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Topics and sample problems for Math 017 Final Exam
(long version)
Below there are 59 categories:
9 “arithmetic” categories
13 “easy algebra” categories
24 “basic algebra” categories
9 “difficult algebra” categories
4 “very difficult algebra” categories
Each version of the exam will be created as follows:
First, 45 categories will be selected: All 9 arithmetic categories will be included. From the
algebraic categories, 8 categories will be selected at random from 13 “easy” categories, all
24“basic algebra” categories will be represented, 3 out of the 9 “difficult” categories, and 1
from the“very difficult” category. After this process, we will have 45 categories. One question
out of each such category will be selected at random. Thus, the test will consist of 45 questions.
ARITHMETIC PART: 9 QUESTIONS
Select randomly one question from each category.
1A
Addition, subtraction, multiplication, or division of integers (using both: 
and fraction bar as a division sign) - up to 3 integers.
Evaluate, if possible. Otherwise, write “undefined”:  3  5 
Compute, if possible. Otherwise, write “undefined”:  7  1 
Evaluate, if possible. Otherwise, write “undefined”: 3 10 
Compute, if possible. Otherwise, write “undefined”:  3  (2)  (11) 
Evaluate, if possible. Otherwise, write “undefined”: (1)( 4) 
Compute, if possible. Otherwise, write “undefined”:  2  (1)  (1) 
Evaluate, if possible. Otherwise, write “undefined”:
 10
2
1
Compute, if possible. Otherwise, write “undefined”: 10  2
Evaluate, if possible. Otherwise, write “undefined”:
 48
8
Compute, if possible. Otherwise, write “undefined”:  3  (3)
Evaluate, if possible. Otherwise, write “undefined”: 2(3)(7)
Compute, if possible. Otherwise, write “undefined”: 4  (100)  100
Evaluate, if possible. Otherwise, write “undefined”:  2  5  9
Compute, if possible. Otherwise, write “undefined”:  6(1)( 2)
Evaluate, if possible. Otherwise, write “undefined”: 4(10)( 100)
Compute, if possible. Otherwise, write “undefined”:  5(1)(8)
Evaluate, if possible. Otherwise, write “undefined”:  6(8)
Compute, if possible. Otherwise, write “undefined”:  5  4
Evaluate, if possible. Otherwise, write “undefined”: 2  (9)
Compute, if possible. Otherwise, write “undefined”:  10  (3)
Compute, if possible. Otherwise, write “undefined”:
9
0
Evaluate, if possible. Otherwise, write “undefined”: 0  10
Compute, if possible. Otherwise, write “undefined”:  10  0
Compute, if possible. Otherwise, write “undefined”:
0
5
2A
Order of operations, integers only, two operations (only integers as an answer,
no exponents).
2
Compute, if possible. Otherwise, write “undefined”: 7  2  3 
Compute, if possible. Otherwise, write “undefined”: 4  (7  8) 
Evaluate, if possible. Otherwise, write “undefined”:
5  10

5
Evaluate, if possible. Otherwise, write “undefined”:  1  1(1) 
Compute, if possible. Otherwise, write “undefined”:  4  4(4)
Evaluate, if possible. Otherwise, write “undefined”: (6  1)( 2  9)
Compute, if possible. Otherwise, write “undefined”: 4  (12  12)
Evaluate, if possible. Otherwise, write “undefined”: 2  (1  3)
Compute, if possible. Otherwise, write “undefined”: 2  (4  5)  (3)
Evaluate, if possible. Otherwise, write “undefined”: [3  (7)]( 2)
Compute, if possible. Otherwise, write “undefined”: 4  (1  3  2)
Evaluate, if possible. Otherwise, write “undefined”:  4  (7)  (2)
Compute, if possible. Otherwise, write “undefined”:  8(2)  8
Evaluate, if possible. Otherwise, write “undefined”:
37
44
Compute, if possible. Otherwise, write “undefined”:  2  3  (2)
Evaluate, if possible. Otherwise, write “undefined”:  12  3  2
Compute, if possible. Otherwise, write “undefined”: 8  (9  2)
Evaluate, if possible. Otherwise, write “undefined”: 4  (2  5)
Compute, if possible. Otherwise, write “undefined”:  12  (3  2)
Evaluate, if possible. Otherwise, write “undefined”:  5  3  4
Evaluate, if possible. Otherwise, write “undefined”: 4  ( 2  2)
3
Compute, if possible. Otherwise, write “undefined”:
 12
 6  (6)
Evaluate, if possible. Otherwise, write “undefined”: (5  5)  10
3A
Addition and subtraction of fractions or of fractions and integers (positive
and negative).
Compute: 
2 4
 
3 5
3  5
Compute:     
5  6
Evaluate: 2 
3
4
Evaluate: 1
4
3
Evaluate:
Evaluate:
Evaluate:
Evaluate:
Evaluate:
Evaluate:
Evaluate:
Compute:
Compute:
Compute:
2  3
  
3  7
5 3
 
4 5
4 5
 
9 12
3 9

14 2
3
  1 
8
 5
    3
 6
7

2
23
 4  2
     
 7  9
3 10
 
4 9
7 4


18  9 
4
Compute:
Compute:
Compute:
Compute:
Compute:
Compute:
 5
1  
 7
3 5
 
14 7
2 3
 
3 11
2 3
 
9 7
2
6
3
4 5
 
9 12
4A
Multiplication of fractions and of fractions and integers (positive and
negative).
Compute:
3
 (10) 
5
Evaluate:

Compute:
Evaluate:
Compute:
345 1 10
 

5
2 345
2
12

 15 
8
 (18)
9
11 4 1
x x
7 3 11
Evaluate:

5 4  4
  
2 6  35 
Compute:

5  3   4
x      
12  10   8 
Evaluate:
 24   1  4 
    
 7  6  32 
5
Compute:
7
 321 
x 2  

321
 6 
Evaluate:
 2
12  
 15 
Compute:
100 
Evaluate:
 24  19 
    
 3  8 
Compute:
7  4
x  
2  21 
Evaluate:
 10  22 
    
 11  5 
Compute:
9 4
x
7 3
Evaluate:
 5  6 
   
 4  10 
Compute:
5 8
 
2 15
Evaluate:
 8 10
x
7 9
Compute:
7  2
x  
3  3
Evaluate:
  5  9 

 
 4  2 
23
27
5A
Division (using both  and fraction bar) of fractions and of fractions and
integers (positive and negative).
6
Evaluate, if possible. Otherwise, write “undefined”:
Compute, if possible. Otherwise, write “undefined”:
3
4
1
2

3 2

5 7
Evaluate, if possible. Otherwise, write “undefined”:
3
4
2
Compute, if possible. Otherwise, write “undefined”:
4
2
7
9 8

7 14
2 5

Compute, if possible. Otherwise, write “undefined”:
5 3
Evaluate, if possible. Otherwise, write “undefined”:

16
 ( 8)
9
2
Compute, if possible. Otherwise, write “undefined”:  20  

 5 
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
2
15
6
4
11
10

Compute, if possible. Otherwise, write “undefined”:
8
9
5
6
9
12

13

Evaluate, if possible. Otherwise, write “undefined”:
Compute, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Compute, if possible. Otherwise, write “undefined”:
3
10
1
5
2
1
5
7
14
3
7
9
2

21
1

14

Evaluate, if possible. Otherwise, write “undefined”:
Compute, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Compute, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Compute, if possible. Otherwise, write “undefined”:
2
1 

 5 
5  2
  
3  3
 15   10 
    
 4 6
 9
     3
 13 
6A
“Zero” category(number/0; 0/number; addition of opposite numbers;
multiplication by 0). “Ten” category: Multiplication and division of decimals
and integers by powers of ten.
Compute, if possible. Otherwise, write “undefined”: 234 1000 
Evaluate, if possible. Otherwise, write “undefined”: 0.13 10 
Compute, if possible. Otherwise, write “undefined”: 0.2 100 
Evaluate, if possible. Otherwise, write “undefined”: 0 237
Compute, if possible. Otherwise, write “undefined”: 0  23
Compute, if possible. Otherwise, write “undefined”:  2667  2667
 17
Evaluate, if possible. Otherwise, write “undefined”:
0
Evaluate, if possible. Otherwise, write “undefined”: 0.2  10
Evaluate, if possible. Otherwise, write “undefined”: 3.21  100
Evaluate, if possible. Otherwise, write “undefined”: 6 100
Evaluate, if possible. Otherwise, write “undefined”: 112  10
8
4.5
100
Evaluate, if possible. Otherwise, write “undefined”: 0.234 100
Compute, if possible. Otherwise, write “undefined”: 7.2 1000
0
Compute, if possible. Otherwise, write “undefined”:
100
Compute, if possible. Otherwise, write “undefined”: 0  0
Compute, if possible. Otherwise, write “undefined”: 101000
Compute, if possible. Otherwise, write “undefined”: 10 67.5
Compute, if possible. Otherwise, write “undefined”:  98  (98)
Compute, if possible. Otherwise, write “undefined”: 100 0.3
Evaluate, if possible. Otherwise, write “undefined”:
7A
Addition, subtraction, multiplication and division of decimals (and integers).
Evaluate, if possible. Otherwise, write “undefined”: 0.1  2.1 
Compute, if possible. Otherwise, write “undefined”: 0.1  0.999 
Compute, if possible. Otherwise, write “undefined”:  3  0.02 
Compute, if possible. Otherwise, write “undefined”: 0.1 0.01 
Evaluate, if possible. Otherwise, write “undefined”: 0.33  (0.011) 
2

 0 .2
1 .2
Compute, if possible. Otherwise, write “undefined”:
 0.018
 49
Compute, if possible. Otherwise, write “undefined”:
0.21
Compute, if possible. Otherwise, write “undefined”: −0.5  (−0.0001)
Evaluate, if possible. Otherwise, write “undefined”: 3.2 − (− 1.4)
Evaluate, if possible. Otherwise, write “undefined”: − 0.8 − (−0.04)
Compute, if possible. Otherwise, write “undefined”: − 0.9 − 0.2 − 0.4
Compute, if possible. Otherwise, write “undefined”: 7.22 + (− 0.002)
Compute, if possible. Otherwise, write “undefined”: − (− 0.8) + (− 0.2)
Compute, if possible. Otherwise, write “undefined”: 0.71  (5.1)  3.4
Compute, if possible. Otherwise, write “undefined”: −3.5  (−2)
Compute, if possible. Otherwise, write “undefined”: − 79.3  0.001
Compute, if possible. Otherwise, write “undefined”: 4  (−0.005)
Compute, if possible. Otherwise, write “undefined”: −0.2  (−0.3)  ( −0.4)
Evaluate, if possible. Otherwise, write “undefined”:
9
Compute, if possible. Otherwise, write “undefined”: 2  (−0.2)  0.03
8A
Exponential notation (fractions, integers, decimals)
Evaluate, if possible. Otherwise, write “undefined”:  16 
Compute, if possible. Otherwise, write “undefined”: (0.1) 3 
2
 2
Evaluate, if possible. Otherwise, write “undefined”:     
 3
Evaluate, if possible. Otherwise, write “undefined”: (0.9) 2
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
(0.4) 3
 (1.1) 2
 (0.01) 4
(0.06) 2
(0.1) 7
 24
(1)14
 (7) 2
(1) 25
Evaluate, if possible. Otherwise, write “undefined”:  (3) 3
 9
Evaluate, if possible. Otherwise, write “undefined”:   
 8
2
1
Evaluate, if possible. Otherwise, write “undefined”:   
2
43
Evaluate, if possible. Otherwise, write “undefined”:
9
4
2
Evaluate, if possible. Otherwise, write “undefined”:  

 3 
 52
Evaluate, if possible. Otherwise, write “undefined”:
6
3
 1
Evaluate, if possible. If not possible, write “undefined”:   
 2
4
9A
10
Addition, subtraction, multiplication and division including mixed number
representations. Order of operation (up to 3 operations).
1
5
Compute, if possible. Otherwise, write “undefined”: 2  1 
3
6
3
2
Compute, if possible. Otherwise, write “undefined”: 2  5 
5
7
1 6
Compute, if possible. Otherwise, write “undefined”:  2  1 
3 7
2
1
Evaluate, if possible. Otherwise, write “undefined”: 1  (1 ) 
3
9
Evaluate, if possible. Otherwise, write “undefined”:  2 
1
 22 
2
1 2
Evaluate, if possible. Otherwise, write “undefined”: (2  3)(   ) 
3 3
3  1 3
Evaluate, if possible. Otherwise, write “undefined”:    
5  2 5
3
4 17
 
Evaluate, if possible. Otherwise, write “undefined”:
20 17 5
2
2
Evaluate, if possible. Otherwise, write “undefined”:
8 
5
3
5 3
Evaluate, if possible. Otherwise, write “undefined”:    2
6 4
2
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
Evaluate, if possible. Otherwise, write “undefined”:
 2 14 
  
