42
41. 96
42. 235
43. 12
1
46. }
6
44. 35
3
47. }
5
45. 256
1
50. 72 5 } (16)x; 9 cm
2
Problem Solving
53. 1046.6 ft
54. 2.5 mm
55. 11 ft
56. 245 lb
4
57. a. } x 5 200
7
b. Plants; if you solve the
equation in part (a) you
find the there are 350 species
of birds.
58. Sample answer: Each member of
the drama club needs to sell
tickets to the upcoming play. If
there are 15 members and you
want to sell 135 tickets, how many
tickets should each member sell?
x 5 9; each member should sell
9 tickets.
59. a.
33. 21.1
52. 49
51. 20.5
t
D
1
6.5
2
13
3 19.5
4
26
5 32.5
48. 22
Algebra 1
Answer Transparencies for Checking Homework
31
59. b.
"
35
30
4 sec
25
20
15
10
5
0
0 1 2 3 4 5 !
c. 26 5 6.5t; 4 sec
60. a. 18 ft 2
b. 6 ft
61. a. 171 hits
b. 215 hits
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
c. No; if Mueller had fewer hits
than Wells but had a higher
batting average, he must have
had fewer at bats than Wells.
62. 47.5 ft 2
63. $2.60
Algebra 1
Answer Transparencies for Checking Homework
32
LESSON
2.3
Answers for the lesson “Solve Two-Step
Equations”
Skill Practice
36. 1, 2, 3; 4; the output of each
1. like terms
2. First, subtract 7 from each side to
get 4x 5 8, then divide each side
by 4 to get x 5 2.
Problem Solving
3. 4
4. 3
5. 2
37. 28 classes
6. 8
7. 23
8. 1
39. 5 half-side advertisements
38. 1.5 h
10. 20
11. 40
40. B
12. 212
13. 18
14. 21
41. Yes; the equation
15. 4
16. 6
17. 9
18. 6
19. 24
20. 22
9. 6
21. D
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
equation is increased by 3 because
x is multiplied by 3, each solution
increases by 1.
22. Unlike terms were combined;
5
23x 5 5, x 5 2}3
23. The division of 22x 1 x by 22 is
done incorrectly. Sample answer:
If like terms are combined as the
first step, the second line would
be 2x 5 10 and the final result
would be x 5 210.
24. y 5 3x 1 7; 25
$542 5 $50 1 6x gives the
monthly cost of a guitar that
costs $542. Solving the equation
gives x 5 $82 per month, so you
can afford the guitar.
42. a. y 5 2,896,112 1 1600x
b. 1163 days
c. Sample answer: Estimate the
number of days to be about
1200;
3,000,000 1 1600(1200) 5
3,000,000 1 1,920,000 5
4,920,000 so the solution
makes sense.
25. y 5 2x 1 4; 27
26. y 5 10x 2 9; 2
27. 4
28. 8
29. 5
30. 10
31. 0.5
32. 1.5
33. 15.9
34. 3.5
35. 6.9
Algebra 1
Answer Transparencies for Checking Homework
34
43. a. y 5 12x
b.
x(h) Marissa Ryan
1
5
7
12
2
10
14
24
3
15
21
36
4
20
28
48
5
25
35
60
c.
about 4.5 h
"
60
48
36
24
12
0
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Total
0 1 2 3 4 5 !
44. 5 days; the restaurant serves
400 1 120 5 520 customers
each day, use the equation
2600 5 520x to find x 5 5.
45. 200 sec
Algebra 1
Answer Transparencies for Checking Homework
35
LESSON
2.4
Answers for the lesson “Solve Multi-Step
Equations”
Skill Practice
36. You should divide each side of the
5
1. }
3
2. Use the distributive property to
get 12y 2 21 5 6, then add 21 to
each side to get 12y 5 27, divide
each side by 12 to get y 5 2.25.
a. 12
b. 2.5
3. 3
4. 3
5. 6
37. 16, 18, 20
6. 25
7. 22
8. 25
Problem Solving
9. 28
10. 27
11. 28
38. 6 tickets
12. 2
13. 4
14. 3
39. 0.75 ft
15. 29
16. 5
17. 219
40. 8 min
41. a. 34 months
18. C
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
equation by the number outside
the parentheses when the number
is a factor of the number on the
other side of the equal sign or it is
a fraction.
