Chapter 2, Problem Set #1
Name: ___________________
Class: _____________________
Date: _____________
The Rules of Algebra
Instructions

Please do these questions on a separate sheet of paper… do not write on this problem
sheet.

Restate key information in each problem before you start solving it.

Show all of your work.
Review and new words
Note: words in boldface are new words that you need to learn. Please write definitions of the
words in your journal.
1. Write three examples of a statement and three examples of expressions. What is the
difference between a statement and an expression?
2. Your book defines the word “statement” as “an equation or inequality that is either true
or false.”
a. Write two true statements.
b. Write two false statements.
3. Consider this equation: 3x  4  31 .
a. What value of x makes this sentence true?
b. What is the solution to this equation? Describe your solution using an algebraic
statement using x.
4. Consider this inequality: 4x  20 for x ° (x is a real number).
a. Solve this inequality.
b. What values of x make this statement false?
c. What is the solution to this inequality? Describe your solution using an algebraic
statement using x.
5. A term is the product of real numbers and variables. Which of the following expressions
are also terms?
a. 3x
b. 4 2
c. 3  7x
d. x 2
Chapter 2, Problem Set #1
Class: _____________________
e.
b  b2  4ac
2a
f.
4
xy
3
g.
7x
2y
Name: ___________________
Date: _____________
h. 82
i. yzq
6. Like terms are terms that have exactly the same variables raised to the same exponent.
3
5x, x , 32 x , and 7x are all like terms because the variables are the same and the
2
exponents are the variables are the same. 17y and 17 y 2 are not like terms, because the
exponents on the variables are different. 8x and 3y are not like terms because the
variables are different.
7. In the Algebra 1 textbook, do problems 10–13, 17, and 18 on page 55.
8. In each case, determine whether the two expressions are equivalent.
a. Example: 2x  3  3  2x Answer: The expressions 2x  3 and 3  2x are
equivalent expressions.
b. (a 2  b2 )  (a  b)(a  b)
c. (a 2  b2 )  a 2  b2
d. 3x  2  x  1  x 2  3x 3  2
e. 3x  2  x  1 x 2  x 2  2x  3
f.
2(x  3)  2x  3
g. 2(x  3)  2x  6
h. x 2  2x  1  (x  1)2
9. Think about multiplication. Which of the following are always true?
a.
x y  yx
b. (x  y)  z  x  ( y  z)
c. 1 x  x 1  x
d. x  0  0  x  0
Chapter 2, Problem Set #1
Name: ___________________
Class: _____________________
Date: _____________
Properties
In the problem sets for chapter 1, we learned about several different sets of numbers:
 {1, 2, 3,...} , Natural numbers,
W  {0, 1, 2, 3}   {0} , Whole numbers,
 {...,  3,  2,  1, 0, 1, 2, 3,...} , the Integers
Q, the rational numbers,
the irrational numbers, and
R, the Real numbers.
In this section, you’ll begin to think about the properties of these sets… how they are the same,
and how they are different.
10. For each of these statements, x, y, z  (x, y and z are Natural Numbers). Which of
these statements are true for any x, y and z? (If the statement is false, provide a counterexample.)
a.
x  y  y  x and x  y  y  x
b. ( x  y )  z  x  ( y  z ) and ( x  y )  z  x  ( y  z )
c. The sum of any two Natural Numbers is a Natural Number ( x  y 
)
d. The product of any two Natural Numbers is a Natural Number ( x  y 
e. The quotient of any two Natural Numbers is a Natural Number (
x

