Zillions of Practice Problems
for
Beginning Algebra
Stanley F. Schmidt, Ph.D.
Polka Dot Publishing
What This Book Is All About
n the Life of Fred: Beginning Algebra book, there areYour Turn to Play
sections after each topic is presented. And eachYour Turn to Play offers
complete solutions to each question.
At the end of each chapter are six problem sets. The answers to
each question are either given in the book or in the study guide, Fred’s
Home Companion: Beginning Algebra.
In addition, Fred’s Home Companion: Beginning Algebra offers
additional problems with their answers.
I
I thought that all those problems would be enough.
I still think all those problems are enough.
As they say in German: Genug ist genug.✶
{
{
{
Some of my readers have written to me, “Enough is not enough.
We want more. We want tons of problems along with completely worked
out solutions. We need the drill-and-kill approach. My kids are not crying
when the read Fred—how could they be learning math without crying?”
If you are a part of those some readers, then this Zillions of
Practice Problems was written for you.
HOW THIS BOOK IS ORGANIZED
Life of Fred: Beginning Algebra is in twelve chapters. This book
is in twelve chapters.
As you work through each chapter in Life of Fred: Beginning
Algebra, you can do as many of the problems as you like in the
corresponding chapter in this book.
✶
“Enough is enough.” I learned this nifty phrase when I studied German in high
school.
7
CHAPTERS IN THIS BOOK
Each chapter in this book is divided into two parts.
✯ The first part takes each topic and offers a zillion problems.
✯ The second part is called the Mixed Bag. It consists of a variety of
problems from the chapter and review problems from the beginning of the
book up to that point.
COMPLETE SOLUTIONS
Every problem (gasp!) will receive a detailed solution—not just the
answer.
ELIMINATING TEMPTATION
The solutions and answers are all given in the back half of the
book. The first question in this book is numbered “45.” The second one is
“888.” In most ordinary practice books, they are numbered, “1, 2, 3 . . .”
which is really silly when you think about it. In those books, when you
look up the answer to “1” you might accidentally see the answer to “2” and
that would spoil all the fun.
8
Contents
Chapter 1
Numbers and Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
finite and infinite sets
natural numbers, whole numbers, integers
set notation
negative numbers
ratios
the empty set
Chapter 2
The Integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
less than (<) and the number line
multiplication
proportion
π
coefficients
Chapter 3
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
solving equations with continued ratios
formulas from geometry
consecutive numbers
set builder notation
reflexive, symmetric and distributive properties
rational numbers
order of operations
distance = (rate)(time) problems
Chapter 4
Motion and Mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
motion problems
price and quantity problems
age problems
mixture problems
Chapter 5
Two Unknowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
solving two equations, two unknowns by elimination
union of sets
transposing
graphing of points
mean, mode, and median averages
graphing linear equations
graphing any equation
word problems in two unknowns
Chapter 6
Exponents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
solving two equations, two unknowns by graphing
exponents: xmxn, (xm)n and xm ÷ xn
solving two equations, two unknowns by substitution
inconsistent and dependent equations
commutative laws
factorials and ordering
negative and zero exponents
volumes
9
Chapter 7
Factoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
multiplying binomials and trinomials
polynomials
first kind of factoring: common factors
factoring easy trinomials: x² + bx + c
factoring a difference of squares
factoring by grouping
factoring harder trinomials: ax² + bx + c
solving quadratic equations by factoring
Chapter 8
Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
solving equations containing fractions
simplifying fractions
adding and subtracting fractions
multiplying and dividing fractions
complex fractions
“job” problems
Chapter 9
Square Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
simplifying square roots
solving pure quadratic equations
principal square roots
the real numbers
the irrational numbers
cube roots and indexes
fractional and negative exponents
solving radical equations
rationalizing the denominator
Chapter 10
Quadratic Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
solving quadratic equations by completing the square
the quadratic formula
long division of polynomials
Chapter 11
Functions and Slope.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
definition of a function
domain, codomain, image
range of a function
slope
slope-intercept (y = mx + b) form of the line
Chapter 12
Inequalities and Absolute Value. . . . . . . . . . . . . . . . . . . . 64
graphing inequalities in two dimensions
division by zero
solving linear inequalities with one unknown
All the answers worked out in complete detail . . . . . . . . . . . . . . . . . . . . . 68
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10
Chapter Four
Mixture problems
6179. Joe wants to make 30 pounds of chili seasoning that contains 30%
cayenne pepper. He has one sack of mild chili seasoning that is 10%
cayenne and one sack of hot chili seasoning that is 40% cayenne.
