Download Algebra 2 Honors

Algebra 2 Honors
Summer Review Packet
There are a total of four files that constitute the summer Algebra 2 Honors assignment for
students at Beckman High School—01 Algebra Overview (this file), 02 Equations and Absolute
Value, 03 Coordinate Geometry, and 04 Cartesian Connection MCQ.
These problems were chosen to make sure you are comfortable with Algebra 1 concepts and will
help provide a smooth transition into Algebra 2 Honors (after all, it has been a while since you’ve
taken Algebra 1). In light of our adoption of the Common Core Standards, there are some
topics—notably coordinate geometry topics—that we must insist that students have mastered
before entering this course.
Answers are provided for the first file—the overview, but they are not provided for the remaining
three files. You will be provided with answers in the Fall so you can check your answers at that
time.
For 01 Algebra Overview, please work neatly on separate sheet(s) of paper and attach (staple)
your work behind this packet. Please do not attempt to put your work on the Algebra Overview
itself—this file is designed as a list of problems and is not intended to have enough space for
students to show sufficient work. However, the remaining three files have more space between
problems and some students may be able to use the actual assignments themselves as space for
their work. If you are unable to use the allotted space to present clear and well organized work,
please use separate sheet(s) of paper and attach them to the back of those other assignments.
Have your work clearly labeled and well organized and have this packet ready to turn in when
you return in the Fall. Also be prepared to take a short quiz of problems based on what you have
seen in this packet during the first week of school.
We look forward to working with you next year.
Student Name
Beckman High School
Last Modified June 27, 2013
Page 1
Solve each of the following. If the equation is an identity (e.g. has all real numbers as solutions)
or has no solutions, state that fact.
9
1. 9  (1  x)  2
2. 3k  
5
a
3. 7t  2t  36
4.   7
3
5. 6(q  3)  24
6. 3 x  7   19  3x
7.
8x  3 4 x  5  2 x  11
8.
5  x
 2x
8
9. Nine more than a number is 90. What is the number?
10. Two thirds of a number is 52. What is the number?
11. Three times a number is 18 more than the number. What is the number?
12. The perimeter of a rectangle is 42 meters. The length of the rectangle is 3 meters less than
twice the rectangle’s width. Find the length and width of the rectangle.
13. A postal clerk sold some fifteen cent stamps and some twenty five cent stamps. Altogether,
10 stamps were sold for a total cost of $1.70. How many of each type of stamp was sold?
14. Stan has 52 quarters and nickels combined. If he has three times as many nickels as quarters,
how much money does he have?
15. Find the slope and the y – intercept of a line with equation 7 x  y  5 .
16. Find the slope and the y – intercept of a line passing through the points (6, 8) and (3, 5).
For problems 17 and 18, change each equation into slope-intercept form and then graph each line
in the space below using only the line’s slope and it’s y – intercept.
y
y
18. 3x  2 y  4
17. 3 = x – y
x
x
Solve the systems of equations in problems 19 through 21 graphically in the spaces below.
19.
y  4  3x
y
20.
y  x4
x
yx
y
y  x  4
x
Page 2
y
21.
x y3
y  x 1
x
Set up problems 22 and 23 by setting up two equations with two variables and then solve each
resulting system.
22. The sum of two numbers is 45. Three times the smaller number exceeds twice the larger
number by 5. Find both numbers.
23. Theater admission for a group of three children and four adults costs $62. If there were one
more adult in the group the cost would have been $73. What is the price of admission for
each child?
Simplify.
24. 8 + 3 – 1 + 5
25. 8 + 3 – (1 + 5)
26. 8  7  15  3
27. 34  6  7  3   7
Match the correct letter of the most appropriate algebraic property on the right with each of the
five statements on the left. (See page 5 of this packet for reference)
28. 2a  3b  1  2a 1  3b 
A.
Commutative Property of Multiplication
29. If a  b  a , then b  0
B.
Identity Property of Addition
30. 2  c  n   2c  2n
C.
Inverse Property of Addition
31. a  6  (6)  a  0
D.
The Distributive Law
E.