19. 12
20. 1
21. 22
b. 307 ft per month
22. 5
23. 29
24. 22
c. After the work crews merged;
25. 23 times 26 is 18, not 218;
5x 2 3x 1 18 5 2, 2x 1 18 5 2,
2x 5 216, x 5 28.
1
26. Multiply each side by 2, not };
2
2x 2 10 5 8, 2x 5 18, x 5 9.
27. 2
28. 0.5
29. 3
30. 0.1
31. 25
32. 1.2
33. 2
1
34. 5 }
3
35. 9.5 in., 6 in.; if you use the
perimeter formula P 5 2* 1 2w
and substitute 3.5 1 w for *, the
solution is w 5 6.
before the work crews merged
they were working at a rate
of 115 1 137 5 252 feet per
month, and after merging at a
rate of 307 feet per month.
42. a. y 5 74.5x 1 750; 21 squares
b. Squares Cost (dollars)
5
1122.50
10
1495
15
1867.50
20
2240
25
2612.50
Algebra 1
Answer Transparencies for Checking Homework
37
43. Let q be the number of
quarters, then q 1 8 is the number
of dimes. The equation 25q 1
10 (q 1 8) 5 280 represents the
situation, but the solution of the
5
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
equation is 5}7 . The number of
quarters must be a whole number,
so Jan must be mistaken.
Algebra 1
Answer Transparencies for Checking Homework
38
LESSON
2.5
Answers for the lesson “Solve Equations with
Variables on Both Sides”
Skill Practice
28. When the equation is 0 5 0, it
1. identity
2. A number plus 3 can’t be equal
to itself plus 1. If you solve the
equation, you get 0 5 2.
3. 22
4. 1
5. 24
6. 3
7. 27
8. 26
9. 8
12. 5
10. 3
11. 24
13. 23
14. 7
15. C
16. D
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
17. Sample answer: Distribute the 3
to get 6z 2 15 5 2z 1 13, then
subtract 2z from each side to get
4z 2 15 5 13, next add 15 to each
side to get 4z 5 28, finally divide
each side by 4 to get z 5 7.
18. no solution
19. 2
20. no solution
21. 27
22. 233
23. no solution
24. identity
25. no solution
26. identity
means that it is true for all values
of y, not just 0; the equation is an
identity.
29. Sample answer: 5x 1 4 5 5x; the
number 5x cannot be equal to
4 more than itself.
1
30. 2}
2
31. 2
32. no solution
33. 24
34. 10
35. 6
37. identity
39. 2
36. 6
38. 3
40. 224
41. 10
42. 24
43. identity
44. 120
45. 60
46. 136
47. 4
48. 5
Problem Solving
49. 9 nights
50. 4 months
51. about 4 yr
27. The 3 was not distributed to both
terms; 3x 1 15 5 3x 1 15,
15 5 15, so the equation is an
identity.
Algebra 1
Answer Transparencies for Checking Homework
40
52. a. 360 1 4x 5 16x; 30 visits
2. 13 lb;
b.
1
3
NonMembers’ members’
Visits
cost
cost
(dollars)
(dollars)
5
380
80
10
400
160
15
420
240
20
440
320
25
460
400
30
480
480
35
500
560
53. a. 23.4t 5 24(t 2 0.3); 12 sec
3. a. 4 games
c. Yes; if Paul bowls only 3
games, he will have $3.75
left over.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
b. about 4.4 sec
c. No; it would take 12 seconds
for the sheepdog to catch up to
the collie and it only takes
4.4 seconds for the collie to
complete the last leg.
54. 20, 8
b. 3 games
4. 4;
4
55. 11, 5
Mixed Review of Problem
Solving
1. a. C 5 0.25 1 0.07(t 2 1)
b. 26 min
Algebra 1
Answer Transparencies for Checking Homework
41
5. 30 min; if you make 40 cranes in
an hour, it will take 2.5 hours to
make 100 cranes, so you spent
30 minutes not making cranes.
6. 4 visits; use an equation to find
when the cost of skiing with a
pass is the same as skiing without
a pass. Solve 90 1 22.50x 5 45x
to find that x 5 4.
7. 9677 ft; if you add 1313 feet to
the current elevation, you will get
the elevation before the eruption.