How much of each sack should he use?
6704. Darlene has some pancake mix that contains 2% sugar. She calls
that her bitter mix. She has some that contains 5% sugar. She calls that
her too-sweet mix.
She wants to make 60 pounds of pancake mix that has 3% sugar.
How much of each mix should she use?
8975. You are a veterinarian and you need to give
a marten a shot to ease its pain. You need to give
it 9 cc that contains 6% morphine. You have
some 4% and some 10%. How much of each
should you mix together?
(Side note: It is often nice in medical work
to get the mathematics right. Too little pain killer and the marten remains
in pain. Too much morphine and the martin will look like:
dead
marten
201. We need to ship 600 cars to Freedonia, and the order specifies that
28% of those cars should have tape players.
The 40% of the older cars have tape players, and 15% of the newer
cars do. How many older cars should we ship?
1850. Darlene has weak dishwashing liquid that contains 17% soap. She
also has strong dishwashing liquid that contains 37% soap.
Darlene just purchased a dishwashing
robot. When she read the directions, she found
out that the robot would only work if you gave it
444 ounces of dishwashing liquid that contained
22% soap.
How much of each dishwashing liquid
should Darlene use?
(Joe never washes his dishes because he
only uses paper plates.)
27
Chapter Four
Second part: the Mixed Bag: a variety of problems from
this chapter and previous material
2226. Joe likes his cereal with exactly 8% flour in it. (The rest is sugar
and artificial flavorings.) He asked Darlene take his two
boxes of cereal—Captain Mousebait (7% flour) and
Sergeant Sugar (11% flour)—and mix them together to
get 300 ounces.
How much of each cereal should Darlene use?
1928. Haydn wrote the “Sun” quartets five times as long ago as Robert
Frost wrote “In the Clearing.”
Sixteen years from now, the “Sun” quartets will be four times as
old as “In the Clearing.”
How old is “In the Clearing” today?
2258. Joe decided one night to read “In the Clearing.” He first read the
poem at the rate of 40 words/minute. Then he decided to read it
backwards. Reading backwards slowed his reading rate down to 5
words/minute.
It took him a total of 27 minutes to read the poem in both
directions. How long did it take Joe to read the poem in the normal
direction?
2280. Joe liked to eat jelly beans while he watched television every
evening. He bought red jelly beans at a cost of $5/lb. He bought green
jelly beans at $6/lb. He spent a total of $488. He bought 8 more pounds
of the green than of the red. How many pounds of red jelly beans did he
buy?
4507. This particular cat likes some bones in his fish. (It’s a good source
of calcium.) This cat prefers a fish mixture
with 7% bones.
How many pounds of 5% bones
should he mix with 8% bones to obtain 36
pounds of fish with 7% bones?
28
Chapter Five
First part: Problems on Each Topic
Solving Two Equations, Two Unknowns by Elimination
1891. Solve
2292. Solve
2904. Solve
4010. Solve
4505. Solve
4519. Solve
7x + 3y = 43
 4x – 3y = 1
5x + 3y = 18
 –5x – 10y = 10
10x + 6y = 62
 8x – 2y = 2
4x + 3y = 3
 12x – 7y = –71
6x + 6y = –18
 7x + 8y = –25
6x + 3y = 6
 9x + 10y = 42
Union of Sets
3328. What is the union of {3, 4, 5} and {4, ♡, w}?
4280. {8} ∪ {1, 2} = ?
4500. {x | x is an integer and x > 4} ∪ {y | y is a whole number and y < 6}
600. {1, 2, 3} ∪ {3, 2, 1}
800. { } ∪ {✇}
924. For any set A, it is always true that A ∪ { } = A.
For any number x, it is always true that x times 1 equals x.
{ } is called the identity element for union.
1 is called the identity element for multiplication.
What is the identity element for addition?
1893. { y | y is an integer and y < –4} ∪ { z | z is a natural number}
29
The Complete Solutions and Answers
6179.
Joe wants to make 30 pounds of chili seasoning that contains 30% cayenne pepper. He
has one sack of mild chili seasoning that is 10% cayenne and one sack of hot chili seasoning that is
40% cayenne.
How much of each sack should he use?
❶ Using the Let x = approach:
Let x = the number of pounds of mild seasoning he uses.
(Since we
want to find out two different things, we can let x equal either one of them.)
Then 30 – x = the number of pounds of hot seasoning he uses.
(If he
uses x pounds of the mild and 30 pounds altogether, then he has to use 30 – x of the hot.)
The first trick for doing mixture problems is to compute the
amount of the “ingredient” in the sacks you are combining.