Commutative Property of Addition
32.
 5  6 x  a  b    a  b 5  6 x 
Evaluate each of the following expressions if x = –5 and y = 8.
33.
 x  y 2
34.
35.
x2  y 2
36. 2x  x  y 
 y  x  y  x 
37.  xy  4 x  5 y
38.
x3  3 y 2  x
x y2
x
40.
x y
x y
39.
Page 3
Solve each of the following equations.
41. 16   3x  8  3
42. 7c  12  16  3c
43. 3  5  t  4   4
44. 6   3a  7   2a  a  6
45.
2
3
 y  9    y  8
3
4
46. 2c  c  5  c  2c  5  5
For problems 47 through 49, substitute the given value of y and then solve for x.
47. 2 xy  y  5   y 2  2  y  x   13


y=2
48.  y 3x  y 2  x 1  y   y
y = –3
49. y equals the average of –4, 2, and x
y=6
Express your answer for each of the following in terms of the given variable. Give answers in
simplest form.
50. A cashier has two one-dollar bills and x five-dollar bills. How much money is that?
51. Each side s of a regular pentagon is lengthened by 3 cm. What is the perimeter of the
resulting new pentagon?
52. A number n is two more than four times another number. What is the sum of those two
numbers?
Set up and solve the following.
53. A change purse contains 14 coins consisting of nickels and dimes and having a total value of
$1.10. How many nickels are in the change purse?
54. At 10:00 AM two cars leave the same location and travel in opposite directions. One car’s
speed is 50 miles per hour and the other car’s speed is 55 miles per hour. At what time of day
are the two cars 273 miles apart?
Factor each of the following completely. If the polynomial cannot be factored into products with
integer coefficients, state that the polynomial is prime.
55.
x 2  7 x  12
57. 2 x 2  13x  15
59.
y 2  y  30
61. 4 y 2  100
63.
xy  3 y  5x  15
56. 3x 2  21x  36
58. 4r 2  9
60.
x3  x 2  12 x
62. 4 x 2  12 x  7
64.
Page 4
x3  x  3x 2  3
The 18 Postulates of Algebra
Just as geometry had postulates (axioms) for ideas which we needed to accept without proof,
algebra also has its postulates for ideas that form the underpinning of nearly everything we do. To
some students, these ideas are so intuitively obvious that they wonder why we stop to delineate
them. However, to truly be rigorous, all (or at least as nearly all as possible) of our assumptions
must be spelled out precisely.
A good part of modern mathematics (and yes, mathematics is a modern study) stems from taking
our assumptions and envisioning what would happen if they were not held. Just as there are nonEuclidean geometries where parallel lines can meet, there are non-Abelian algebras where
ab  ba 1. It is impossible to experiment with and to challenge our assumptions unless we first
know what those assumptions are.
To that end, you are tasked this summer with knowing the postulates below. Given an algebraic
statement, you need to be able to fully name the postulate that it demonstrates, or given a
postulate, you need to be able to formulate an example statement for it. Problems 28 through 32
earlier in this assignment were a good set of example questions, the next page follows with more
sample questions.
The 18 Postulates of Algebra
1. The Associative Property of Addition:
2. The Associative Property of Multiplication:
3. The Commutative Property of Addition:
4. The Commutative Property of Multiplication:
5. The Identity Property of Addition:
6. The Identity Property of Multiplication:
7. The Inverse Property of Addition:
8. The Inverse Property of Multiplication:
9. The Distributive Law:
10. The Substitution Principle:
11. The Zero Product Property:
12. The Reflexive Property of Equality:
13. The Symmetric Property of Equality:
14. The Transitive Property of Equality:
15. The Additive Property of Equality:
16. The Multiplicative Property of Equality:
17. Closure of the Reals under Addition:
18. Closure of the Reals under Multiplication:
1
a + (b + c) = (a + b) + c
a(bc) = (ab)c
a+b=b+a
ab = ba
a + 0 = a and 0 + a = a
a  1  a and 1  a  a
a + (–a) = 0 and –a + a = 0
a  1a  1 and 1a  a  1
a(b + c) = ab +ac
If a = b, then any statement true for a is also
true for b.