 7 5
 0. 2
 20.1
 0.01
(0.7  0.3)( 0.02)
(0.2)( 5)  (0.2)(0.3)
0.9 1.8 10
 4.2  (3.9  4.6)
3
 2 1
7
1
 5  (2) 
3
3
22
9
11
Evaluate, if possible. Otherwise, write “undefined”:
1
3
4
3
5
ALGEBRA PART: 36 QUESTIONS
EASY CATEGORY: select randomly 8 categories (1 question from each category)
1E
Writing phrases (involving only one operation) as algebraic expressions.
Write a direct translation of the following phrases as an algebraic expression. DO NOT
SIMPLIFY: The sum of a and  b
Write a direct translation of the following phrases as an algebraic expression. DO NOT
SIMPLIFY:  x raised to m -th power
Write a direct translation of the following phrases as an algebraic expression. DO NOT
SIMPLIFY: The product of a and  b
Write a direct translation of the following phrases as an algebraic expression: The opposite of C
Let m represent mass, and c speed of light. Use m and c to write the following phrase as an
algebraic expression: the product of mass and the square of speed of light.
Use any letter to represent a number and write the following statement as an algebraic
expression:
Double a number
Use any letter to represent a number and write the following statement as an algebraic
expression:
Two thirds of a number
Use any letter to represent a number and write the following statement as an algebraic
expression:
A number increased by 3
Write a direct translation of the following phrases as an algebraic expression. DO NOT
SIMPLIFY:  n subtracted from m .
Write a direct translation of the following phrases as an algebraic expression. DO NOT
SIMPLIFY: The product of  a and  b .
Write a direct translation of the following phrases as an algebraic expression:  n divided by
 m . Do not simplify.
12
Write a direct translation of the following phrases as an algebraic expression: The sum of x ,
p and y.
Write a direct translation of the following phrases as an algebraic expression. DO NOT
b
SIMPLIFY:
raised to the tenth power.
a
Write a direct translation of the following phrases as an algebraic expression. DO NOT
SIMPLIFY: 2ab squared.
Write a direct translation of the following phrases as an algebraic expression. DO NOT
2a
SIMPLIFY: The opposite of 
.
7
Write a direct translation of the following phrases as an algebraic expression. DO NOT
 2a
SIMPLIFY: The opposite of
. Do not simplify.
7
Use the letter x to represent a number and write the following phrase as algebraic expression:
One-eighth of a number
Use the letter x to represent a number and write the following phrase as an algebraic expression:
A number multiplied by  3
Use the letter x to represent a number and write the following phrase as an algebraic expression:
A number subtracted from  3
Use the letter x to represent a number and write the following phrase as an algebraic expression:
The product of AB and a number.
Let a be a variable representing the length of the side of a square. Use a to write the following
statement as an algebraic expression: Four times the side of a square.
Let a be a variable representing the length of the side of a square. Use a to write the following
statement as an algebraic expression: The side of a square raised to the second power.
Let a be a variable representing the length of the side of a square. Use a to write the following
statement as an algebraic expression: The side of a square doubled.
John has x dollars in his bank account. How much money will John have in the bank, after he
withdraws $100? Give your answer in the form of an algebraic expression.
John’s salary is x dollars. He got a new job and his salary doubled. How much does John earn in
his new job? Give your answer in the form of an algebraic expression.
13
If a family has x children, and each of those children has 3 children of his own, how many
grandchildren are there? Give your answer in the form of an algebraic expression.
Let t be a variable representing temperature on a given day. Use the variable to write the
following statement as an algebraic expression: The temperature on the given day increased by 5
degrees.
2E
Ability to distinguish between an algebraic expression and equation.
Determine whether the following examples represent an equation or an algebraic expression.
Circle ALL equations.
a) 4x  2
b) 4x  2  7
c) A  0
Determine whether the following examples represent an equation or an algebraic expression.
Circle ALL equations.
2
2
a) 4x  x  2  9x b) 4x  2  7
c) x  3 y  4  x
Determine whether the following examples represent an equation or an algebraic expression.
Underline the example that represents an equation.
a) 4 x  2  5 y
b) x 3  2 x
c) 4 x  2  5 y  9
Determine whether the following examples represent an equation or an algebraic expression.
Circle ALL algebraic expressions.
a) 3x  7 y  5
b) x  0
c) 9 x  7 y  3c  8
Determine whether the following examples represent an equation or an algebraic expression.
Circle ALL algebraic expressions.
a) x 2  2 y 2  0
b)  4 x  5  y  z
c) x
3E
Solving linear equations (all variables on one side, no parentheses involved,
no need for collecting terms, no fractions or decimals unless they appear as a
solution).
Solve the following equation:
98  2 x
Solve the following equation:
 7  x  2
Solve the following equation:
3x  7  4
14
Solve the following equation:
x0
Solve the following equation:  2a  5
Solve the following equation: x  5  1
Solve the following equation:
27 y
Solve the following equation: 4 x  8
Solve the following equation:
3  b  2
Solve the following equation:  9  x  8
Solve the following equation:
 2  x
Solve the following equation:
5  2x  8
Solve the following equation:
 7 x  21  0
Solve the following equation:
0  1 x
Solve the following equation:  3  6w  2
Solve the following equation:
2x  2  4
Solve the following equation:  2  x  5
Solve the following equation:  2x  3  1
Solve the following equation:
7  3 x
15
4E
Solving a linear inequality (all variables on one side, no parentheses involved,
no need for collecting terms, no fractions or decimals unless they appear as a
solution).
Solve the following inequality:
10  2x
Solve the following inequality:
9  x 2
Solve the following inequality:
x0
Solve the following inequality: x  3  4
Solve the following inequality:
 7x  0
Solve the following inequality:
9  x
Solve the following inequality:
 4  3x
Solve the following inequality:
9  2  x
Solve the following inequality:
x  2  1
Solve the following inequality: 4 x  2
Solve the following inequality:
 2x  5
Solve the following inequality:
3 5 x
Solve the following inequality:
 2  7a  5
Solve the following inequality:
2  3x  1
16
Solve the following inequality:
6  4 x
,,,
Solve the following inequality:
7 x  7
Solve the following inequality:
 9  2  x
Solve the following inequality:
0  3a  1
Solve the following inequality:
8  6x  3
Solve the following inequality:
6  x  1
5E
Graphing sets on a number line.
Graph the following number set on a number line. Assume that the distance between all marks is
the same.
x  1
Graph the following number set on a number line. Assume that the distance between all marks is
the same.
x3
Using inequality symbols, describe the set that is graphed below:
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x3
17
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x
1
2
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x
5
2
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x
1
2
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x  2
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x
1
3
18
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x
2
3
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x  2
Graph the following inequality on the number line. Assume that the distance between all marks
is the same.
x  3
Using inequality symbols, describe the set that is graphed below:
Using inequality symbols, describe the set that is graphed below:
Using inequality symbols, describe the set that is graphed below:
Using inequality symbols, describe the set that is graphed below:
19
6E
Checking if a given integer is a solution of a linear equation (inequality) in one
variable.
Is the number  5 a solution of the following inequality. x  2  4 ?
Is the number  5 a solution of the following inequality.  x  8  13 ?
Is the number 2 a solution of the following inequality.  x  3  1 ?
For each of the following equations determine if the number 0 is its solution. If 0 is a solution,
circle the equation.
1
a) 3x  0
b)  0
x
For each of the following equations determine if the number 0 is its solution. If 0 is a solution,
circle the equation. a) 3x  22  22
b)  x  0
Does x  7 make the statement 2( x  1)  x  7 true or false?
Does x  2 make the statement  x 2  4 true or false?
Does x  2 make the statement  x 2  4 true or false?
Is x  1 a solution of
Is x  1 a solution of
( x)(  x)  2
xx 2
Is x  2 a solution of
 6 x  36
Is x  2 a solution of
(6) x  36
Is the number 2 a solution of the following inequality.  x  2 .
x
2
Is the number 9 is a solution of the following inequality.
3
Is the number  1
a solution of the following inequality. 4  x  3  8
20
Is the number  3 a solution of the following equation. 4x  7  19 .
Is the number 100 a solution of the following equation.  432x  4320 .
Is the number  1 a solution of the following equation
x 1
0
5
x
0
5
2
0
Is the number 0 a solution of the following equation.
x
Is the number 0 a solution of the following equation.
7E
Understanding “<” , “  ” and “>”, “  ” notation. Using it to describe sets of
numbers (phrases like “at least”, “at most”, “not less”, “not more”).
Circle all of the numbers that satisfy the condition x  2 .
 4,  2.1,  2,  0.9, 0, 1
Name three numbers that satisfy the condition: x  1 .
Write any inequality that is satisfied by  2 and 3.
Write any inequality that is not satisfied by  2 and 3.
Write any inequality that is satisfied by  2, but not by 3.
Describe the following set of numbers using inequality signs: All numbers x that are positive
Describe the following set of numbers using inequality signs: All numbers x that are nonnegative
Describe the following set of numbers using inequality signs: All numbers x that are at most
equal to 9.
Describe the following set of numbers using inequality signs: All numbers x that are at least
equal to 9.
Describe the following set of numbers using inequality signs: All numbers x that are not less
than 9.
21
Describe the following set of numbers using inequality signs: All numbers x that are not more
than  1 .
Describe the following sets of numbers using inequality signs: All numbers x that are negative.
Describe the following sets of numbers using inequality signs: All numbers x that are nonpositive.
Describe the following sets of numbers using inequality signs: All numbers x that are at least
equal to 6.
Describe the following sets of numbers using inequality signs: All numbers x that are at most
equal to 6.
Describe the following sets of numbers using inequality signs: All numbers x that are not more
than 6
Name three numbers that satisfy the condition: x  4
Name three numbers that satisfy the condition: x  3
Name three numbers that satisfy the condition: x 
1
2
Name three numbers that satisfy the condition: x  0.5
Find a number that satisfies x  8 but does not satisfy x  100
Circle all numbers that satisfy the following inequality x  3
2,
 1,
0,
1,
2,
3,
 3,
4,
5
Circle all numbers that do not satisfy the inequality: x  3
2,
 1,
0,
1,
2,
 3,
3,
4,
5
Circle all numbers that satisfy the inequality: x  2 :
2,
 1,
0,
1,
2,
 3,
3,
4,
5
8E
22
Rewriting the expression by replacing the variable with its value (only
integers) and evaluating it, if possible. One operation needed for the
evaluation.
Let x  100 . Rewrite the expression replacing the variable with its value and evaluate it, if
possible. Otherwise, write “not defined” : 3x
Let x  1 . Rewrite the expression replacing the variable with its value and evaluate it, if
possible. Otherwise, write “not defined” : x  2
Let x  0 . Rewrite the expression replacing the variable with its value and evaluate it, if
4
possible. Otherwise, write “not defined” :
x
Let x  100 . Rewrite the expression replacing the variable with its value and evaluate it, if
x
possible. Otherwise, write “not defined” :
700
Let x  100 . Rewrite the expression replacing the variable with its value and evaluate it, if
0
possible. Otherwise, write “not defined” :
x
Let x  2 . Rewrite the expression replacing the variable with its value and evaluate, if possible.
Otherwise, write “not defined” : 3 x
Let x  2 . Rewrite the expression replacing the variable with its value and evaluate, if possible.
Otherwise, write “not defined” : x 3
Let x  2 . Rewrite the expression replacing the variable with its value and evaluate, if possible.
Otherwise, write “not defined” : x x
Rewrite the expression  A replacing the variable with its value and then evaluate if A  2 .
Substitute x  6 in the following expression and then evaluate, if possible. Otherwise, write “not
defined” : x  8
Substitute x  6 in the following expression and then evaluate, if possible. Otherwise, write “not
defined” : 10  x
Substitute x  6 in the following expression and then evaluate, if possible. Otherwise, write “not
defined” :  4  x
Substitute x  6 in the following expression and then evaluate, if possible. Otherwise, write “not
defined” : x  6
23
Substitute x  6 in the following expression and then evaluate, if possible. Otherwise, write “not
defined” :  2  x  6
Substitute x  2 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” : 2  x
Substitute x  2 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” : 2  x
Substitute x  2 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  2  x
Substitute x  2 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  5  x  4
Substitute x  2 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” : 6  x  10
Substitute x  10 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” : 3x
Substitute x  10 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  5x
Substitute x  10 in the following expression and then evaluate, if possible. Otherwise, write
 200
“not defined” :
x
Substitute x  10 in the following expression and then evaluate, if possible. Otherwise, write
x
“not defined” : 
2
Substitute x  10 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  50  x
Substitute x  10 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” : x 4
Substitute x  12 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  1000x
Substitute x  12 in the following expression and then evaluate, if possible. Otherwise, write
x
“not defined” :
6
Substitute x  12 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  5x
24
Substitute x  12 in the following expression and then evaluate, if possible. Otherwise, write
6
“not defined” :
x  12
Substitute x  12 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” :  24  x
Substitute x  12 in the following expression and then evaluate, if possible. Otherwise, write
“not defined” : x 2
9E
Solving literal equations (variable one is seeking will not appear in the
denominator). One operation needed. No simplification needed at the end.
Solve for x :
Solve for x :
Solve for x :
Solve for x :
x
b
a
xa b
ax  c
xd 0
Solve for the indicated variable:
Solve for the indicated variable:
Solve for m :
Solve for x :
b
 7c for b
a
a  b  4 for a
m
x
4
x Y  Z
Solve for x :
xc
Solve for x :
x
z
v
Solve for x :
yx z
Solve for x :
4  xm
Solve for x :
3  xy
25
Solve for x :
y  x
Solve for the indicated variable: b  7 y  z for b
Solve for the indicated variable:
Solve for m :
Solve for x :
sa  m for a
xy  3m
x
 c
a
10E
If possible, rewriting algebraic expressions with the use of exponential laws
(non-negative exponents; only one law used) .
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: a 3 a 15
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: a 3  a 15
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression:
b 
4 2
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: a 3 b 4
If possible, eliminate parentheses by applying an exponential law. If it is not possible to simplify,
just rewrite the expression: (ab) 7
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: a 3b15
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
a7
rewrite the expression:
a3
Simplify, using one of the exponential laws, if possible, If not possible, write “not possible” :
m 
10 2
26
Simplify, using one of the exponential laws, if possible, If not possible, write “not possible” :
m3m 4 m
If possible, eliminate parentheses by applying an exponential law. If not possible, write “not
m
possible”:  
n
7
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: xx15
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
s 14
rewrite the expression:
s3
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression:
t 
15 3
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: m 3  m 4
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: x 7 x 15
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
rewrite the expression: aa 3 a
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
q 28
rewrite the expression:
q 14
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
q 14
rewrite the expression:
q 13
Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just
q5
rewrite the expression:
q
11E
27
Removing parentheses in multiplication of a monomial by a polynomial (up to
3 terms). Performing numerical operations might be required but no use of
exponential laws needed. No fractions, no decimals.
Eliminate parentheses in the following expression:
 4( x  3 y  z )
Eliminate parentheses in the following expression:
 ( x  2 y)
Eliminate parentheses in the following expression:
a (b  c  d ) .
Eliminate parentheses in the following expression:
3x  10  2
Eliminate parentheses in the following expression:
2L  W 
Eliminate parentheses in the following expression:
R1  x
Eliminate parentheses in the following expression:
R
Eliminate parentheses in the following expression:
3P1  rt 
Eliminate parentheses in the following expression:
( x 2  7 z )(2)
Eliminate parentheses in the following expression:
 (a  a 2  2)
Eliminate parentheses in the following expression:
 (3  xy) .
Eliminate parentheses in the following expression:
 d ( 2a  c )
Eliminate parentheses in the following expression:
( x  y )( 2 w)
Eliminate parentheses in the following expression:
( x 3  x 2  1)4 y
Eliminate parentheses in the following expression:
(a  b  g )( d ) .
Eliminate parentheses in the following expression:
 (10  a 3 )
2