Then 0.1x = the pounds of cayenne in the mild seasoning that he
uses. (10% = 0.1 He is using x pounds of which 10% is cayenne.)
Then 0.4(30 – x) = the pounds of cayenne contributed by the hot
seasoning. (He is using 30 – x pounds of which 40% is cayenne.)
The second trick in mixture problems is to compute the
amount of the “ingredient” in the final mixture that is desired. In this case,
we want 30 pounds with 30% cayenne. So the amount of the ingredient in
the final mixture is 0.3(30) which is 9 pounds of cayenne.
You will get a lot of practice with decimals and percents in these
mixture problems. Note that to change a percent like 82.357% into a
decimal, you move the decimal two places to the left and get 0.82357. The
old saying: When the percent has left, you move the decimal to the left.
Finally, the equation is 0.1x + 0.4(30 – x) = 9 (since the amount
of ingredient (cayenne) contributed by the mild is 0.1x and by the hot is
0.4(30 – x), and the total amount in the desired mixture is 9.
distributive property
0.1x + 12 – 0.4x = 9
combine 0.1x and –0.4x
–0.3x + 12 = 9
add 0.3x to both sides
12 = 9 + 0.3x
subtract 9 from both sides
3 = 0.3x
divide both sides by 0.3
10 = x
Joe uses 10 pounds of the mild seasoning and 20 pounds (which is 30 – x)
of the hot seasoning. (The question asked how much of each sack.)
❷ When you fill in the Six Pretty Boxes . . .
218
lbs. of cayenne
% of cayenne in the
seasoning
lbs. of each seasoning
used
mild seasoning
0.1x
10%
x
hot seasoning
0.4(30 – x)
40%
30 – x
Index
age problems
#680. . . . . . . . . . . . . . . . . . .
#6122. . . . . . . . . . . . . . . . . .
#5570. . . . . . . . . . . . . . . . . .
#1928. . . . . . . . . . . . . . . . . .
#5600. . . . . . . . . . . . . . . . . .
#4716. . . . . . . . . . . . . . . . . .
#4610. . . . . . . . . . . . . . . . . .
area rectangle
#6411. . . . . . . . . . . . . . . . . .
#7304. . . . . . . . . . . . . . . . . .
#5304. . . . . . . . . . . . . . . . . .
cardinality of a set
#3000. . . . . . . . . . . . . . . . . .
#8888. . . . . . . . . . . . . . . . . .
#3700. . . . . . . . . . . . . . . . . .
#245. . . . . . . . . . . . . . . . . . .
circumference
#7103. . . . . . . . . . . . . . . . . .
coefficients
#3302. . . . . . . . . . . . . . . . . .
#5700. . . . . . . . . . . . . . . . . .
#2898. . . . . . . . . . . . . . . . . .
#4278. . . . . . . . . . . . . . . . . .
#7023. . . . . . . . . . . . . . . . . .
#6421. . . . . . . . . . . . . . . . . .
commutative laws
#243. . . . . . . . . . . . . . . . . . .
#402. . . . . . . . . . . . . . . . . . .
#541. . . . . . . . . . . . . . . . . . .
#501. . . . . . . . . . . . . . . . . . .
#22. . . . . . . . . . . . . . . . . . . .
#4711. . . . . . . . . . . . . . . . . .
consecutive numbers
#7041. . . . . . . . . . . . . . . . . .
#8695. . . . . . . . . . . . . . . . . .
#6274. . . . . . . . . . . . . . . . . .
#6408. . . . . . . . . . . . . . . . . .
#8332. . . . . . . . . . . . . . . . . .
#9081. . . . . . . . . . . . . . . . . .
#7777. . . . . . . . . . . . . . . . . .
26
26
26
28
37
44
44
20
20
44
14
18
43
49
18
18
18
18
19
32
38
35
35
35
35
35
37
21
21
21
21
21
21
21
#6124. . . . . . .
#5231. . . . . . .
#6748. . . . . . .
continued ratios
#7020. . . . . . .
#6148. . . . . . .
#8912. . . . . . .
#4503. . . . . . .
#2001. . . . . . .
cube roots and indexes
#303. . . . . . . .
#6344. . . . . . .
#6505. . . . . . .
#7225. . . . . . .
#867. . . . . . . .
#2890. . . . . . .
#646. . . . . . . .
#8212. . . . . . .
distance = (rate)(time)
#8870. . . . . . .
#525. . . . . . . .
#590. . . . . . . .
#6127. . . . . . .
#9303. . . . . . .
#8989. . . . . . .