If ab = 0, then either a = 0, b = 0, or both.
a=a
If a = b, then b = a
If a = b and b = c, then a = c
If a = b, then a + c = b + c and c+ a = c + b
If a = b, then ac = bc and ca = cb
If a and b are real numbers then a + b exists
and is also a real number
If a and b are real numbers then ab exists
and is also a real number
“Basic Algebra” does not mean easy, it means considering algebra at its most core and fundamental level.
For the purposes of this course, we will not hypothesize algebra systems without the postulates given
above. That type of advanced exploration into mathematics typically follows three semesters of study in
Calculus and is most commonly seen in a college program of study.
Page 5
Match the expression on the left with the best corresponding postulate on the right. Write the
letter of the postulate on the space next to each number.
_____ 65. 3x 1  1  3 x
A. The inverse property of multiplication
_____ 66. If x  7  3 , then x  10
B. The multiplicative property of equality
_____ 67. If a  xy , then 2a  2  xy
C. The identity property of multiplication
_____ 68. If a  3x  1 , then 3x 1  a
D. The commutative property of multiplication
_____ 69. If 3  (2  12 )x , then 3  1x
E. The symmetric property of equality
_____ 70. If y  3x 1 , then y  3x
F. Closure of the real numbers under addition
Write the postulate that best describes the expression given.
71. If 2 x 2 (3x  1)  0 , then 2 x 2  0 or 3 x  1  0
72. If 5 x  2  0 then 5 x  2  2  0  2
73. If m  2b and y  mx  b , then y  2b  b
74. If 4 x  (2  3) y  24 , then 4 x  6 y  24
Consider the statement a  0  0 . Since 0 is a very special real number, in particular it is “the
additive identity,” we cannot simply claim this is true because of the closure of the real numbers
under multiplication. We can verify this statement however, as a theorem in the following way:
Write the postulate that best describes the expression given.
a  0  a  (0  0)
The identity property of addition
 a0  a0
75. ________________________
Thus,
(a  0)  a  0  (a  0)  a  0  a  0
The additive property of equality
0  (a  0)  a  0  a  0
76. ________________________
 [(a  0)  a  0]  a  0
The associative property of addition
 0  a0
77. ________________________
 a0
78. ________________________
Finally, since 0  a  0 , we have a  0  0 .
79. ________________________
Page 6
Answers to Summer HW Packet
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
33.
35.
37.
39.
41.
43.
45.
47.
49.
51.
53.
55.
57.
59.
61.
63.
65.
67.
69.
71.
73.
75.
77.
79.
6
–4
–7
13
81
9
8 stamps at 15¢, 2 stamps at 25¢
slope = –7, y – int. at (0, 5)
y  x 3
(2, –2)
(2, 1)
$6.00
5
3
B
C
9
–39
20
3
7
 21
5
–144
x  12
x  20
5s  15
6 nickels
( x  3)( x  4)
(2 x  3)( x  5)
( y  6)( y  5)
4( y  5)( y  5)
( y  5)( x  3)
D
B
A
The zero product property
The substitution principle
The distributive law
The inverse property of addition
The symmetric property of equality
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
22.
24.
26.
28.
30.
32.
34.
36.
38.
40.
42.
44.
46.
48.
50.
52.
54.
56.
58.
60.
62.
64.
66.
68.
70.
72.
74.
76.
78.
Page 7
 53
–21
no solution
 13
78
13 meters and 8 meters
$5.20
slope = 1, y – int. at (0, 2)
y   32 x  2
(2, 2)
19 and 26
15
51
E
D
A
39
–30
–322
 133
7
7
6
1
x6
5x  2
Let the two numbers be n and y, where
n  2  4 y . Then the sum is represented
2
both by 5n
or by 5 y  2
4
12:36 PM
3( x  3)( x  4)
(2r  3)(2r  3)
x( x  4)( x  3)
prime
( x  3)( x  1)( x  1)
F
E
C
The additive property of equality
Closure of the real numbers under mult.
The inverse property of addition
The identity property of addition