 r2 A.
12E
Collecting like terms (2 terms only), if possible. No fractions, no decimals.
28
If possible, collect like terms. If not possible, just rewrite the expression  3x  4x
If possible, collect like terms. If not possible, just rewrite the expression a 2  6a 2
If possible, collect like terms. If not possible, just rewrite the expression  mn  m
If possible, collect like terms. If not possible, just rewrite the expression xy  yx
If possible, collect like terms. If not possible, just rewrite the expression 4x  2x
If possible, collect like terms. If not possible, just rewrite the expression a 2  a
If possible, collect like terms. If not possible, just rewrite the expression y  y
If possible, collect like terms. If not possible, just rewrite the expression xy  yx
If possible, collect like terms. If not possible, just rewrite the expression a  ab
If possible, collect like terms. If not possible, just rewrite the expression st  ts
If possible, collect like terms. If not possible, just rewrite the expression  x  9x
If possible, collect like terms. If not possible, just rewrite the expression 4 y 2  9 y 2
If possible, collect like terms. If not possible, just rewrite the expression  m 3  m 3
If possible, collect like terms. If not possible, just rewrite the expression xyz  6 zyx
If possible, collect like terms. If not possible, just rewrite the expression 3cd  4dcs
If possible, collect like terms. If not possible, just rewrite the expression 2ab  ba
If possible, collect like terms. If not possible, just rewrite the expression  3x 3  4 x 3 y
If possible, collect like terms. If not possible, just rewrite the expression  3a 22  6a 22
2
If possible, collect like terms. If not possible, just rewrite the expression st  t
If possible, collect like terms. If not possible, just rewrite the expression 2mnk  5kmn
13E
29
Factoring an indicated expression (a variable or an integer). No use of
exponential laws needed. No instances when “an entire term is factored” and
thus “1” becomes a term after factorization.
Factor x from the following expression: 3 x  2 xy
Factor 4 from the following expression: 4 y 2  8x
Factor 5 from the expression: 5 x  5 y
Factor 7 from the expression: 21 49a
Factor c from the expression: 3c  2ca
Factor y from the expression:  8 xy  7 y
Factor 2 from the 2lw  2lh  2wh
Factor x from the expression: 4 xz  yx  5 x
Factor x from the following expression:
yx  zx  vx
Factor 10 from the following expression: 100 yxz  10v
Factor 9 from the expression: 18 x  27 y
Factor 2 from the expression: 20  4a  2b
Factor c from the expression: 3cd  2c
Factor y from the expression:  y  14 xy
Factor 2 from the expression 2 x 4 yz  12bc
Factor x from the expression: 7 xz  4 xy  5 x
Factor x from the following expression: xy 2  8 xy
Factor 4 from the following expression: 12 xy 2  4 z
Factor m from the expression: 5mn  3m
Factor s from the expression: st  49s
30
BASIC CATEGORY: 1 question from each category of 23 categories
1B
Simplifying an expression by performing numerical operations when possible.
(may need to change the order of terms or factors). Performing an operation
on fractions or on decimals might be needed.
Simplify if possible.
 2  x 1
If no numerical operation can be performed, write
“not possible”:
Simplify if possible. If no numerical operation can be performed, write “not possible”:
2 1
 Y
3 6
Simplify if possible. If no numerical operation can be performed, write “not possible”:
Simplify if possible.
12  3x
If no numerical operation can be
performed write
4x
8
“not possible”:
Simplify if possible. If no numerical operation can be performed, write “not possible”:
(2  3) m
Simplify if possible. If no numerical operation can be performed, write “not possible”:
2  3m
Simplify if possible. If no numerical operation can be performed, write “not possible”:
2 x ( 3) y
Simplify if possible. If no numerical operation can be performed, write “not possible”:
 ( x)
Simplify if possible.
 0.2  x  0.03
If no numerical operation can be performed, write
“not possible”:
Simplify if possible. If no numerical operation can be performed, write “not possible”:
Simplify if possible.
4a(8)
If no numerical operation can be performed, write
x
1
“not possible”:
Simplify if possible. If no numerical operation can be performed, write “not possible”:
8a
4
31
Simplify if possible. If no numerical operation can be performed, write “not possible”:
1
1
 x
2
2
Simplify if possible. If no numerical operation can be performed, write “not possible”:
3bd
9ac
Simplify if possible. If no numerical operation can be performed, write “not possible”:
2 x ( 3) y
Simplify if possible.
2
4  x
5
If no numerical operation can be performed, write
“not possible”:
Simplify if possible. If no numerical operation can be performed, write “not possible”:
12 y
2 xz
Simplify if possible. If no numerical operation can be performed, write “not possible”:
1
2x 2  
2
Simplify if possible. If no numerical operation can be performed, write “not possible”:
x
10 
5
Simplify if possible. If no numerical operation can be performed, write “not possible”:
 0.1x(10 y 2 )
Simplify if possible. If no numerical operation can be performed, write “not possible”:
0.5(0.3m)
Simplify if possible. If no numerical operation can be performed, write “not possible”:
0.5  0.6  m
Simplify if possible. If no numerical operation can be performed, write “not possible”:
0 .4 y
 0.02
2B
32
Simplifying algebraic expressions by applying exponential laws (only nonnegative exponents; one variable, up to 2 types of operations. Performing an
operation on fractions or on decimals might be needed.
Simplify
Simplify:
2( x 2 ) 3 x
a3
aa 2
Perform the indicated operation and simplify:
Perform the indicated operation and simplify:
(3x 5 )( 4 x 2 )
3x 4
6x 2
Perform the indicated operation and simplify:
3x(4 x 2 )
Perform the indicated operation and simplify:
(2 x 6 ) 3
Perform the indicated operation and simplify:
Perform the indicated operation and simplify:
8x 5
6x 2
 2x3 x5
x4
Perform the indicated operation and simplify:
a2 2
a
a4
Perform the indicated operation and simplify:
1 7
  v
v
Perform the indicated operation and simplify:
(3x 2 ) 3
x4
Perform the indicated operation and simplify:
3x 5
(3 x ) 2
2
Simplify
(3aa 2 ) 2
Simplify:
a3
aa 2
Perform the indicated operation and simplify:
(4 x) 3
x
33
(a) 3
2a 2
Perform the indicated operation and simplify:
Perform the indicated operation and simplify:
 (x 2 )5 x 7
Perform the indicated operation and simplify:
(2 x 6 ) 3
Perform the indicated operation and simplify:
(3x 2 )(4 x 5 x)
Perform the indicated operation and simplify:
(3x)(2)( x 3 )
3B
The Distributive Law: removing parentheses in multiplication of a monomial
by a polynomial with up to 3 terms. No collecting terms needed at the end but
numerical operations or application of exponential laws needed in the process.
Performing an operation on fractions or on decimals might be needed.
Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if
possible.
x(2  x 2 )
Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if
possible. a(a  a 3  a 4 ) .
Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if
1

possible.  3 x  2 x
2

Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if
possible.  2( x  3 y  4 z )
Remove parentheses and simplify: ( x 2  2 x) x 4
2
Remove parentheses and simplify:  (3c  33d )
3
Remove parentheses and simplify: a(a 2  ab  ab 2 )
Remove parentheses and simplify: (2 x 3 y 
3
 xy 4 ) x
7
34
2
Remove parentheses and simplify: (a  b)( a 3 )
5
Remove parentheses and simplify:
1
x( 4 x 5  8)
4
Remove parentheses and simplify: 3 y(6 y 4  8 y 3 )
Remove parentheses and simplify: x 2 ( x 5  2 x 3 )
Remove parentheses and simplify: 2 y 2 (7 y  y 2 )
Remove parentheses and simplify:  a 4 (9a 2  a 2 b)
Remove parentheses and simplify: ab(a 2 b  ab 5 )
Remove parentheses and simplify:  0.2c(0.3c  100c 6 )
Remove parentheses and simplify:  x10 (4 x 5  xy)
Remove parentheses and simplify: mn 2 (m  n)
Remove parentheses and simplify:

x
2x5  4x
2

Remove parentheses and simplify: 0.7(0.3  n)
Remove parentheses and simplify: (2.3x  0.06)(100)
Remove parentheses and simplify:  5(0.06 x  2 y )
Remove parentheses and simplify: 0.6(0.2n  4)
4B
The Distributive Law: removing parentheses in multiplication of a binomial
by a binomial. No collecting terms needed at the end but numerical
operations or application of exponential laws needed in the process (no
operations on fractions or decimals).
35
Eliminate parentheses in the following expression. Simplify: (a  b)( 2a  3)
Eliminate parentheses in the following expression. Simplify: (2b  c)(c  c 5 )
Eliminate parentheses in the following expression. Simplify ( x  y)( zx 2  w)
Eliminate parentheses in the following expression. Simplify (a  3b)(10  a 3 )
Eliminate parentheses in the following expression. Simplify (a  g )(ca  dg 2 )
Eliminate parentheses in the following expression. Simplify: ( x 3  x 2 )( yx  1)
Eliminate parentheses in the following expression. Simplify: ( yx  1)( y  1)
Eliminate parentheses in the following expression. Simplify (2ab 5  1)(1  ba)
Eliminate parentheses in the following expression. Simplify (4 x  8)( 2 yx 3  3x 9 )
Eliminate parentheses in the following expression. Simplify (m 2  m)(3m 5  1)
Eliminate parentheses in the following expression. Simplify: (4a  b)(3ab 2  3a 2 b 3 )
Eliminate parentheses in the following expression. Simplify: (b 4  c)(c  cb 5 )
Eliminate parentheses in the following expression. Simplify (3x 2 y  xy)(2 x  y 3 )
Eliminate parentheses in the following expression. Simplify (2b  a 3 )(2a  b)
Eliminate parentheses in the following expression. Simplify ( x  y 2 )( y  3)
Eliminate parentheses in the following expression. Simplify: ( x 2  1)(a  2 x)
Eliminate parentheses in the following expression. Simplify: (3m  n)( x  2my)
Eliminate parentheses in the following expression. Simplify ( x  3 y)(3x 2  2 y)
Eliminate parentheses in the following expression. Simplify (3x  xy)(1  3x)
Eliminate parentheses in the following expression. Simplify (2  z )( z  v)
36
5B
Factorization of an indicated monomial (factorization of “an entire term” is
and thus getting “1” as a term after factorization possible). No operations on
fractions or decimals.
Factor x 2 from the following expression: 3x 5  2 x 3  x 2
Factor 4a from the following expression: 4a  8 xa 4
Factor x 2 from the following expression: 3x 5  yx 2
Factor xy 2 from the following expression: 4 x 2 y 2  xy 4  3x 3 y 3
Factor 2 xy from the following expression:
2 xy  2 xy2  4 x 2 y 2
Factor a 3b 3 from the following expression: a 3b 4  5b 3 a 7  a 3b 3
Factor 7 xy from the following expression: 7 x 5 y 3  14 x 3 y 2  21xy
Factor 7a 3 b 3 from the following expression:  14b 3 a 7  7a 3b 3
Factor 4ac 3 from the following expression:  16ac 3  8ac 7  12ac 8 d
Factor 7ab 2 from the following expression: 35ab 3  14a 2 b 4  7ab 2
Factor 2 y from the following expression: 2 x 2 y  4 y 4
Factor x 2 from the following expression: 2 x 2  3 x 4
Factor 2 x 2 y from the following expression: 6 x 2 y  14 y 7 x 3
Factor 3abc 2 from the following expression: 3abc 2  12a 6 bc 5  6a 2 b 2 c 2
Factor z 3 from the following expression: 3z 5  z 4  2 z 3
Factor x 2 from the following expression: 2 x 3  x 2
Factor 3ay from the following expression: 3a 3 y  3x 2 ay  6a 4 y 2
37
Factor 3ab from the following expression:
6ab 5  3ba
Factor a 2 b from the following expression: a 3b  a 2 b  4a 7 b 2
Factor 3z 2 y 3 from the following expression: 3z 4 y 6  9 y 4 z 2
6B
Simplification of rational expressions. Factorization of a numerator OR a
denominator (but not both) needed.
a  2ab
a
xy
Simplify, if possible. If not possible, write “not possible”: 2
x y  3xy 2
Simplify, if possible. If not possible, write “not possible”:
Simplify, if possible. If not possible, write “not possible”:
a  4d
 4d  a
Simplify, if possible. If not possible, write “not possible”:
2
2x  2 y
Simplify, if possible. If not possible, write “not possible”:
xy  xz
3x
Simplify, if possible. If not possible, write “not possible”:
3x  3x 2
6x
4 x  5x 2
Simplify, if possible. If not possible, write “not possible”:
x
m  3n
Simplify, if possible. If not possible, write “not possible”:
3n  m
Simplify, if possible. If not possible, write “not possible”:
a 3b  4b 4 a 4
a
Simplify, if possible. If not possible, write “not possible”:
3x
3x  9
Simplify, if possible. If not possible, write “not possible”:
3x
3x  9 x 2
38
Simplify, if possible. If not possible, write “not possible”:
x2
2x 2  x5
Simplify, if possible. If not possible, write “not possible”:
abc
ab
Simplify, if possible. If not possible, write “not possible”:
xy
xy  xy 2
Simplify, if possible. If not possible, write “not possible”: 
12 x  4
3x
Simplify, if possible. If not possible, write “not possible”:
u  2v  s
s  2v  u
Simplify, if possible. If not possible, write “not possible”:
4uv 2  2vu
2uv
Simplify, if possible. If not possible, write “not possible”:
xy  xy 2
3 yx
Simplify, if possible. If not possible, write “not possible”:
 4 x  12 y  8 z
4
Simplify, if possible. If not possible, write “not possible”:
8z
4x  8z
7B
Understanding what it means that a number(s) is a solution of a given
equation (“relatively” complicated evaluation needed). Performing an
operation on fractions or on decimals might be needed.
Ability to rewrite the expression and evaluating it, if possible (one instance of
a variable). If evaluation is not possible, recognizing that it is not possible to
evaluate. Performing an operation on fractions or on decimals might be
needed.
Circle all of the numbers below that are solutions of the equation x 3  6  x .
39
x2
x  2
Determine if x  1, y  2 is a solution of the following equation. Show your work.
3x  2 y  1.
Determine if x  1, y  2 is a solution of the following equation. Show your work.
x y 3
Determine if x  1, y  2 is a solution of the following equation. Show your work.
y
 2
x
Does a 
1
, b  2 make the following statement true or false? Show your work.
2
ab  1
Does a 
1
, b  2 make the following statement true or false? Show your work.
2
ab  1
1
Does a  , b  2 make the following statement true or false? Show your work.
2
a
1