#443. . . . . . . .
#6405. . . . . . .
#957. . . . . . . .
#979. . . . . . . .
#1100. . . . . . .
#6303. . . . . . .
distributive property
#6902. . . . . . .
#8898. . . . . . .
#527. . . . . . . .
#351. . . . . . . .
#6149. . . . . . .
division by zero
#1906. . . . . . .
#2444. . . . . . .
#6924. . . . . . .
. . . . . . . . . . 24
. . . . . . . . . . 37
. . . . . . . . . . 38
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20
20
20
24
49
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51
51
51
51
53
57
61
66
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22
23
23
23
23
23
23
23
24
24
48
53
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21
22
22
22
24
. . . . . . . . . . 64
. . . . . . . . . . 64
. . . . . . . . . . 65
257
equations containing fractions
#8008. . . . . . . . . . . . . . . . . .
#2566. . . . . . . . . . . . . . . . . .
#4360. . . . . . . . . . . . . . . . . .
#7803. . . . . . . . . . . . . . . . . .
#4537. . . . . . . . . . . . . . . . . .
#2727. . . . . . . . . . . . . . . . . .
#1203. . . . . . . . . . . . . . . . . .
#2108. . . . . . . . . . . . . . . . . .
exponents
#1277. . . . . . . . . . . . . . . . . .
#9012. . . . . . . . . . . . . . . . . .
#8418. . . . . . . . . . . . . . . . . .
#51. . . . . . . . . . . . . . . . . . . .
#5712. . . . . . . . . . . . . . . . . .
#6277. . . . . . . . . . . . . . . . . .
#6311. . . . . . . . . . . . . . . . . .
#6420. . . . . . . . . . . . . . . . . .
#7788. . . . . . . . . . . . . . . . . .
#6742. . . . . . . . . . . . . . . . . .
#6800. . . . . . . . . . . . . . . . . .
#7030. . . . . . . . . . . . . . . . . .
#7306. . . . . . . . . . . . . . . . . .
#9306. . . . . . . . . . . . . . . . . .
#7116. . . . . . . . . . . . . . . . . .
#498. . . . . . . . . . . . . . . . . . .
#1583. . . . . . . . . . . . . . . . . .
#648. . . . . . . . . . . . . . . . . . .
#7791. . . . . . . . . . . . . . . . . .
factorials and ordering
#1316. . . . . . . . . . . . . . . . . .
#1375. . . . . . . . . . . . . . . . . .
#1602. . . . . . . . . . . . . . . . . .
#1667. . . . . . . . . . . . . . . . . .
#805. . . . . . . . . . . . . . . . . . .
#871. . . . . . . . . . . . . . . . . . .
#1725. . . . . . . . . . . . . . . . . .
#4531. . . . . . . . . . . . . . . . . .
#4468. . . . . . . . . . . . . . . . . .
#3850. . . . . . . . . . . . . . . . . .
factoring— common factors
#5592. . . . . . . . . . . . . . . . . .
#4911. . . . . . . . . . . . . . . . . .
#5100. . . . . . . . . . . . . . . . . .
#5201. . . . . . . . . . . . . . . . . .
258
45
45
45
45
48
48
58
62
14
33
33
33
33
33
33
33
33
33
33
33
33
33
37
58
58
62
67
35
35
35
35
35
35
36
37
37
48
40
40
40
40
#5302. . . . . . . . . . . . . . .
#5450. . . . . . . . . . . . . . .
#358. . . . . . . . . . . . . . . .
#4835. . . . . . . . . . . . . . .
factoring— difference of squares
#1635. . . . . . . . . . . . . . .
#1680. . . . . . . . . . . . . . .
#1726. . . . . . . . . . . . . . .
#1750. . . . . . . . . . . . . . .
#1898. . . . . . . . . . . . . . .
#1939. . . . . . . . . . . . . . .
#2000. . . . . . . . . . . . . . .
#2050. . . . . . . . . . . . . . .
#2310. . . . . . . . . . . . . . .
#2390. . . . . . . . . . . . . . .
#2506. . . . . . . . . . . . . . .
#4030. . . . . . . . . . . . . . .
#4125. . . . . . . . . . . . . . .
factoring— easy trinomials
#1003. . . . . . . . . . . . . . .
#1099. . . . . . . . . . . . . . .
#1200. . . . . . . . . . . . . . .
#1280. . . . . . . . . . . . . . .
#1387. . . . . . . . . . . . . . .
#1520. . . . . . . . . . . . . . .
#407. . . . . . . . . . . . . . . .