y
)
)
f. The difference of any two Natural Numbers is a Natural Number ( x  y 
)
g. Given any Natural Number x, you can find another Natural Number y such that
x y 0
h. Given any Natural Number x, you can find another Natural Number y such that
x y 1
11. As you have discovered, not all of the statements above are true for . Which of the
above statements are true for ? for ? for ? (split the work among your group
members).
12. Here are some new “operations.” Let x, y, z 
the work between your group members.)
(x, y and z are Real Numbers). (Split
a. Let the operation * be defined as x * y  xy  x . Find the value of x * y for the
following ordered pairs (x y): (1, 2), (2, 1), (4, 3), (–4, –3). Example:
1*2  1 2 1  2 1  3 , 2*1  2 1  2  4 . Consider each one of the following
statements. Is the statement always true, true with some restrictions, or false?
i. x * y  y * x ?
Chapter 2, Problem Set #1
Class: _____________________
Name: ___________________
Date: _____________
ii. (x * y) * z  x * ( y * z) ?
iii. x *1  x
iv. x * 0  0
v. x * 0  x
vi. 0 * x  x
b. Let the operation & be defined as x & y  (x  1)( y  2) . Find the value of x & y
for the following ordered pairs: (1, 2), (2, 1), (4, 3), (–4, –3). Consider each one
of the following statements. Is the statement always true, true with some
restrictions, or false?
i. x & y  y & x
ii. (x & y) & z  x & ( y & z)
iii. x &1 x
iv. 1& x  x
v. x &0  0
vi. 0 & x  0
c. Let the operation @ be defined as x @ y  x  y  xy . Find the value of x @ y for
the following ordered pairs: (1, 2), (2, 1), (4, 3), (–4, –3). Consider each one of
the following statements. Is the statement always true, true with some
restrictions, or false?
i. x @ y  y @ x
ii. (x @ y)@ z  x @( y @ z)
iii. x @1  x
iv. 1@ x  x
v. x @ 0  0
vi. 0@ x  0
13. As a group, make up your own operation, denoted #. As above, check and see what
properties your operation has.
14. Exercise 12 showed examples of some important properties. Although they weren’t
always true for the operations above, they are, in general, true for addition and
multiplication. For each of the following statements, find the name of the property that it
corresponds to. A table of properties can be found in section 2.6 in your textbook.
a.
x y  y x
b. x  y  y  x
Chapter 2, Problem Set #1
Name: ___________________
Class: _____________________
c.
Date: _____________
x  ( y  z)  ( x  y)  z
d. x  ( y  z )  ( x  y )  z
e. There is a number 0, called __________ such that a  0  0  a  a
f. There is a number 1, called __________ such that a 1  1 a  a
g. ( x  y )  z  z  ( x  y ) Careful! This is a trick question.
h. ( x  y )  z  ( y  x)  z Careful! This is a trick question, too.
15. The table below lists several properties that we have explored. See if you can match the
property statement to the questions in the previous problem. Copy this table to your
journal and fill in the blanks.
Properties of Equality
Name of property
Commutative
Property (addition
and multiplication)
Statement
a b  ba
a b  b a
Order of multiplication or
addition doesn’t matter
Associative
Properties (addition
and multiplication)
a  (b  c)  (a  b)  c
a  (b  c)  (a  b)  c
You can group consecutive
additions or multiplications
any way you want.
Additive identity
There is a number 0, called
the “additive identity”,
such that a  0  0  a  a
Multiplicative
identity
There is a number 1, called
the “multiplicative
identity” such that
1 a  a 1  a
Multiplication
property of zero
0 a  a  0  0
Example
Chapter 2, Problem Set #1
Name: ___________________
Class: _____________________
Date: _____________
Properties of Equality
Name of property
Statement
Multiplicative
inverse
For every a  0 , there is a
1
number such that
a
1
a 1
a
Quotient Property
a
Example
If 3  x then x  3
1 a

b b
16. Look back at the operations @, & and * above. Which ones of them (if any) have an
inverse? An identity?
17. Algebra basics: p. 55: 17-19, 26, 27, 35–38, 40–42, 44, 50, 51, 56–59
a. Be sure you know what each of the following words means: equation, inequality,
expression, statement, solve, solution, like terms, simplified form. If there are any
you don’t know, be sure to write them in your journal.
Chapter 2, Problem Set #2
Name: ___________________
Class: _____________________
Date: _____________
Properties of Equality and Zero
1. Show how you would solve these equations step by step. Describe each step in words.
a. Example: Solve 3x  7  13
3 x  7  13
(3 x  7)  7  (13)  7
Subtract 7 from both sides
3x  6
(3 x) (6)