b
4
1
Does a  , b  2 make the following statement true or false? Show your work.
2
3
ab  
2
Is x  1, y  0 a solution of 2 xy  2 ? Show your work.
Is x  1, y  0 a solution of 2 xy  0 ? Show your work.
Is x  1, y  0 a solution of y  x  1 ? Show your work.
Is x  1, y  0 a solution of
x
 0 ? Show your work.
y
1
, y  2 a solution of
2
x
 4 ? Show your work.
y
Is x 
40
Is x  1, y  0 a solution of
y
 4 ? Show your work.
x
Is x  1, y  0 a solution of y  x 
3
? Show your work.
2
Is x  1, y  0 a solution of xy  y  2 x ? Show your work.
Let x  4 . Rewrite the expression replacing the variable with its value and evaluate:
 2x
Let x  4 . Rewrite the expression replacing the variable with its value and evaluate:
(2) x
Let x  8 . Rewrite the expression replacing the variable with its value and evaluate:
( x) 2
Let x  100 . Rewrite the expression replacing the variable with its value and evaluate:
Substitute x 
 x2
2
5
x
in the following expression and then evaluate:
3
3
2
in the following expression and then evaluate:
3
2
Substitute x  in the following expression and then evaluate:
3
Substitute x 
x
1
5
x
2
7
Substitute x 
2
in the following expression and then evaluate:
3

5
x
12
Substitute x 
2
in the following expression and then evaluate:
3
2 x
Substitute x 
2
in the following expression and then evaluate:
3
 x 3
Substitute x  
3
in the following expression and then evaluate:
5
3
x
10
Substitute x  
3
in the following expression and then evaluate:
5

1
x
7
41
Substitute x  
3
in the following expression and then evaluate:
5
1
2 x
5
Substitute x  
3
in the following expression and then evaluate:
5
1
1  x
4
Substitute x  
3
in the following expression and then evaluate:
5
 x3
Substitute x 
2
in the following expression and then evaluate:
7
2x
Substitute x 
2
in the following expression and then evaluate:
7
 7x
Substitute x 
2
in the following expression and then evaluate:
7

Substitute x 
2
in the following expression and then evaluate:
7
5
x
28
Substitute x 
2
in the following expression and then evaluate:
7
5
x
Substitute x 
2
in the following expression and then evaluate:
7
x
2
Substitute x  
3
in the following expression and then evaluate:
4
1
2
14
x
3
x2
Substitute x  
3
in the following expression and then evaluate:
4
3
Substitute x   in the following expression and then evaluate:
4
4
x
3
x
2
1
3
3
in the following expression and then evaluate:
4
3
Substitute x   in the following expression and then evaluate:
4
1
1  x
8
x
3
Substitute x  
Substitute x  
3
in the following expression and then evaluate:
4
0
x
42
Substitute x  0.2 in the following expression and then evaluate:
x  3.21
Substitute x  0.2 in the following expression and then evaluate:
35.01  x
Substitute x  0.2 in the following expression and then evaluate:
x
4
Substitute x  0.2 in the following expression and then evaluate:
 40x
Substitute x  0.2 in the following expression and then evaluate:
0.3x
Substitute x  0.2 in the following expression and then evaluate:
x
0.04
Substitute x  0.6 in the following expression and then evaluate:
 x  4.5
Substitute x  0.6 in the following expression and then evaluate:
 2.7  x
Substitute x  0.6 in the following expression and then evaluate:
1 .2
x
Substitute x  0.6 in the following expression and then evaluate:
 600x
Substitute x  0.6 in the following expression and then evaluate:
0.001x
Substitute x  0.6 in the following expression and then evaluate:
x3
Substitute x  1.5 in the following expression and then evaluate:
x  0.08
Substitute x  1.5 in the following expression and then evaluate:
 3  x  0.4
Substitute x  1.5 in the following expression and then evaluate:
x  0.15
Substitute x  1.5 in the following expression and then evaluate:
 0.2x
Substitute x  1.5 in the following expression and then evaluate:
 30
x
8B
43
Recognizing equivalent expressions by applying the following
a) the order of terms of an expression can be changed
b) the order of factors of an expression can be changed
c)
b c bc
 
a a
a
d)
ab
b
1
 a   ab 
c
c
c
a
b
e)  
a
a

b
b
Circle all expressions that are equal to 2 x  5 y  4 z :
2x  4z  5 y
 5 y  4z  2x
 5 y  2x  4z
Rewrite the expression yx(a  b) in two equivalent forms by multiplying its factors in a
different order.
Rewrite the following expression in its equivalent form as a single fraction:
6
a
11
Rewrite the following expression in its equivalent form as a single fraction: 2
x
y
Rewrite the following expression in its equivalent form as a single fraction:
1 2
(a  1)
3
Rewrite the following expression in its equivalent form as a single fraction:
a
2b

x2 x2
Fill in the blanks to make a true statement:
3x
 x  _____
y
Fill in the blanks to make a true statement:
3x 1
  ____
y
y
ab 1
  _____
y
y
2b
Circle all expressions that are equivalent to
c
b2
2 b
1
b2
 ,
( 2  b) ,
,
c
c c
c
c
Fill in the blanks to make a true statement:
Circle all expressions that are equivalent to  m
m
m
m(1) ,
 ,
,
1
1

1
m
44
Circle all expressions that are equivalent to m  n :
m(  n ) ,
 n  m,
n  m,
 nm ,
Circle all expressions that are equivalent to
2
xy ,
9
2
xy
,
9
2 xy 
1
,
9
Circle all expressions that are equivalent to
1
xy  2 ,
9
xy
2,
9
2xy
9
2xy
9
2x
y
9
Circle all expressions that are equivalent to  a  c  b  d .
 a  b  c  d,
 a  b  c  d,
d  b  c  a,
Circle all expressions that are equivalent to 
a
,
b
1
a ,
b
2a
,
2b
a
.
b
1
b ,
a
Circle all expressions that are equivalent to 

acbd
a
,
b
a
b
a
.
b
Circle all expressions that are equivalent to  abc
cba
b(a)c
a  bc
1
Circle all expressions that are equivalent to  abc
bca
ab(c)

ab  c
1
Circle all expressions that are equivalent to
5x
.
6
45
5
5
x,
6
x
,
6
1
 5x
6
Circle all expressions that are equivalent to
5
x ,
6
5
,
6x
5x
.
6
10 x
,
12
Circle all expressions that are equivalent to
x
,
2x ,
2 x,
2
x2 .
Circle all expressions that are equivalent to
4x
x
x2,
,
8
2
x2 .
Circle all expressions that are equivalent to
11a ,
8a  3 ,
3  a 8 ,
3  8a .
Circle all expressions that are equivalent to
b  3a
,
6
1
(3a  b),
6
 b  3a
,
6
(3a  b)
mn
,
3
3a  b
6
1
6
Circle all expressions that are equivalent to
n m
 ,
3 3
3a  b
.
6
3a
 b,
6
Circle all expressions that are equivalent to
3a b
 ,
6 6
11  a ,
m n
 .
3 3
1
1
 m  n,
3
3
By placing the minus sign differently, write the following expression in two additional
2a

equivalent ways.
b
46
By placing the minus sign differently, write the following expression in two additional
2ac
equivalent ways.
 2d
m 7n
Rewrite the following expression as of a single fraction: 
4 4
7m n 2

t
t
5m
2n 2
Rewrite the following expression as of a single fraction:

4c  2 4c  2
Rewrite the following expression as of a single fraction:
Rewrite the following expression as of a single fraction:
m
3
t


s 1 s 1 s 1
Using the rule for multiplication of fractions, rewrite the following expression in its equivalent
m
form as a single fraction: 3 
n
Using the rule for multiplication of fractions, rewrite the following expression in its equivalent
b
form as a single fraction: a

 4 
Using the rule for multiplication of fractions, rewrite the following expression in its equivalent
b
form as a single fraction:  a

 4 
Using the rule for multiplication of fractions, rewrite the following expression in its equivalent
t
form as a single fraction: ( s  4)
n
Using the rule for multiplication of fractions, rewrite the following expression in its equivalent
a
form as a single fraction: 4 
n 1
Write the following expressions in its equivalent form as a single fraction:
2
4
x y
3
3
Write the following expressions in its equivalent form as a single fraction:
2
1
x y
3
3
1
y
Write the following expressions in its equivalent form as a single fraction: 2 x   7 
3
3
Write the following expressions in its equivalent form as a single fraction:
3
1
x y
t
t
47
Write the following expressions in its equivalent form as a single fraction:
Write the following expression in its equivalent form as a single fraction:
x
y
3
t
t
1
n

2
k t
k t
2
9B
Expanding exponential expression; Rewriting expressions using exponential
notation whenever possible. Understanding that the exponent pertains only to
the closest expression.
Evaluation of exponential expressions raised to the zero-th power.
Identifying bases, exponents and numerical coefficients of exponential
expressions.
Expand (i.e. write without using exponential notation). Do not simplify.
a(bc) 2
Expand (i.e. write without using exponential notation). Do not simplify.
5A 3
Rewrite using exponential notation whenever it is possible:
Rewrite using exponential notation whenever it is possible:
Rewrite using exponential notation whenever it is possible:
Rewrite using exponential notation whenever it is possible:
Evaluate
7mmm
 aaaa  aa
zzz
zzzz
( z )(  z )(  z )
zzz
(2 x) 0
Evaluate 2x 0
Evaluate 4  (8  m 7 ) 0
 3a 27
In the following expression, identify base, exponent and numerical coefficient:
Base___________ exponent__________ numerical coefficient_______________
In the following expression, identify base, exponent and numerical coefficient:
 yz 
 
 x 
9
48
Base___________ exponent__________ numerical coefficient_______________
In the following expression, identify base, exponent and numerical coefficient:  x 4
Base___________ exponent__________ numerical coefficient_______________
2x3
In the following expression, identify base, exponent and numerical coefficient:
3
Base___________ exponent__________ numerical coefficient_______________
In the following expression, identify base, exponent and numerical coefficient:  (a  bc) 2
Base___________ exponent__________ numerical coefficient_______________
x
In the following expression, identify base, exponent and numerical coefficient:  
 y
Base___________ exponent__________ numerical coefficient_______________
m
( x  y) 7
In the following expression, identify base, exponent and numerical coefficient:
4
Base___________ exponent__________ numerical coefficient_______________
3  3x  z 