#873. . . . . . . . . . . . . . . .
#522. . . . . . . . . . . . . . . .
#544. . . . . . . . . . . . . . . .
#807. . . . . . . . . . . . . . . .
#951. . . . . . . . . . . . . . . .
#4607. . . . . . . . . . . . . . .
#990. . . . . . . . . . . . . . . .
factoring— grouping
#2608. . . . . . . . . . . . . . .
#2800. . . . . . . . . . . . . . .
#2949. . . . . . . . . . . . . . .
#3009. . . . . . . . . . . . . . .
#3162. . . . . . . . . . . . . . .
#3215. . . . . . . . . . . . . . .
#3260. . . . . . . . . . . . . . .
#3380. . . . . . . . . . . . . . .
#4824. . . . . . . . . . . . . . .
.
.
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.
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.
40
40
40
44
.
.
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41
41
41
41
41
41
41
41
41
41
41
43
43
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41
41
41
41
41
41
41
41
41
41
41
41
44
58
.
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.
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.
41
41
41
42
42
42
42
42
43
factoring— harder trinomials
#3407. . . . . . . . . . . . . . . . .
#3505. . . . . . . . . . . . . . . . .
#3606. . . . . . . . . . . . . . . . .
#4000. . . . . . . . . . . . . . . . .
#4260. . . . . . . . . . . . . . . . .
#4290. . . . . . . . . . . . . . . . .
#4350. . . . . . . . . . . . . . . . .
#4477. . . . . . . . . . . . . . . . .
#4533. . . . . . . . . . . . . . . . .
#4831. . . . . . . . . . . . . . . . .
#5009. . . . . . . . . . . . . . . . .
#547. . . . . . . . . . . . . . . . . .
#599. . . . . . . . . . . . . . . . . .
fractional and negative exponents
#9275. . . . . . . . . . . . . . . . .
#9073. . . . . . . . . . . . . . . . .
#8919. . . . . . . . . . . . . . . . .
#1003. . . . . . . . . . . . . . . . .
#8210. . . . . . . . . . . . . . . . .
#1225. . . . . . . . . . . . . . . . .
fractions— adding and subtracting
#$3393. . . . . . . . . . . . . . . .
#1060. . . . . . . . . . . . . . . . .
#2954. . . . . . . . . . . . . . . . .
#3222. . . . . . . . . . . . . . . . .
#3900. . . . . . . . . . . . . . . . .
#3424. . . . . . . . . . . . . . . . .
#3610. . . . . . . . . . . . . . . . .
#3870. . . . . . . . . . . . . . . . .
#6021. . . . . . . . . . . . . . . . .
#3960. . . . . . . . . . . . . . . . .
#3955. . . . . . . . . . . . . . . . .
#3411. . . . . . . . . . . . . . . . .
fractions— complex
#1753. . . . . . . . . . . . . . . . .
#1835. . . . . . . . . . . . . . . . .
#2051. . . . . . . . . . . . . . . . .
fractions— multiplying and dividing
#3965. . . . . . . . . . . . . . . . .
#4401. . . . . . . . . . . . . . . . .
#5117. . . . . . . . . . . . . . . . .
#5240. . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
42
42
42
42
42
42
42
42
42
44
44
49
62
.
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.
.
.
51
51
51
51
52
52
.
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46
46
46
46
46
46
46
46
46
48
48
48
. 47
. 47
. 49
.
.
.
.