8. a. 40 daffodil bulbs
b. 9 in.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
c. 20 daffodil bulbs. Sample
answer: If she plants bulbs
6 inches apart she will need
360 inches 4 6 inches 5 60,
so she needs 60 2 40 5 20
more bulbs.
9. Sample answer: You want to buy
a video game for $47, if you have
$15 and plan on saving money
from a paper route each week for
4 weeks, how much money should
you save each week? x 5 8; you
need to save $8 each week for
4 weeks to have enough money to
buy the video game.
Algebra 1
Answer Transparencies for Checking Homework
42
LESSON
2.6
Answers for the lesson “Write Ratios and
Proportions”
4
3
43. Yes. Sample answer: } 5 }
8
6
Skill Practice
1. ratios
2
2. Sample answer: }, 2 : 5, 2 to 5
5
3. no; 7 to 9
4. yes
5. yes
6. no; 7 to 8
6
7. }
5
8. 64
10. 6
11. 48
12. 5
13. 15
14. 4
15. 40
16. 48
17. 12
18. 56
19. C
9. 22
Problem Solving
7
2
45. }
46. }
145
8
3
2
1
47. }
48. }
49. }
5
5
2
50. 21 pages
1.85
1
b.
Width (ft)
Length (ft)
20
37
25
46.25
30
55.5
35
64.75
40
74
side by 10 and the other side by
50
20; 10 + }
5 10 + }
, m 5 25.
10
20
x
3
23. } 5 }; 12
32
8
8
x
25. } 5 }; 2
16
4
7
b
27. } 5 }; 35
2
10
d
12
29. } 5 }; 18
27
18
a
5
24. } 5 }; 35
49
7
y
9
26. } 5 }; 36
5
20
n
4
28. } 5 }; 1
3
12
40
t
30. } 5 }; 30
28
21
31. 1.8
32. 3.3
33. 2.4
34. 2
35. 4
36. 6
37. 4
38. 4
39. 2
40. 19
41. 3.5
42. 4
x
38
} 5 }, 70.3 ft
22. Multiply each side by 10, not one
m
51. 45 goals
52. a. Sample answer:
20. B
1
21. Multiply each side by 6, not };
6
3
x 1
}
}
}
6 + 4 5 6 + 6 , 4 2 5 x.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
44. Yes; multiply each side by b to get
cb
the equation a 5 }
, then divide
d
b
a
each side by c to get }c 5 }d .
10
53. a. }
23
b. 110 lift tickets
c. 40 snowboarders
Algebra 1
Answer Transparencies for Checking Homework
44
x
10
54. a. Sample answer: } 5 }
11
3
x
5
b. Sample answer: } 5 }
14
2
c. Your friend; you will wait in
2
line for 36 }3 minutes and
your friend will wait only
35 minutes.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
55. 37.5 mi, 22.5 mi
Algebra 1
Answer Transparencies for Checking Homework
45
Answers for the lesson “Solve Proportions
Using Cross Products”
LESSON
2.7
Skill Practice
1. cross product
2. Measure the distance in
centimeters in the drawing and
then substitute the value in the
distance in drawing
1
proportion, }3 5 }}
.
actual distance
decrease.
Problem Solving
3. 6
4. 7
5. 24
33. 5 c
34. 18 min
6. 6
7. 1
8. 6
35. 90 km
36. 48 km
10. 7
11. 2
37. 7.5 km
38. 70.5 km
13. 12
14. 26
39. 17.728 m
9. 249
12. 27
15. B
40. a. 213.75 ft
16. D
17. Use the cross products property to
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
32. Decreases; as k gets larger, the
k
value of }
increases, so for the
14
5
value of } to increase, h must
h
multiply 4 by x and 16 by 3;
4 + x 5 3 + 16, 4x 5 48, x 5 12.
18. Distribute the 14 to both b and 2;
18b 5 14b 1 28, 4b 5 28, b 5 7.
19. 15
20. 2
21. 10
22. 22
23. 5.5
24. 22.5
25. 23.4
26. 5.2
27. 4.2
28. 7.4
29. 25.9
30. 27.1
31. a. Multiplication property of
equality
b. Multiply
b.
Shell
diameter
(in.)