4 w 
Base___________ exponent__________ numerical coefficient_______________
7
In the following expression, identify base, exponent and numerical coefficient:
(ab) 5
2
Base___________ exponent__________ numerical coefficient_______________
In the following expression, identify base, exponent and numerical coefficient: 
Write using exponential notation whenever it is possible: 3a  3a  3a  3a
Write using exponential notation whenever it is possible:  xyxyxxx
Write using exponential notation whenever it is possible:  a  aaaa
Write using exponential notation whenever it is possible: xxy yyx
Write using exponential notation whenever it is possible: (a  b)( a  b)( a  b)
49
Write using exponential notation whenever it is possible: (2t 3 )( 2t 3 )( 2t 3 )( 2t 3 )
Write using exponential notation whenever it is possible: m  nn  m
Write using exponential notation whenever it is possible:
kkk
k
n
x
Write using exponential notation whenever it is possible: x   x  x
2
Write using exponential notation whenever it is possible: (3  x)(  x  3)(3  x)
 2a  a  2a 
Write using exponential notation whenever it is possible:   2  
 b  b  b 
 3   3  3 
Write using exponential notation whenever it is possible:   


 x  x   x 
1
Write using exponential notation whenever it is possible:
( w  2v)( 2v  w)( 2v  w)
 x y  x  y 
Write using exponential notation whenever it is possible:   

 m m  m 
Write using exponential notation whenever it is possible: (m  n)( m  n)m  n
Write using exponential notation whenever it is possible: (m  p  n)( p  m  n)( n  m  p)
Write the following expression without using exponential notation: (m) 3
Write the following expression without using exponential notation:  m 3
Write the following expression without using exponential notation: (xy) 3
Write the following expression without using exponential notation: 2a 3
Write the following expression without using exponential notation: (a  b) 2
Write the following expression without using exponential notation: a  b 2
Simplify by raising to the indicated power: (3x) 0
Simplify by raising to the indicated power:
3x 0
50
Simplify by raising to the indicated power: 3 0 x
Simplify by raising to the indicated power:
a(b  c) 0
Simplify by raising to the indicated power:
abc 0
Simplify by raising to the indicated power: (abc) 0
Simplify by raising to the indicated power:
ab  c 0
Simplify by raising to the indicated power:
a 0b 0 c 0
10B
Determining if rational expressions can be simplified and simplifying them.
Factorization not needed.
Simplify, if possible. If not possible, write “not possible”:
3 xy
9 yx
Simplify, if possible. If not possible, write “not possible”:
 a2
b2a2
Simplify, if possible. If not possible, write “not possible”:
a 2 (b  c)
2a
Simplify, if possible. If not possible, write “not possible”:
v  4z
4z
Simplify, if possible. If not possible, write “not possible”:
 12 xy
9x
Simplify, if possible. If not possible, write “not possible”:
2abc
8ab
Simplify, if possible. If not possible, write “not possible”:
 2mn 2
4n 2
Simplify, if possible. If not possible, write “not possible”:
5 xy 4
20 y 3
51
Simplify, if possible. If not possible, write “not possible”:
15 x(a  b)
25 x
Simplify, if possible. If not possible, write “not possible”:
a 2 (b  c)
bc
Simplify, if possible. If not possible, write “not possible”:
bc(b  e)
b
Simplify, if possible. If not possible, write “not possible”:
4 x(5 x 2 )
x
Simplify, if possible. If not possible, write “not possible”:
ab
7( a  b)
Simplify, if possible. If not possible, write “not possible”:
2x  3
2x
Simplify, if possible. If not possible, write “not possible”:
4x 2 y 3
12 x
Simplify, if possible. If not possible, write “not possible”:
a  3b
3b  a
Simplify, if possible. If not possible, write “not possible”:
ab  c
ab
Simplify, if possible. If not possible, write “not possible”:
 2( x  3 y )
3y  x
Simplify, if possible. If not possible, write “not possible”:
2x 2 z
7 xyz
Simplify, if possible. If not possible, write “not possible”:
3
9(mn  k )
11B
Collecting like terms: more than one type of like terms in the expression
(fractions and decimals used but not as coefficients of variables).
Simplify by collecting like terms:
3x  a  4x  a
52
Simplify by collecting like terms:
7 y  5  12 y  y  6
Simplify by collecting like terms:
a 2 b 3  4b 2 a 3  6a 2 b 3
Simplify by collecting like terms:
 0.2  x  x  0.8
Simplify by collecting like terms:
2
1
 xy  y  y   yx
3
3
Simplify by collecting like terms:  3 x  8 y  8 x  2 x
Simplify by collecting like terms:  5a  3b  2b  7a
Simplify by collecting like terms:
 2ab  4ba  2  3ab  1
Simplify by collecting like terms:
3 j2  4 j2  2 j
Simplify by collecting like terms:
a 2 b 3  4b 2 a 3  6a 2 b 3
Simplify by collecting like terms:
0.2  2z  5z  z  0.4
Simplify by collecting like terms:
 x3  x4  x3
Simplify by collecting like terms:
Simplify by collecting like terms:
Simplify by collecting like terms:
1
 2x  1
2
 3y  7x  x  4 y
2
2  7m  4 
5
x
Simplify by collecting like terms:
ab  ba  0.7  0.9  ba
Simplify by collecting like terms:
x 4 y  2 x  3 yx 4  5x
Simplify by collecting like terms:
2
1
 cd   dc  d  c
3
3
Simplify by collecting like terms:
2  a  2a  2.3
Simplify by collecting like terms:
2x  y  x  2 y
12B
53
Solving linear equations with a variable on both sides. No fractions or
parentheses involved. “No solution” or “all real numbers” as a solution
possible. No operations on decimals needed.
Solve the following equation:
3x  5  4x
Solve the following equation:
4x  5  15  x
Solve the following equation:
x 8  x 9
Solve the following equation:
2  a  a
Solve the following equation:
3j 4j  2j
Solve the following equation:
2  7m  4  m
Solve the following equation:
z 1   z  z
Solve the following equation:
7 x  8x
Solve the following equation:
 4x  x  12
Solve the following equation:
6x  15x
Solve the following equation:
3  7x  x
Solve the following equation:
3x  5  4x
Solve the following equation:
 4x  2  3x
Solve the following equation:
3x  7  1  4x
Solve the following equation:
4x  3  12 x
Solve the following equation:
 6  6x  2x  5
Solve the following equation:
30  x  9  21  x
Solve the following equation:
 x  3  3x  2x
Solve the following equation: x  4  x  2
Solve the following equation:
 2x  3   x  4
54
13B
Solving linear inequalities (no parentheses, no fractions, no decimals involved;
if a variables is only on one side its coefficient will be negative). “No solution”
or “all real numbers” as a solution possible.
Solve the following inequality: x  x
Solve the following inequality: x  x
Solve the following inequality:
x4
Solve the following inequality:
 x  2 1
Solve the following inequality:
2  3x  14
Solve the following inequality:
8  2  3x
Solve the following inequality:
 7 x  1
Solve the following inequality:
4x  1  2x  5
Solve the following inequality:
2  6x  8
Solve the following inequality:
 1  7 x  5
Solve the following inequality:
 x  1  2
Solve the following inequality:
x 8  x 9
Solve the following inequality:
3 a 5  4
Solve the following inequality:
3x  1  3x  11
Solve the following inequality:
2  a  4a  5a  2
Solve the following inequality:
 5x  2  3x  2
Solve the following inequality:
3  2a  6a
Solve the following inequality:
3  2  5x
Solve the following inequality:
2x  5  6x  16
Solve the following inequality:  3x  18
55
14B
“The difference of squares” factorization (formula not given).
Factor the following expression: 36 x 2  1
Factor the following expression: x 2  0.09
Factor the following expression: m 4  n 2
Factor the following expression:
1
 9x 2
4
Factor the following expression: y 2  100a 2
Factor the following expression: x 2 y 2  25
Factor the following expression: 25s 2  64t 2
Factor the following expression:
a2
 0.01
b2
Factor the following expression: m 2  81n 2
Factor the following expression: (2a) 2  9a 2
Factor the following expression: 4 x 2  36 y 2
Factor the following expression:
x2
1
36
Factor the following expression: 1  m 2 n 2
Factor the following expression: x 4 
9
25
Factor the following expression: 9m 4 
1
16
Factor the following expression: 25 x 2  (7 y ) 2
Factor the following expression: x 6  16
56
Factor the following expression: 0.04  s 4
Factor the following expression: 4  s 12
Factor the following expression: 25 x 10  4
15B
Rewriting algebraic expression in terms of another variable. Only a direct
substitution needed; “relatively easy” simplification.
Rewrite the expression
3a
in terms of x , given that a  2 x (i.e. substitute 2x for a ) Simplify.
2
Express  2 P in terms of m , when P  m (i. e. substitute  m for P ). Write your final answer
without parentheses.
Express  2 P in terms of m , when P  m 2  m (i. e. substitute m 2  m for P ). Write your
final answer without parentheses.
Express A3 in terms of x , when A  2x (i.e. substitute 2x for A). Write your answer without
parentheses. Simplify.
Express A3 in terms of x , when A   x (i.e. substitute  x for A). Write your answer without
parentheses. Simplify.
Rewrite the expression
a 1
in terms of x , if it is given that a  3x (i.e. substitute 3x for a).
3x  1
Simplify.
Express A3 in terms of x , when A  x 7 (i.e. substitute x 7 for A). Write your answer without
parentheses. Simplify.
Express A3 in terms of x , when A   x 3 (i.e. substitute  x 3 for A). Write your answer without
parentheses. Simplify.
Write the expression  a  2b in terms of x , if a  5x and b   x (i.e. substitute 5x for a and
 x for b ). Eliminate parentheses and simplify.
57
Write the expression a 2  4a in terms of x , if a  x  2 (i.e. substitute x  2 for a). Eliminate
parentheses and simplify.
x y
in terms of a and b if x  a  b , y  a  b (i.e. substitute a  b for x and
x y
a  b for y and simplify). . Eliminate parentheses and simplify.
Express
Express the following expression in terms of x (i. e. substitute x for y). Simplify the new
expression:
x y
if y  x
x y
Let P   x  2, Q  4 x  1 . Express P  Q in terms of x (i.e. substitute  x  2 for P and
4x  1 for Q ). Write your final answer without parentheses, in a simplified form.
Let P   x  2, Q  4 x  1 . Express P  Q in terms of x (i.e. substitute  x  2 for P and
4x  1 for Q ). Write your final answer without parentheses, in a simplified form.
Let P   x  2, Q  4 . Express PQ in terms of x (i.e. substitute  x  2 for P and 4 for Q ).
Write your final answer without parentheses, in a simplified form.
Let Q  4 x  1 . Express Q 2 in terms of x (i.e. substitute 4x  1 for Q ). Write your final answer
without parentheses, in a simplified form.
Let A  x  2 , B  4x . Express A  B in terms of x (i.e. substitute  x  2 for A and 4x for
B). Write the expression without parentheses. Simplify.
Let A  x  2 , B  4x . Express  AB in terms of x (i.e. substitute  x  2 for A and 4x for
B). Write the expression without parentheses. Simplify.
Express A 2  AB in terms of x , if A  x, B   x (i.e. substitute x for A and  x for B). Write
your answer without parentheses. Simplify.
Let P   x 3 and R  2x 2 . Write the expression PR in terms of x (i.e. substitute  x 3 for P
and 2x 2 for R). Simplify.
Let P   x 3 and R  2x 2 . Express the expression
P
in terms of x (i.e. substitute  x 3 for P
R
and 2x 2 for R). Simplify.
58
Rewrite the expression a 2  b 2 in terms of x , if a  5x and b  1 (i.e. substitute 5x for a and
1 for b). Write the new expression without parentheses. Simplify.
Rewrite the expression a 2  b 2 in terms of x , if a  x and b   x (i.e. substitute x for a and
 x for b). Write the new expression without parentheses. Simplify.
Rewrite the expression a 2  b 2 in terms of a, if a  b. (i.e. substitute b for a). Simplify.
Rewrite the expression a 2 b in terms of x , if a  x 3 , b  x 4 (i.e. substitute x 3 for a and x 4 for
b). Write the new expression without parentheses. Simplify.
Rewrite the expression a 2 b in terms of a, if a  b. (i.e. substitute b for a). Simplify.
Let C  3x, D 
x
. Express the expression CD in terms of x (i.e. substitute  3x for C and
2
x
for D ). Simplify.
2
Let C  3x, D 
x
. Write the expression C  D in terms of x (i.e. substitute  3x for C and
2
x
for D ). Simplify.
2
Rewrite the expression a 2  2ab  b 2 in terms of x , if
x for b. Simplify.
a  1, b  x , i.e. substitute 1 for a and
Rewrite the expression a 2  2ab  b 2 in terms of x , if a  3x, b  2 x , i.e. substitute 3x for a
and 2x for b. Simplify.
Rewrite the expression a 2  2ab  b 2 in terms of x , if a 
and  x for b. Simplify.
x
x
, b   x , i.e. substitute
for a
2
2
Rewrite the expression a 2  2ab  b 2 in terms of x , if a  b  x , i.e. substitute x for a and for
b. Simplify
Express 2 x  y in terms of s , if x  s 3 , y  2 s , i.e. substitute s 3 for x and 2s for y .
Simplify, if possible.
59
Express x 2 y in terms of s , if x  s 3 , y  2 s . i.e. substitute s 3 for x and 2s for y . Simplify,
if possible.
Express 2 x  y 3 in terms of s , if x  s 3 , y  2 s . i.e. substitute s 3 for x and 2s for y .
Simplify, if possible.
xy 2
in terms of s , if x  s 3 , y  2 s . i.e. substitute s 3 for x and 2s for y . Simplify,
4
if possible.
Express
Express ab in terms of x , if a 
5x
126
5x
126
, i.e. substitute
for a and
for b .
, b
126
5
126
5
Simplify.
x t
x
t
x
t
 2 y , i.e. substitute  y, for
Express  in terms of y, if   y ,
and 2 y for .
z z
z
z
z
z
Simplify.
2
2
x
t
x
 x  t 
 2 y , i.e. substitute 3y, for
Express      in terms of y, if  3y,
and  2 y for
z
z
z
z z
t
. Simplify.
z
16B
Simplifying algebraic expressions by removing parentheses first, and
then collecting like terms. Performing an operation on fractions or on
decimals might be needed (no expressions of the form (a  b) 2 )
Eliminate parentheses and simplify by collecting like terms:  2(t  5)  4t
Eliminate parentheses and simplify by collecting like terms: 3( x 5  2)  (4  x 5 )
Eliminate parentheses and simplify by collecting like terms: (q  6)( 8)  21q
Eliminate parentheses and simplify by collecting like terms: (a  3)( a  4)
Eliminate parentheses and simplify by collecting like terms:
2x
 ( 2 x  1)
5
Eliminate parentheses and simplify by collecting like terms: 2(a  3)  (a  4)
2x
 ( 2 x  1)
Eliminate parentheses and simplify by collecting like terms:
5
Eliminate parentheses and simplify by collecting like terms: 2 x  5 y  4(3x  y )
60
Eliminate parentheses and simplify by collecting like terms:
 3xy  7 yx  ( xy  3)
Eliminate parentheses and simplify by collecting like terms: (6 x 4  3x 3  1)  (3x 3  3x 2  3)
Eliminate parentheses and simplify by collecting like terms: a  0.1(2  3a)
Eliminate parentheses and simplify by collecting like terms:  3cb  abc  2(a  bc)
1
1
Eliminate parentheses and simplify by collecting like terms:
a  b  (a  2b)
3
3
2
1
Eliminate parentheses and simplify by collecting like terms:  6 ( d  a )  a  d
3
2
2
Eliminate parentheses and simplify by collecting like terms: 3a  1  (4a 2  2)
4
Eliminate parentheses and simplify by collecting like terms: 3a  9b  (4a  b)
5
Eliminate parentheses and simplify by collecting like terms:  (2q  6)  2q  5q
Eliminate parentheses and simplify by collecting like terms: ( x  1)(  x)  x
Eliminate parentheses and simplify by collecting like terms:  ( x  1)  x
Eliminate parentheses and simplify by collecting like terms:  2(t  3)  t  1
Eliminate parentheses and simplify by collecting like terms: 0.2 y  3( y  0.3)
Eliminate parentheses and simplify by collecting like terms:  (6a  2)  12a
Eliminate parentheses and simplify by collecting like terms: 4a  7a  2(8a  1)
17B
Collecting like terms (if possible) involving operations on fractions, or
decimals.
If possible, collect like terms. If not possible, write “not possible”
 0.03x  0.4 x 2
If possible, collect like terms. If not possible, write “not possible”
1 2 3 2
a  a
2
7
If possible, collect like terms. If not possible, write “not possible”
0.5mn  0.7m
If possible, collect like terms. If not possible, write “not possible”
ab 1
 ba
3 6
61
m m