46
46
46
46
functions, domains, codomains, images
#1047. . . . . . . . . . . . . . . . . 59
#1122. . . . . . . . . . . . . . . . . 59
#869. . . . . . . . . . . . . . . . . . 59
#644. . . . . . . . . . . . . . . . . . 59
#1527. . . . . . . . . . . . . . . . . 59
#993. . . . . . . . . . . . . . . . . . 59
#1205. . . . . . . . . . . . . . . . . 60
#1390. . . . . . . . . . . . . . . . . 60
#1263. . . . . . . . . . . . . . . . . 61
#275. . . . . . . . . . . . . . . . . . 62
#258. . . . . . . . . . . . . . . . . . 63
#1140. . . . . . . . . . . . . . . . . 66
#6955. . . . . . . . . . . . . . . . . 66
going from 4 to 11 problems
#593. . . . . . . . . . . . . . . . . . 13
#917. . . . . . . . . . . . . . . . . . 13
#445. . . . . . . . . . . . . . . . . . 13
#494. . . . . . . . . . . . . . . . . . 13
#864. . . . . . . . . . . . . . . . . . 13
#532. . . . . . . . . . . . . . . . . . 13
#980. . . . . . . . . . . . . . . . . . 13
#685. . . . . . . . . . . . . . . . . . 13
#732. . . . . . . . . . . . . . . . . . 13
#207. . . . . . . . . . . . . . . . . . 14
#1585. . . . . . . . . . . . . . . . . 14
#4965. . . . . . . . . . . . . . . . . 54
graphing any equation
#5709. . . . . . . . . . . . . . . . . 31
#1721. . . . . . . . . . . . . . . . . 31
#6150. . . . . . . . . . . . . . . . . 32
#7800. . . . . . . . . . . . . . . . . 32
#1832. . . . . . . . . . . . . . . . . 32
#2777. . . . . . . . . . . . . . . . . 37
#595. . . . . . . . . . . . . . . . . . 58
graphing inequalities in two dimensions
#1050. . . . . . . . . . . . . . . . . 64
#1136. . . . . . . . . . . . . . . . . 64
#3022. . . . . . . . . . . . . . . . . 64
#1250. . . . . . . . . . . . . . . . . 64
#1529. . . . . . . . . . . . . . . . . 64
#1730. . . . . . . . . . . . . . . . . 64
#2527. . . . . . . . . . . . . . . . . 64
#4180. . . . . . . . . . . . . . . . . 64
#1241. . . . . . . . . . . . . . . . . 65
259
graphing linear equations
#1633. . . . . . . . . . . . . . . . . .
#208. . . . . . . . . . . . . . . . . . .
#9087. . . . . . . . . . . . . . . . . .
#9000. . . . . . . . . . . . . . . . . .
#470. . . . . . . . . . . . . . . . . . .
#530. . . . . . . . . . . . . . . . . . .
#999. . . . . . . . . . . . . . . . . . .
#9011. . . . . . . . . . . . . . . . . .
#5234. . . . . . . . . . . . . . . . . .
#997. . . . . . . . . . . . . . . . . . .
#255. . . . . . . . . . . . . . . . . . .
#2304. . . . . . . . . . . . . . . . . .
graphing points
#4447. . . . . . . . . . . . . . . . . .
#4520. . . . . . . . . . . . . . . . . .
#4900. . . . . . . . . . . . . . . . . .
#918. . . . . . . . . . . . . . . . . . .
#3303. . . . . . . . . . . . . . . . . .
#8915. . . . . . . . . . . . . . . . . .
inconsistent and dependent equations
#1894. . . . . . . . . . . . . . . . . .
#1935. . . . . . . . . . . . . . . . . .
#2100. . . . . . . . . . . . . . . . . .
#2305. . . . . . . . . . . . . . . . . .
#5225. . . . . . . . . . . . . . . . . .
integers
#518. . . . . . . . . . . . . . . . . . .
#726. . . . . . . . . . . . . . . . . . .
#3116. . . . . . . . . . . . . . . . . .
integers— addition
#986. . . . . . . . . . . . . . . . . . .
#188. . . . . . . . . . . . . . . . . . .
#855. . . . . . . . . . . . . . . . . . .
#240. . . . . . . . . . . . . . . . . . .
#333. . . . . . . . . . . . . . . . . . .
#977. . . . . . . . . . . . . . . . . . .
#714. . . . . . . . . . . . . . . . . . .
#460. . . . . . . . . . . . . . . . . . .
#640. . . . . . . . . . . . . . . . . . .
#616. . . . . . . . . . . . . . . . . . .
#944. . . . . . . . . . . . . . . . . . .
#441. . . . . . . . . . . . . . . . . . .
#651. . . . . . . . . . . . . . . . . . .
#2203. . . . . . . . . . . . . . . . . .
260
31
31
31
31
31
31
31
32
44
61
63
66
30
30
30
30
30
32
34
34
34
34
34
12
12
15
12
12
12
12
12
12
12
12
12
12
12
14
14
18
#1836. . . . . . . . . . . . .
#3855. . . . . . . . . . . . .
#58. . . . . . . . . . . . . . .
integers— multiply and divide
#4444. . . . . . . . . . . . .
#6121. . . . . . . . . . . . .
#1409. . . . . . . . . . . . .
#1111. . . . . . . . . . . . .
#1748. . . . . . . . . . . . .
#1830. . . . . . . . . . . . .
#3256. . . . . . . . . . . . .
#1625. . . . . . . . . . . . .
#1371. . . . . . . . . . . . .
#1001. . . . . . . . . . . . .
#6181. . . . . . . . . . . . .
#5525. . . . . . . . . . . . .
#9009. . . . . . . . . . . . .