Burst
diameter
(ft)
2
90
3
135
4
180
5
225
6
270
41. 80 yd; find the actual length of
the field by using the ratio
1 in. : 20 yd, then use the number
to find the width of the soccer
field by using the ratio 3 : 2.
c. Simplify
Algebra 1
Answer Transparencies for Checking Homework
47
42. a. 4.5 moles
b. 3.4 moles
c. Nearly equal. Sample answer:
The ratio of moles of hydrogen
to moles of carbon is about
1.33 to 1, the ratio of moles of
hydrogen to moles of
oxygen is about 1.32 to 1. The
two ratios are nearly the same.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
43. Sample answer: I would assign
11 seats to the freshmen, 8 seats
to the sophomores, 7 seats to the
juniors, and 4 seats to the seniors.
The actual numbers obtained from
the proportions are 10.464, 8.4,
6.72, and 4.32. If I round each of
those numbers in the traditional
way, the sum is only 29. I rounded
10.464 up to 11 to get a total of
30 seats. I chose 10.464 because
of the three numbers with decimal
parts less than 0.5, its decimal
part is the greatest.
Algebra 1
Answer Transparencies for Checking Homework
48
LESSON
2.8
Answers for the lesson “Rewrite Equations
and Formulas”
Skill Practice
23. C
1. literal equation
2. Divide each side by pr to get
I
} 5 t.
pr
c
3. x 5 };22
b2a
c
4. x 5 } 2 b; 3.5
a
5. x 5 bc 2 a; 9
ab
6. x 5 }; 3
c
29. Sample answer: You want to find
how long it will take to drive
150 miles if you drive at an
average rate of 55 miles per hour.
9. b should have been subtracted
from both sides, not added;
2b
ax 5 b, x 5 }
.
a
10. Both sides should have been
divided by (a 2 b), not
d12
32. a. n 5 }
4
b. 1.75 in., 3.5 in., 4.5 in., 5.5 in.
C 2 25
33. a. x 5 }
12
b. 10 league nights;
13 league nights;
15 league nights
c
multiplied; x 5 }
.
a2b
11. y 5 7 2 2x
5
5
12. y 5 } 2 } x
2
4
13. 4 2 3x 5 y
14. y 5 213 1 9x
15.
17.
19.
21.
by
31. a 5 }
y 2 bx
Problem Solving
b1d
8. x 5 }; 2
c2a
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5
1
25. y 5 18 2 5x 26. y 5 2} x 1 }
2
2
S
27. * 5 }
pr 2 r; 13.03 cm
A
28. p 5 }; 8.01 ft
4pw
b1c
30. a 5 }
bx 2 1
7. x 5 a(c 2 b); 28
6
2 1 }7 x 5 y
9
}x 2 6 5 y
5
1
1
y 5 }2 x 1 }3
S 2 2B
h5}
P
24. y 5 2.1x 2 8.4
16.
18.
20.
22.
5
y 5 x 2 }8
3
2 2 }2 x 5 y
V
w5}
*h
*
f5}
24
P 2 2pr
34. a. x 5 }
2
b.
r
x
50
173
51
170
52
167
53
164
Algebra 1
Answer Transparencies for Checking Homework
50
34. c.
"
173
161 ft
171
169
167
165
163
0
3
3
4. Solve Q 5 } for r to ger r 5 };
r
Q
3
substitute }
for r in the formula
Q
6
d 5 2r to get d 5 }
.
Q
5. 140 attempts;
1
4
0
0 50 51 52 53 54 !
35. Divide each side by the total bill,
a
b, to get } 5 p%.
b
C
36. a. r 5 }F
2p
b. 1.1 ft; 1.3 ft; 1.4 ft
C
c. Solve for r using }F
5 r,
2p
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
then substitute r into the
equation A 5 pr 2.
11t
37. g 5 }
4
Mixed Review of Problem
Solving
x
7
1. a. Sample answer: } 5 }
1209
31
b. 273 students
5
2. a. }
9
b. 120 females
3. $70. Sample answer: Use the
x
7
} to find that
proportion }
5
700
154
6. a. $12,480
b. $12,300
c. 0.6%; find that $1230 is 10%
of $12,300, then find $1230 is
what percent of $205,000.
7. Multiply each side by 2 and
divide each side by d2 to get
2A
d1 5 }
.
d
2
8. Sample answer: Increase the
length, 16 feet by 4 feet.
you will need about 35 gallons of
gas. Multiply 35 by $2 per gallon.
Algebra 1
Answer Transparencies for Checking Homework
51