4 6
If possible, collect like terms. If not possible, write “not possible”:
If possible, collect like terms. If not possible, write “not possible”: 
2m 2m

38 38
If possible, collect like terms. If not possible, write “not possible”:  0.7abc  0.08cba
If possible, collect like terms. If not possible, write “not possible”: ab  8.2ba
2
3s 3
If possible, collect like terms. If not possible, write “not possible”:  s 3 
11
22
If possible, collect like terms. If not possible, write “not possible”:
1
2
x x
3
7
If possible, collect like terms. If not possible, write “not possible” : 0.2mn 2  7nm 2
If possible, collect like terms. If not possible, write “not possible”: 0.3xy  0.5 yx
7
2x
x
9
5
3x
If possible, collect like terms. If not possible, write “not possible”: 2 x 
10
x 2
If possible, collect like terms. If not possible, write “not possible”:  x
5 3
1
1
If possible, collect like terms. If not possible, write “not possible”: x  x
2
2
If possible, collect like terms. If not possible, write “not possible”: 
If possible, collect like terms. If not possible, write “not possible”: 0.24 xy  0.17 x
If possible, collect like terms. If not possible, write “not possible”:
1
3
x x
2
2
If possible, collect like terms. If not possible, write “not possible” : 0.5c 3  10c 4
If possible, collect like terms. If not possible, write “not possible”:  0.4 x 2 y  0.8 yx 2
18B
Solving linear equations (inequalities) when the removal of parentheses is
needed; no fractions involved. “No solution” or “all real numbers” as a
solution is a possibility. Operations on decimals might be needed.
62
Solve the following equation:
3x  2  4( x  1)
Solve the following equation:
6(2 x  3)  x  11x
Solve the following equation:  ( x  3)  x  (2 x  1)
Solve the following equation: 0.2( x  3)  0.2 x  3(0.1  x)
Solve the following equation:
 2(4 x  1)  7 x  4  x
Solve the following equation:
4( y  2)  (1  y )
Solve the following equation: 0  2(1  x)
Solve the following equation: 4x  3  12
Solve the following equation:
 6  2( x  5)
Solve the following equation:
 0.7  2( x  0.3)   x  0.1
Solve the following equation:
32m  4  6m  6
Solve the following inequality:  6  2( x  5)
Solve the following inequality: 3x  1  11  3( x  4)
Solve the following inequality: 4 x  1  7 x  (5  2 x)
Solve the following inequality:  x  1  6( x  2)
Solve the following inequality: 4 x  1  2( x  5)
Solve the following inequality:  3( x  2)  3x  2
Solve the following inequality: 3(2 x  5)  6 x  16
Solve the following inequality: 3  23a  5  6a
Solve the following inequality: 32 x  1  2x  2
Solve the following inequality: 0.1(10 x  100)  x
63
19B
Solving equations for a given variable. Factorization of this variable might be
needed in the process. Variable might be on both sides of the equation.
Variable could be in the denominator. No simplification needed at the end.
More than one operation might be needed in the process.
Solve for x :
x
b
ac
Solve for x :
2x  a  b
Solve for x :
ax  b  c
Solve for x :
ax  bx  c  0
Solve for x :
a
b
x
Solve for the indicated variable: ax  b  4 for a
Solve for the indicated variable:
m2
 a for u .
u
Solve for the indicated variable: 2 x  y  t  3s for y .
Solve for the indicated variable: AX  A  1 for A.
Solve for the indicated variable: x 3  sy  2 x for y.
Solve for the indicated variable: 3v  t  s  v for v .
Solve for x :  x  a .
Solve for x : ax  by  3 x .
2m
 4k
n2
b
Solve for b :   c
a
Solve for m :
Solve for b: abc 2  1  c
Solve for x :
m  2n
a
x
64
Solve for s: sx  s  t
Solve for x : ax  2x  1
Solve the following equation
x
 d  e for y .
y
20B
Writing phrases as algebraic expressions (involving 2 operations).
Use the letter x to represent a number and write the following as algebraic expression:
Seven more than one third of a number
Use the letter x to represent a number and write the following as algebraic expression:
A quantity decreased by 9, and then multiplied by A
Use the letter x to represent a number and write the following as algebraic expression:
A number cubed, and then decreased by y
Write a direct translation of the following phrases as an algebraic expression.
Multiply 3 by x , and then add y.
Write a direct translation of the following phrases as an algebraic expression.
Multiply the sum of a and b by 4.
Write a direct translation of the following phrases as an algebraic expression. Do not simplify.
The opposite of x , then raise it to the sixth power.
Write a direct translation of the following phrases as an algebraic expression. Do not simplify.
Subtract 3 from y, and then multiply it by z .
Write a direct translation of the following phrases as an algebraic expression.
Raise x to the third power, and then multiply by 9.
Write a direct translation of the following phrases as an algebraic expression. Do not simplify.
Multiply x by 9, and then raise the result to the third power.
Write a direct translation of the following phrases as an algebraic expression.
The difference of a and b, then divided by c
Write a direct translation of the following phrases as an algebraic expression.
Divide 3 by y, and then add x
Write a direct translation of the following phrases as an algebraic expression.
65
Raise  x to the third power, raise y to the seventh power, and then add them together
Let C be a variable representing the temperature in Celsius. Write the following phrase as an
algebraic expression: Nine fifths of the Celsius temperature plus 32.
Let m represents mass, and c speed of light. Use m and c to write the following phrase as an
algebraic expression: the product of mass and the square of speed of light.
Use the letter X to represent a number and write the following as an algebraic expression:
A number decreased by 7, and then the result is doubled.
Use the letter Y to represent a number and write the following as an algebraic expression:
The opposite of a number, then raised to one hundred and twenty first power
Use the letter C to represent a number and write the following as an algebraic expression:
A number, first divided by 2, and then raised to the third power.
Use the letter s to represent a number and write the following as an algebraic expression:
Multiply a number by 9, and then subtract it from c.
Use the letter m to represent a number and write the following as an algebraic expression:
The difference between a number and 4, then multiplied by y.
Use the letter X of your choice to represent a number and write the following as an algebraic
expression: Take one fourth of a number, and then subtract 5.
21B
Identifying linear equations and writing them in a standard form.
Recognizing the value of parameters in the representation.
Rewriting algebraic expressions to match a prescribed format and identifying
the values of given variables.
Determine if the following equation is linear in one variable. If it is not linear, write “not linear”.
If it is linear, express it in the form ax  b  0 , a  0 . Determine the values of a and b in your
x
representation:  1  0
4
Determine if the following equation is linear in one variable. If so, express it in the form
ax  b  0 , a  0 . Determine the values of a and b in your representation: 7 x  2  3x
66
Determine if the following equation is linear in one variable. If it is not linear, write “not linear”.
If it is linear, express it in the form ax  b  0 , a  0 . Determine the values of a and b in your
representation: 2 x( x  3)  2 x 2
Determine if the following equation is linear in one variable. If it is not linear, write “not linear.
If it is linear, express it in the form ax  b  0 , a  0 . Determine the values of a and b in your
representation: 3x 2  1  0
Write the expression ( x  2) 2  ( y  1) 2  r 2 in the form: ( x  p) 2  ( y  q) 2  r 2
are the values of p and q?
What
Write the following expression in the form A 2 , where A is any algebraic expression. Identify
81x 2
A:
Write the following expression in the form A 2 , where A is any algebraic expression. Identify
x8
A:
Write the following expression in the form A 2 , where A is any algebraic expression. Identify
0.09 x 2
A:
y6
Write the following expression in the form Ax  By , where A , B are any numbers. Identify A
 x  8y
and B in your representation:
4
Write the following expression in the form A5 , where A is any algebraic expression. Identify
A:
x 10
Write the following expression in the form A 2 , where A is any algebraic expression. Identify
A:
36 x 2 y 2
Write the following equation in the form y  mx  b , where m and b are any numbers. In the
y  2x  4
expression you obtained, identify m and b :
Write the following equation in the form y  mx  b , where m and b are any numbers. In the
yx0
expression you obtained, identify m and b :
Write the following equation in the form y  mx  b , where m and b are any numbers. In the
2 y  4 x  3
expression you obtained, identify m and b :
Write the expression x 24 in the form a 4 . Identify a in your representation.
Write the expression x 24 in the form a 6 . Identify a in your representation.
67
Rewrite the following expression in the form ax 3  b , where a and b are any numbers. Identify
a and b in your representation:  x 3  3
Rewrite the following expression in the form ax 3  b , where a and b are any numbers. Identify
a and b in your representation: 2 x 3  3
Rewrite the following expression in the form ax 3  b , where a and b are any numbers. Identify
4x3  3
a and b in your representation:
2
Rewrite the following expression in the form ax 3  b , where a and b are any numbers. Identify
(2 x) 3  4
a and b in your representation:
Write the following expression in the form a 3 , where a is any algebraic expression or a number.
State what a is equal to: 27
Write the following expression in the form a 3 , where a is any algebraic expression or a number.
8
State what a is equal to:
125
Write the following expression in the form a 3 , where a is any algebraic expression or a number.
State what a is equal to: 64x 3
Write the following expression in the form a 3 , where a is any algebraic expression or a number.
State what a is equal to:
y6
22B
Applying laws of exponents in order to evaluate an expression (up to two
operations). Performing an operation on fractions or on decimals might be
needed.
Or
Factoring -1
Evaluate:
9 81
9 80
Evaluate:
47  46
411
68
15 248
15 247
(12 3 ) 6
Evaluate:
 1217
3 21  310
Evaluate:
3 29
10
0.5  0.5 6
Evaluate:
(0.5 7 ) 2
Evaluate:
(64 3 )11
Evaluate:
64 32
215 216
Evaluate:  27
2
Factor  1 from the following expression: 2 x  y  z .
X
Factor  1 from the following expression:
 y.
3
Factor  1 from the following expression:  a  b  1
x yz
Factor  1 from the following expression: a 
2
Factor  1 from the following expression: 4 x 3  5 y 2  5 z
1
1
s
Factor  1 from the following expression:
ab
ab
2 2
4
Factor  1 from the following expression: x y  z
3
3
Factor  1 from the following expression:  2( x  2 y)  ( x  2 z ) z 2
23B
Removing parentheses from expressions of the form (a  b) 2 and simplifying
them by collecting like terms.
Eliminate parentheses and simplify by collecting like terms: (2  3x) 2
Eliminate parentheses and simplify by collecting like terms: (3ab  1) 2
Eliminate parentheses and simplify by collecting like terms: (2 x  y ) 2
Eliminate parentheses and simplify by collecting like terms: ( xy  3) 2
69
Eliminate parentheses and simplify by collecting like terms: (3a  b) 2
Eliminate parentheses and simplify by collecting like terms: (2  x 3 ) 2
Eliminate parentheses and simplify by collecting like terms: (0.1  x) 2
Eliminate parentheses and simplify by collecting like terms: (a 2 b  1) 2
Eliminate parentheses and simplify by collecting like terms: (7  2m) 2
Eliminate parentheses and simplify by collecting like terms: ( xyz  1) 2
Eliminate parentheses and simplify by collecting like terms: (0.5  y 5 ) 2
Eliminate parentheses and simplify by collecting like terms: (0.2 x  0.1y) 2
Eliminate parentheses and simplify by collecting like terms: (8  x 2 y) 2
Eliminate parentheses and simplify by collecting like terms: (3m  n) 2
Eliminate parentheses and simplify by collecting like terms: (2s  3t ) 2
24B
Evaluation of an algebraic expression when the value of a part of the algebraic
expression is given (rather than the value of the variables). No rewriting the
expression in its equivalent form is needed.
If A 5  3 , evaluate:
 2A5
If A 5  3 , evaluate:
 (2 A5 ) 2
If A 5  3 , evaluate:
 2  2 A5
2
a
a
 3
Evaluate the expression   , if
b
b
a
a
 3
Evaluate the expression  , if
b
b
ab
ab
 2 , if
 2
Evaluate the expression
c
c
70
ab
Evaluate the expression  