#6300. . . . . . . . . . . . .
#6500. . . . . . . . . . . . .
#6288. . . . . . . . . . . . .
#436. . . . . . . . . . . . . .
integers— subtraction
#742. . . . . . . . . . . . . .
#356. . . . . . . . . . . . . .
#521. . . . . . . . . . . . . .
#959. . . . . . . . . . . . . .
#585. . . . . . . . . . . . . .
#655. . . . . . . . . . . . . .
#682. . . . . . . . . . . . . .
#729. . . . . . . . . . . . . .
#989. . . . . . . . . . . . . .
#840. . . . . . . . . . . . . .
#866. . . . . . . . . . . . . .
#891. . . . . . . . . . . . . .
#925. . . . . . . . . . . . . .
#913. . . . . . . . . . . . . .
#1313. . . . . . . . . . . . .
#907. . . . . . . . . . . . . .
#5606. . . . . . . . . . . . .
job problems
#3430. . . . . . . . . . . . .
#2403. . . . . . . . . . . . .
#2960. . . . . . . . . . . . .
#875. . . . . . . . . . . . . .
#7700. . . . . . . . . . . . .
. . . . 18
. . . . 43
. . . . 49
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15
15
16
16
16
16
16
16
16
16
18
18
19
19
38
38
49
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12
12
12
12
12
12
12
12
12
12
12
12
12
12
14
14
38
.
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.
.
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.
.
.
.
47
47
47
49
54
#4177. . . . . . . . . . . . . . . .
#1130. . . . . . . . . . . . . . . .
#1692. . . . . . . . . . . . . . . .
#1420. . . . . . . . . . . . . . . .
less than <
#1362. . . . . . . . . . . . . . . .
#1600. . . . . . . . . . . . . . . .
#1663. . . . . . . . . . . . . . . .
#1717. . . . . . . . . . . . . . . .
#1921. . . . . . . . . . . . . . . .
#2270. . . . . . . . . . . . . . . .
#2275. . . . . . . . . . . . . . . .
#3333. . . . . . . . . . . . . . . .
#5585. . . . . . . . . . . . . . . .
#960. . . . . . . . . . . . . . . . .
#6418. . . . . . . . . . . . . . . .
#6215. . . . . . . . . . . . . . . .
#360. . . . . . . . . . . . . . . . .
#7791. . . . . . . . . . . . . . . .
linear inequalities
#2303. . . . . . . . . . . . . . . .
#2525. . . . . . . . . . . . . . . .
#2635. . . . . . . . . . . . . . . .
#2896. . . . . . . . . . . . . . . .
#5099. . . . . . . . . . . . . . . .
#5454. . . . . . . . . . . . . . . .
#5608. . . . . . . . . . . . . . . .
long division of polynomials
#3015. . . . . . . . . . . . . . . .
#992. . . . . . . . . . . . . . . . .
#642. . . . . . . . . . . . . . . . .
#6498. . . . . . . . . . . . . . . .
#6831. . . . . . . . . . . . . . . .
#251. . . . . . . . . . . . . . . . .
#5022. . . . . . . . . . . . . . . .
mean, median, and mode averages
#6190. . . . . . . . . . . . . . . .
#8900. . . . . . . . . . . . . . . .
#8340. . . . . . . . . . . . . . . .
#6735. . . . . . . . . . . . . . . .
#5222. . . . . . . . . . . . . . . .
#5707. . . . . . . . . . . . . . . .
#6130. . . . . . . . . . . . . . . .
#8977. . . . . . . . . . . . . . . .
#6302. . . . . . . . . . . . . . . .
.
.
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.
58
62
63
67
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15
15
15
15
15
15
15
15
15
15
32
38
49
67
.
.
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.
.
.
.
.
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65
65
65
65
65
65
65
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57
57
57
57
57
57
67
.
.
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
30
30
30
30
30
30
30
30
30
#6417. . . . . . . . . . . . . . . . .
#4522. . . . . . . . . . . . . . . . .
#4282. . . . . . . . . . . . . . . . .
#1222. . . . . . . . . . . . . . . . .
#5203. . . . . . . . . . . . . . . . .
mixture problems
#1850. . . . . . . . . . . . . . . . .
#201. . . . . . . . . . . . . . . . . .
#6179. . . . . . . . . . . . . . . . .
#6704. . . . . . . . . . . . . . . . .
#4507. . . . . . . . . . . . . . . . .
#2226. . . . . . . . . . . . . . . . .
#5107. . . . . . . . . . . . . . . . .
#6323. . . . . . . . . . . . . . . . .