Divide both sides by 3
3
3
x2
Solution
Now, look at the example and explain how each step helped you find a solution.
b. Example: Solve 3x  2  4x  7
3x  2  4 x  7
Original equation
(3x  2)  3x  (4 x  7)  3x Subtract 3x from both sides
2  4 x  3x  7 Combine like terms
2  x7
(2)  7  ( x  7)  7
Subtract 7 from both sides
5  x
Now, explain how each step helped solve the problem.
Note: When you do something to both sides of an equation, surround the original
part in parentheses first.
c. Solve 2x  3  4x  x  2
d. Solve 3x  2  2x  2 (this does have a solution)
e. Solve x 2  3x 2  8
f. Solve 3( x  4)  2(3 x  9)  12
g. Solve 5x 10  20x . Although there are many ways to solve this, I’d like you to
start by dividing both sides by 5. When you divide, be sure to put parentheses
around each side of the equation first..
2. Use the properties of equality to solve these equations for a variable. When you solve
for a variable, you rewrite the equation with that variable alone on one side of the
equation, and all other variables and constants on the other side. You do not have to
find a value for x and y.
Chapter 2, Problem Set #2
Name: ___________________
Class: _____________________
Date: _____________
a. Example: Solve 3x  4 y  x  3  2 y for y.
3x  4 y  x  3  2 y
Original equation
(3 x  4 y )  3x  ( x  3  2 y )  3 x
4 y  3  2 y  2x
Subtract 3 x from both sides
Simplify
4 y  2 y  (3  2 y  2 x)  2 y
Subtract 2 y from both sides
2 y  3  2x
(2 y )  2  (3  2 x)  2
Simplify
Divide both sides by 2
y
3  2x
2
Simplify/rewrite
b. 4 x  10 y  7  3x  y
3. To get more practice solving equations, try these problems in your textbook: p. 61: 21,
22, 25, 26, 33, 35, 46, 47, 52, 54.
3x  2 y  5
4. Here are a pair of equations: 
. There is an ordered pair (x, y) that will
4 x  2 y  9
solve both equations. We know that 4 x  2 y  9 . The property of equality allows us
to add equal things to both sides of an equation. So we could add 4 x  2 y to the left
side of an equation and 9 to the right side. Try doing that to the first equation and see
if you can find the ordered pair that solves both equations. Try these:
 x  3 y  19
a. 
x  3y  1
4 x  3 y  10
b. 
 3x  3 y  4
4 x  y  7
c. 
(this one’s a little different)
y  2x 1