 c 
2
 ab
Evaluate the expression  

c 

2
, if
ab
 2
c
, if
ab
 2
c
If x  y  2 , evaluate ( x  y) 2  4
If x  y  2 , evaluate
If
x y 1

7
3
1
2
1
  , evaluate 
A
7
A
Evaluate the expression
1
, if GHJ  1 :
GHJ
Evaluate the expression 2 
Evaluate the expression 
a
a
 10 , if
3
b
b
xy
, if
z
xy
1

z
3
Evaluate the following expression a  b  c  d if a  b  2
cd 3
Evaluate the following expression 9(a  b) 2  c  d if a  b  2
Evaluate the following expression
ab
if a  b  9
cd
cd 3
cd 3
Evaluate the expression ( x  z ) x 2 y if x 2 y  0.2, x  z  0.6
Evaluate the expression
xz
if x 2 y  0.2, x  z  0.6
2
x y
Evaluate the expression  ( x  z )  x 2 y if x 2 y  0.2, x  z  0.6
Evaluate the expression  4 xyzt , if xy  1, zt  3 .
Evaluate the expression  xy  zt , if xy  1, zt  3 .
71
Evaluate the expression
xy  zt
, if xy  1, zt  3 .
xy
Evaluate the following expression  y 4 x 3 z 6 , if x 3 y 4 z 6  
Evaluate the following expression
2
.
3
1
, if x 2  y 2  0.1
x  y2
2
Evaluate the following expression y 2  x 2  0.3 , if y 2  x 2  0.1
DIFFICULT CATEGORY: randomly select 4 categories (1 question from each category)
1D
Ability to rewrite the expression by substituting (up to 2 substitutions) value
of variable(s) and then evaluating the expression, if possible. If evaluation is
not possible, recognizing that it is not possible to evaluate. “More challenging”
examples.
The expression
all of them.
a) x  3,
3 x
cannot be evaluated for which of the following values of x and y? Circle
y 5
y  5
The expression 3x
y5
c) x  3,
y0
d) x  0,
y5
1
cannot be evaluated for which of the following values of x and y?
2 y
Circle all of them.
a) x  3, y  2
The expression
b) x  3,
b) x  3,
y  2
c) x  3,
y  0 d) x  0,
y  2
3y
cannot be evaluated for which of the following values of x and y? Circle
2 x
all of them.
a) x  0,
y2
The expression
3x
cannot be evaluated for which of the following values of x and y? Circle
yx
all of them.
a) x  3,
y3
b) x  3,
b) x  5,
y0
y5
c) x  2,
c) x  0,
y2
d) x  0,
y  5 d) x  2,
y0
y0
Rewrite the expression by substituting values of variables and then evaluate it, if possible. If not
possible, write “undefined”:  ( A  3B ) , if A  1, B  1
72
Rewrite the expression by substituting values of variables and then evaluate it, if possible. If not
possible, write “undefined”:  ( A  3B ) , if A  0.1, B  0.2
Rewrite the expression by substituting values of variables and then evaluate it, if possible. If not
2
1
possible, write “undefined”:  ( A  3B ) , if A  , B  
7
6
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
possible, write “undefined”: 2a 2  (2a) 2 if a  1
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
1
2
possible, write “undefined”: A 2  B 2 when A  , B  
3
3
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
A B
1
2
possible, write “undefined”:
when A  , B  
A B
3
3
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
A( B)
1
2
possible, write “undefined”:
when A  , B  
AB
3
3
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
0.1x
possible, write “undefined”:
 22 when x  2, y  0.02
y
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
x y
possible, write “undefined”:
 when x  0.1, y  0.2
y x
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
possible, write “undefined”: m  (n  4) 2 when m  1, n  2,
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
A B
5
A  4, B  
possible, write “undefined”:
3
9
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
A
2
possible, write “undefined”: AB   B A  , B  1
B
8
73
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
1
possible, write “undefined”: (ab) 2  a  b a  1, b 
3
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
bn
1
possible, write “undefined”:
b ,n3
bn  1
3
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
possible, write “undefined”: n  0.2  m when m  0.3, n  0.6
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
1
4
possible, write “undefined”: 8m  10n when m   , n 
8
5
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
1
1
4
possible, write “undefined”: n  (  m) when m   , n 
8
8
5
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
1
4
possible, write “undefined”:  8m 2  n when m   , n 
8
5
Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not
possible, write “undefined”: 2(m  3n) when m  2, n  5
2D
Recognizing equivalent expressions. Applying more than one of the following
rules at a time.
a) the order of terms of an expression can be changed
b) the order of factors of an expression can be changed
b c bc
 
a a
a
a a
a

d)  
b
b
b
ab
b
1
 a   ab 
e)
c
c
c
c)
Circle all of the following expressions that are equivalent to m  (n  k ) :
 (n  k )
(  k  n)  m ,
,
,
m
mnk ,
mnk ,
1
74
Circle all of the following expressions that are equivalent to 
 a  4c
,
b
1
 (4c  a) ,
b
1
(1)( a  4c),
b
a  4c
:
b
4c a
 ,
b b
Rewrite the following expression in its equivalent form as a single fraction: x
 a a2  2

4s
4s
Rewrite the following expression in its equivalent form as a single fraction:  7 
Rewrite the following expression in its equivalent form as a single fraction: 
y 1
m
3
1
nm
2
k t
k t
Rewrite the following expression in its equivalent form as a single fraction:
x y zt
2 
( a  b)
t
t
Rewrite the following expression in its equivalent form as a single fraction: 
2a
mn
3
xy
xy
Rewrite the following expression in its equivalent form as a single fraction: 3
a b
cd  1
2
t
t
Circle all of the following expressions that are equivalent to xy  ab(c  d ) :
 ab(d  c)  xy
 (c  d )ab  yx
ab(c  d )(  yx)
Circle all of the following expressions that are equivalent to
b  ad
,
6
1
(b  ad ),
6
da
 b,
6
Circle all of the following expressions that are equivalent to

ad b
 ,
6 6
b  da
,
6
ad  b
:
6
 b  da
6
(ad  b)  6
m
n

:
x x
 ( m  n)
x
Circle all of the following expressions that are equivalent to

m n

,
x
x
,
 1
(m  n)    ,
 x
75
Circle all of the following expressions that are equivalent to
a2
,
3b
 (2  a)
,
3b
a2
3b
1
a b,
cd
,

a2
3b
1
(a  2)( 1)
3b
Circle all of the following expressions that are equivalent to
2
a2
3b
a
2
,

 (3  b)  (3  b)
Circle all of the following expressions that are equivalent to
 (2  a)
,
 (3  b)

2( a  b )
cd
 (b  a)
2,
 (c  d )
2(b  a)  (d  c) ,
cd
,
2( a  b )
Replace  with expression such that the resulting statement is true.
Replace  with expression such that the resulting statement is true:
2x  7 y
7y


ab
ab
mn
   ( m  n)
v
Replace  with expression such that the resulting statement is true: 2(a  2b)(3  d )  (3  d )
Replace  with expression such that the resulting statement is true: da  b  b  d
m
mn
 
2
2
a a

Replace  with expression such that the resulting statement is true:
b

Replace  with expression such that the resulting statement is true:
3D
Simplifying expressions involving exponential expressions (“relatively difficult
examples”). Operations on fractions or decimals might be needed.
20
a
Circle all expressions that are equivalent to   :
3
20
20
12 8
a
a
(a )
a 12 a 8
,
,
,
,
3
3 20
3 20
3 20
(a 4 ) 5
3 20
76
Perform the indicated operations and simplify:
 y5 3
y
y8
Perform the indicated operations and simplify: (2a 2 b)( a 3 )
Perform the indicated operations and simplify:  2a 
Perform the indicated operations and simplify:
a7
(2a) 2
 2x3 x5
x4
(3x 2 ) 3
Perform the indicated operations and simplify:
x4
Perform the indicated operations and simplify: 
a4 2
a
a4
2
1
Perform the indicated operations and simplify:   v 7
v
Perform the indicated operations and simplify: (3aa 2 ) 2
 x2 y 

Perform the indicated operations and simplify: 
 x 
3
Perform the indicated operations and simplify: 2 x(2 x 5 y 3 ) 3
Perform the indicated operations and simplify:  3b(10b 5 ) 2
Perform the indicated operations and simplify:  3a(ab 2 ) 8
 a3 
Perform the indicated operations and simplify:   2 
 4b 
2
Perform the indicated operations and simplify: x 2 y( x 3 ) y 4
Perform the indicated operations and simplify:
s2x
(4s) 2
Perform the indicated operations and simplify:
 ab 7 a
ba 2 b 6
77
Perform the indicated operations and simplify:
2x 2 y 2
y
x
Perform the indicated operations and simplify:
a13b 2 c 4
a 5c 2b
Perform the indicated operations and simplify:
(ab) 8
b
2ab 4
4D
Applying the Distributive Law when removing parentheses in multiplication
of a binomial by a trinomial. Collecting terms is needed.
Eliminate parentheses and simplify: (b  c  2)(b 2  b 3 )
Eliminate parentheses and simplify: ( x 3  2 x 2  2 x)( x 2  x 3 )
Eliminate parentheses and simplify: ( x 2 y  2 x  y)( x 2  3)
Eliminate parentheses and simplify: (a  0.2)( a  0.1b  ab)
Eliminate parentheses and simplify: ( x 3  2 x 2  2 x)( x 2  x 3 )
Eliminate parentheses and simplify: ( x 3 y 2  2 x 2  3)( x 3 y 2  1)
Eliminate parentheses and simplify: (4m  m 2 )(2  m  3m 2 )
Eliminate parentheses and simplify: (2 A  3  4 B)(5 A  B)
Eliminate parentheses and simplify: (m 3  5m 2  2m)(3m 2  m 3 )
Eliminate parentheses and simplify: (ab 3  ab 2  ab)(ab 2  ab 3 )
Eliminate parentheses and simplify: (u  9t )(ut  u 2  5t 2 )
Eliminate parentheses and simplify: ( x 3 y 2  x 2  3)(1  y 3 )
Eliminate parentheses and simplify: (a  b)( a  b  3)
Eliminate parentheses and simplify: ( x  y )( x  y  2)
78
Eliminate parentheses and simplify: (a 2  b 2  0.8)(b 2  a 2 )
Eliminate parentheses and simplify: (m 3 n 2  2m 2 n 3  1)(2  m 2 n 3 )
Eliminate parentheses and simplify: (2s 2  t )(st  st 2  3ts 3 )
Eliminate parentheses and simplify: (u 3  3u 2  4u)(2u 2  10u 3 )
Eliminate parentheses and simplify: (2m 3 n 2  3)(m 3 n 2  1  m 6 n 4 )
Eliminate parentheses and simplify: ( x 7 y  2 x 2  2 yx 5 )( x 2  x 5 y)
5D
Factoring an indicated expression involving parentheses, fractions or
decimals.
Factor (a  2b) from the following expression: 4 x(a  2b)  (a  2b) .
2
from the following expression:
3
Factor
Factor 0.01
Factor
Factor
2 2
4
x y z
3
3
from the following expression:
0.2 x 2  0.03 z
1
1
1
s
from the following expression:
ab
ab
ab
1
from the following expression:
5
1
1
x
5
25
Factor a  b from the following expression: 6(a  b)  x(a  b)
Factor a  b from the following expression: 4(a  b)  3(a  b) 2
Factor
x
2x
x
from the following expression:
a
y
y
y
Factor 0.02 from the 0.2lw  10lh  0.04wh
Factor
11
11 2 33
t  t
from the following expression
2
2
2
79
Factor m  3n from the following expression: 5x(m  3n)  4(m  3n) 5
Factor s  t 2 from the following expression s(s  t 2 )  t ( s  t 2 )
Factor 1.6 from the following expression 1.6 x  3.2 x 2  16 x 3
Factor 0.001 from the following expression 0.1a 4  0.01a 3  0.001a 4
Factor
3
from the following expression:
8
1
2
5
Factor
6
2
Factor
3
9
Factor
5
Factor
from the following expression:
from the following expression:
from the following expression:
from the following expression:

6
9
s  t2
8
8
5 2
1
x yz 
2
2
5
5
5
ab  bc  ac
6
6
6
4 5
 z
3 3
9
27 2
y
y
5
10
6D
Rewriting an algebraic expression in terms of a designated variable when a
rewriting of the expression in an equivalent form is needed.
Evaluation of an algebraic expression when the value of a part of the algebraic
expression is given (rather than the value of the variables) and rewriting the
expression in its equivalent form is needed.
Rewrite the expression a  2b  3c  4d in terms of x , if it is given that a  3c  5x and
4d  2b   x . Simplify.
Rewrite the expression 7 x  7 y in terms of m if x  y  m .
Rewrite the expression  x  y in terms of m if x  y  m .
Rewrite the expression
x y
 in terms of m if x  y  m .
7 7
Rewrite the expression
1
1
 x.
in terms of x if
2
A
A
80
Rewrite the expression  c  b  2a in terms of x if 2a  b  c  x .
Rewrite the expression  2a  b  c in terms of x if 2a  b  c  x .
Rewrite the expression  a  b  c  a in terms of x if 2a  b  c  x .
Rewrite the expression m(2)n(1) p in terms of s if mnp  s . Simplify.
Rewrite the expression m 3 (np) 3 in terms of s if mnp  s .
Rewrite the expression
a2a
a
y .
in terms of y if
3
b
b
x2 y2
xy m

in terms of m if
.
2
z
2
z
Rewrite the expression 
Rewrite the expression  zxty in terms of m if xy  m, zt  3m . Simplify.
Rewrite the expression z 6 (2) y 4 (3) x 3 in terms of s if x 3 y 4 z 6  s . Simplify.
Rewrite the expression
1 2 1 2
x  y in terms of s if x 2  y 2  2s . Simplify.
2
2
 
Write the expression 6 a
Write the expression
b
in terms of m, if ab  m
a
b in terms of m, if ab  m
2
1
Write the expression (2b)(  a ) in terms of m, if ab  m
2
Write the expression a 3b 3 in terms of m, if ab  m
Write the expression
2z
1
x
t
x  t in terms of y, if   y ,
 2 y . Simplify.
z
z
z
z
Write the expression
2x t
x
t
 in terms of y, if   y ,   2 y . Simplify.
z
z
z
z
81
Write the expression
xt
x
t
in terms of y, if   y ,
 2 y . Simplify.
z
z
z
7D
Solving linear equations (inequalities) involving fractions. “No solution” or
“all real numbers” as a solution is a possibility.
Solve the following equation:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following inequality:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
4x
2
1 
5
3
4 x x2

5
2
4
3x  6
 x2
3
x2
 5
4
 y y3

 1
2
3
2a  3
2
 1  5a 
5
3
x
 1  1  x
4
x 1 x 1

4
3
3x
x
 15 
4
8
x
5 
 2x
4
3 x x  2

5
2
3
x2
 5  x
4
x
 3x   1  0
5
x 1
x
 1
3
2
x
 1  8x 
2
82
1
1 3
y 
y
5
5 10
x 2 3
 
Solve the following equation:
2 3 4
3x
3x
 1 
Solve the following equation: 
2
5
2
5
Solve the following equation: 1  a 
3
6
Solve the following equation:
8D
Solving equations for a given variable. Factorization of this variable might be
needed in the process. Variable could be in the denominator. Simplification at
the end required.
Solve for the indicated variable. Simplify your answer whenever possible:
b
 ac for b
a
Solve for the indicated variable. Simplify your answer whenever possible: ax  b  4b for a
Solve for the indicated variable. Simplify your answer whenever possible: abc 2  (ac) 3 for b
Solve for the indicated variable. Simplify your answer whenever possible: AX  A  X  1 for
A
a2
 a for u
Solve for the indicated variable. Simplify your answer whenever possible:
u
Solve for the indicated variable. Simplify your answer whenever possible: xa  a 2  3a 2 for x
Solve for the indicated variable. Simplify your answer whenever possible: ax  3ax  4 for x
Solve for the indicated variable. Simplify your answer whenever possible:
s
 4t for s.
 t3
Solve for the indicated variable. Simplify your answer whenever possible:  2t  8t  9t 2  tx
for t.
Solve for the indicated variable. Simplify your answer whenever possible: 2 ys  xys  y for s.
Solve for the indicated variable. Simplify your answer whenever possible:
Solve for the indicated variable. Simplify your answer whenever possible:
mx
 m 2 n 3 for x.
n
a b
 for a
b3 c
83
Solve for the indicated variable. Simplify your answer whenever possible:
a2
 a for u
u
Solve for the indicated variable. Simplify your answer whenever possible:
2m
 4n for m
n2
Solve for the indicated variable. Simplify your answer whenever possible: 2 x  y  t  3x for y
‘
Solve for the indicated variable. Simplify your answer whenever possible: x 3  xy  2x 3 for y
Solve for the indicated variable. Simplify your answer whenever possible: s( x  1)  s 2 for x
Solve for the indicated variable. Simplify your answer whenever possible: ax  by  3ax  4by
for a
Solve for the indicated variable. Simplify your answer whenever possible: 3(v  s )  s for v
Solve for the indicated variable. Simplify your answer whenever possible: ax  x(a  2)  1 for
x
9D
Evaluating algebraic expressions when the value of a part of it is given (rather
than the value of variables). Rewriting the original expression in some
equivalent form is required to make the needed substitution.
Evaluate the expression a  c  b , if a  b  1 and c  2 .
Evaluate the expression
a b
 , if a  b  1 and c  2 .
c c
Evaluate the expression 2c  (b  a) , if a  b  1 and c  2 .
Evaluate the expression
m2
m
 4.
, if
2
n
n
Evaluate the expression
n
m
 4.
, if
m
n
If x  y  2 , evaluate 7 x  7 y
84
If x  y  2 , evaluate
x y

7 7
If x  y  2 , evaluate  x  y .
Evaluate the expression x 3 ( y 2 z 3 ) 2 , if x 3 y 4 z 6  
2
.
3
Evaluate the expression z 6 (2) y 4 (3) x 3 , if x 3 y 4 z 6  
Evaluate the expression 
2
.
3
x3 y 4 6
2
 z , if x 3 y 4 z 6   .
3
3
Evaluate the expression 2(a  d  c  b) if a  b  2
Evaluate the expression 9(a  b) 2  d  c if a  b  2
cd 3
cd 3
Evaluate the expression,  zxty if xy  1, zt  3 :
Evaluate the expression
1 2 1 2
x  y , if x 2  y 2  0.1
2
2
Evaluate the expression
x2
y2
, if x 2  y 2  0.1

0.2 0.2
Evaluate the expression G 3 (HJ ) 3 , if GHJ  1
Evaluate the expression G (2) H (1) J , if GHJ  1
If
1
2
1
  , evaluate 2
A
7
A
If
1
2
1
  , evaluate
A
7
A
Simplify the expression 4 x  ( z  4 x) , and then evaluate when z  3 :
VERY DIFFICULT CATEGORY: randomly select 1 category (1 question from the category
selected)
85
1VD
Writing phrases as algebraic expressions and then simplifying them:
a) using the laws of exponents
b) removing parentheses and collecting like terms.
Write as a single algebraic expression using parentheses where appropriate Then eliminate the
parentheses and simplify: The product of  x and x 2 , then raised to the sixth power.
Write as a single algebraic expression using parentheses where appropriate. Then eliminate the
parentheses and simplify: The quotient of 4b 3 and b 2 , then raised to the second power
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: Add 3x  2 and  4x  2 and then raise the result to the second power/
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms:
Subtract  x 2  3 x  2 from 4 x 2  4 x  2 .
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: Multiply 2 x 2  3 x  2 and  4x  2
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
1
2
collect like terms: The product of a 
and  2a
3
5
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: The product of 4, 2 x 2  y and 3 y  x 2
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: The sum of  mnk , 4nmk , and  3mn
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: The sum of 3x and 2, then raised to the second power
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: The difference of 2a and b , then raised to the second power
Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and
collect like terms: The product of 2 and  4x  1 , then added to 5x  2
Write as an algebraic expression using parentheses where appropriate, then remove the
parentheses and simplify: The product of  3x and x 2 , then raised to the third power
86
Write as an algebraic expression using parentheses where appropriate, then remove the
parentheses and simplify:
The quotient of 4a 12 and a 2 , then raised to the second power
Write as an algebraic expression using parentheses where appropriate, then remove the
parentheses and simplify:  a 3 raised to seventh power, then multiplied by a
Write as an algebraic expression using parentheses where appropriate, then remove the
parentheses and simplify: xy 7 raised to the fifth power , then divided by y 3
Write as an algebraic expression using parentheses where appropriate, then remove the
parentheses and simplify: 3ab 3 squared, and then multiplied by b .
Write as an algebraic expression, rewrite without parentheses and collect like terms, if possible;
2
y
Subtract  xy  from 3 yx  y
3
5
Write as an algebraic expression, rewrite without parentheses and collect like terms, if possible;
The sum of 3 x 4  x 2  1 and twice the expression  4 x 4  2
2VD
Rewriting algebraic expressions in terms of a designated variable requiring
“challenging” simplifications.
Express (m  n)( m 2  mn  n 2 ) in terms of x , if m  2 , and n  3x . Simplify.
Express (m  n)( m 2  mn  n 2 ) in terms of m , if n  m . Simplify.
Write the expression (mn) 2  mn 2 in terms of m, if n  m . Simplify.
Write the expression (mn) 2  mn 2 in terms of s , if m  2s, n  s . Simplify.
Write the expression m 3 n  2m 2 n 2 in terms of s , if m  s, n  2s . Simplify.
Write the expression m 3 n  2m 2 n 2 in terms of s , if m  2s 2 , n  s 4 . Simplify.
Write the expression m 3 n  2m 2 n 2 in terms of s , if n  m  s . Simplify.
Write the expression (m  n) 2  (m  n) 2 in terms of s , if m  2s, n  3s . Simplify.
Write the expression (m  n) 2  (m  n) 2 in terms of m, if n  m . Simplify.
87
Write the expression (m  n) 2  (m  n) 2 in terms of m, if n  m . Simplify.
Rewrite the expression (a  b)( a  b) in terms of x , if it is given that a  5x and b  2  x .
Simplify.
3VD
“Non standard” factoring of the difference of squares, difference (sum) of
cubes (formula for the difference/sum of cubes given but students need to
know the formula for the difference of squares).
Factor and then simplify the following expression m 2  (1  3m) 2
Factor and then simplify the following expression m 2  (2m  1) 2
Factor and then simplify the following expression ( x  1) 2  (3x  5) 2
Factor and then simplify the following expression (2a  3) 2  9a 2
Factor and then simplify the following expression (a  2b  3) 2  9
Factor and then simplify the following expression ( x 2  3 y 2 ) 2  (2 x 2  1) 2
Factor and then simplify the following expression (3x  2) 2  ( y  5x  1) 2
a 3  b 3  (a  b)( a 2  ab  b 2 ) . Factor the following
The following formula is true:
expression using the above formula: y 3 x 3  64
.
a 3  b 3  (a  b)( a 2  ab  b 2 ) . Factor the following
1
 8y3
expression using the above formula:
3
a
The following formula is true:
a 3  b 3  (a  b)( a 2  ab  b 2 ) . Factor the following
The following formula is true:
expression using the above formula: y 6  1 .
a 3  b 3  (a  b)( a 2  ab  b 2 ) . Factor the following
The following formula is true:
27 x 3  8 y 3
expression using the above formula:
88
The following formula is true:
a 3  b 3  (a  b)(a 2  ab  b 2 ) . Factor the following
expression using the above formula:
y3
 125 x 3
8
.
The following formula is true:
a 3  b 3  (a  b)( a 2  ab  b 2 ) . Factor the following
expression using the above formula:
64  ( xy) 3
a 3  b 3  (a  b)(a 2  ab  b 2 ) . Factor the following
1
27 y 3 
expression using the above formula:
8
The following formula is true:
The following formula is true:
a 3  b 3  (a  b)(a 2  ab  b 2 ) . Factor the following
expression using the above formula:
8  (mn) 3
4VD
Replacing up to 2 variables with an algebraic expression and solving the
resulting equation.
Let a  3x  2 and b   x  2 . Find the value of x so that the following is true:
ab
Let a  3x  2 and b   x  2 . Find the value of x so that the following is true:
a  2b
Let A  2 x, B   x, . Find x , so the following are true:
A B
 1
3 8
Let A  2 x  1, B   x  2, . Find x , so the following are true: A  B  0
Let A  2 x, B   x, . Find x , so the following are true: A  3B  1
Let P  3( x  1), Q  4 x  5 . Find x such that the following is true: P  Q
Let P  3( x  1), Q  4 x  5 . Find x such that the following is true:
Q
P
2
Let x  3a  2,
y  2a  1 . Find a , so the following are true: x  y  3
y z
Let y  2a  1, z  2  a . Find a , so the following are true: 
2 4
89
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