5. Copy this table into your journal and complete it with the four properties of
equality.
Properties of Equality
Name of property
Addition Property
of Equality
Statement
Example
If a  b , then a  c  b  c If x  4 , then
“add the same thing to both x  3  4  3
sides”
Chapter 2, Problem Set #2
Name: ___________________
Class: _____________________
Date: _____________
Properties of Equality
Name of property
Statement
Example
Subtraction
Property of
Equality
Multiplication
Property of
Equality
Division Property
of Equality*
Reflexive Property
of Equality
aa
3.5 = 3.5
Symmetric
Property of
Equality
If a  b then b  a
If 3  x then x  3
Transitive Property
of Equality
If a  b and b  c then
ac
If x  3 and 3  y then
x y
* Note: There is a restriction that you need to think about for the Division Property of
Equality. What is it?
6. As you solved equations in problem 1, you had to decide what to subtract from both
sides of the equation. In most cases, you tried to make one term “move from one side
to the other,” or “disappear from one side of the equation.” To do that, you used an
inverse operation. Look up inverse operation in section 2.2 of your book. Look
back at your solutions to problem 1 and write down what inverse operation you used.
7. Another useful property is the Zero Product Property. Solve the following equations.
What is it about Zero that makes these easy to solve? What other properties do you
need to solve these?
a. Solve ( x  3)( x  4)  0 . There are two solutions.
b. Solve x  ( x  2)3 ( x  1)2  0 . There are three distinct solutions.
c. Solve x 2 (3x  1)  0 . How many solutions are there?
d. Solve ( x 2  2)( x 2  1)  0 . How many solutions are there?
8. Here are some practice problems: p. 66: 25, 28, 31–35, 38, 40, 41, 44, 47, 49–53.
Chapter 2, Problem Set #2
Name: ___________________
Class: _____________________
9. Consider the diagram of the
rectangle on the right. Let x  3 ,
y  5 , and z  2 . Compute the area
of the rectangle BACD. Using only
the variables x, y and z, write a
formula for computing the area of
this rectangle. Try to write the
formula in two different ways that
produce the same result. Hint…
think about the “Distributive
Property”.
Date: _____________
y
z
A
C
B
D
x
10. Each of the following expressions can be used to
compute the area of a rectangle similar to the one
above. For each expression, draw a diagram, label
it appropriately, and write the expression in its
equivalent form.
4
x
3
4x
12
a. Example: 4  ( x  3) . Diagram is to the right,
equivalent form is: 4x  4  3  4x  12 .
b. x  (4  y )
c. 3  (5  7)
d. 2 x  3x
e. 15 y  30 x (there are several ways of doing this… try to find as many as you can)
11. Look up the “substitution property” in your book. Write a definition of it in your
journal, and show an example of how to use it.
12. Important tip: when you use the substitution property, always put parentheses around
the value you substitute. In each of the following, substitute the given value of x into
the equation, and then solve:
a. Example: x  4 y  7 , x  2 y .
Substituting, we get:
x  4y  7
(2 y )  4 y  7
6y  7
7
y
6
Note how parentheses are used around 2y when it is substituted for x.
Chapter 2, Problem Set #2
Class: _____________________
b. Example: 3x  y  10 , x  2  4 y .
Substituting, we get:
3 x  y  10
3(2  4 y )  y  10
3  2  3  4 y  y  10
6  12 y  y  10
6  13 y  10
13 y  16
16
y
13
c.
x  y  15 , x  5
d. 2 x  3 y  11 , x  4
e.
x y  9, x  y 3
f.
2x  3y  6 , x  y  2
Name: ___________________
Date: _____________
Chapter 2, Problem Set #2
Name: ___________________
Class: _____________________
Date: _____________
Look up the Distributive Property in your textbook. Complete this table in your
journal.
Distributive Property
Name of property
Statement
Example
Distributive
Property of
Multiplication over
addition
Distributive
property of
Multiplication over
Subtraction
Summary

Terms: Review all of the new terms that you’ve learned in these problem sets (Chapter
2, Problem Set #1, and Problem Set #2). If there are any you don’t remember (or didn’t
learn), find them and write them down in your journal. Look on page 86 in your
textbook for a list of vocabulary.

Use those properties. Start with the equation x  5 . Use as many of the properties you
have learned to transform this equation into another equation that is also true. Here’s an
example to get you started:
x5
x  y  5  y Addition property of equality
3( x  y )  3(5  y ) Multiplication property of equality
3 x  3 y  15  3 y Distributive property
Chapter 2, Problem Set #3
Class: _____________________
Name: ___________________
Date: _____________
Word Problems
Your textbook gives the following hints for solving word problems:
1. Read and re-read the problem
2. Determine what you need to find
3. Try some numbers, looking for a pattern
4. Use a strategy (guess a solution, write an equation, work backwards, draw a model, try a
simpler problem)
5. Check your answer, and be sure that it makes sense
I’d say that step 5 is required for every problem… always check your answer.
Here are two sets of problems from your textbook.
1. p. 76: 3, 4, 6, 9, 10, 16. Work on these problems first. Even if you can solve them in
your head, or by trial and error, be sure to go through the exercise of doing them
algebraically. Write and solve equations to find or check your answer.
2. p. 76: 7, 8, 11, 12, 15, 17