#3550. . . . . . . . . . . . . . . . .
#2892. . . . . . . . . . . . . . . . .
#6333. . . . . . . . . . . . . . . . .
motion problems
#1709. . . . . . . . . . . . . . . . .
#8970. . . . . . . . . . . . . . . . .
#9065. . . . . . . . . . . . . . . . .
#9225. . . . . . . . . . . . . . . . .
multiplying binomials and trinomials
#2504. . . . . . . . . . . . . . . . .
#2606. . . . . . . . . . . . . . . . .
#2790. . . . . . . . . . . . . . . . .
#2946. . . . . . . . . . . . . . . . .
#300. . . . . . . . . . . . . . . . . .
#3211. . . . . . . . . . . . . . . . .
#3404. . . . . . . . . . . . . . . . .
#3502. . . . . . . . . . . . . . . . .
#3601. . . . . . . . . . . . . . . . .
#4017. . . . . . . . . . . . . . . . .
#7403. . . . . . . . . . . . . . . . .
#7606. . . . . . . . . . . . . . . . .
#4354. . . . . . . . . . . . . . . . .
multiplying by zero
#2939. . . . . . . . . . . . . . . . .
natural numbers
#405. . . . . . . . . . . . . . . . . .
#589. . . . . . . . . . . . . . . . . .
#6210. . . . . . . . . . . . . . . . .
negative and zero exponents
#2500. . . . . . . . . . . . . . . . .
#1834. . . . . . . . . . . . . . . . .
30
32
32
32
44
27
27
27
27
28
28
44
48
54
58
67
25
25
25
25
39
39
39
39
39
39
39
39
39
39
39
39
43
32
11
11
32
36
36
261
#1936. . . . . . . . . . . . . .
#2308. . . . . . . . . . . . . .
#2400. . . . . . . . . . . . . .
#2603. . . . . . . . . . . . . .
#3370. . . . . . . . . . . . . .
#7309. . . . . . . . . . . . . .
#7814. . . . . . . . . . . . . .
#2745. . . . . . . . . . . . . .
#1244. . . . . . . . . . . . . .
number, rate and time problems
#8327. . . . . . . . . . . . . .
#920. . . . . . . . . . . . . . .
#1919. . . . . . . . . . . . . .
#2258. . . . . . . . . . . . . .
#5300. . . . . . . . . . . . . .
#7120. . . . . . . . . . . . . .
#4399. . . . . . . . . . . . . .
#5004. . . . . . . . . . . . . .
#7333. . . . . . . . . . . . . .
order of operations
#1844. . . . . . . . . . . . . .
#8413. . . . . . . . . . . . . .
#6730. . . . . . . . . . . . . .
#8000. . . . . . . . . . . . . .
#1630. . . . . . . . . . . . . .
#870. . . . . . . . . . . . . . .
#5215. . . . . . . . . . . . . .
pi π
#2249. . . . . . . . . . . . . .
#2289. . . . . . . . . . . . . .
#2291. . . . . . . . . . . . . .
#3299. . . . . . . . . . . . . .
#4811. . . . . . . . . . . . . .
#6204. . . . . . . . . . . . . .
#7103. . . . . . . . . . . . . .
#8004. . . . . . . . . . . . . .
#650. . . . . . . . . . . . . . .
polynomials
#4470. . . . . . . . . . . . . .
#4712. . . . . . . . . . . . . .
#4600. . . . . . . . . . . . . .
#4275. . . . . . . . . . . . . .
#4822. . . . . . . . . . . . . .
#4535. . . . . . . . . . . . . .
262
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36
36
36
36
37
38
53
57
66
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25
25
26
28
37
38
43
44
66
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22
22
22
22
22
24
44
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17
17
17
17
17
17
18
32
57
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39
39
39
39
40
44
price and quantity problems
#2272. . . . . . . . . . . . . . . . . 26
#2280. . . . . . . . . . . . . . . . . 28
principal square roots
#4538. . . . . . . . . . . . . . . . . 50
#4611. . . . . . . . . . . . . . . . . 50
#4718. . . . . . . . . . . . . . . . . 50
#4848. . . . . . . . . . . . . . . . . 51
#5252. . . . . . . . . . . . . . . . . 51
#5311. . . . . . . . . . . . . . . . . 51
#5595. . . . . . . . . . . . . . . . . 53
#5692. . . . . . . . . . . . . . . . . 53
#7815. . . . . . . . . . . . . . . . . 65
proportions
#2284. . . . . . . . . . . . . . . . . 16
#3160. . . . . . . . . . . . . . . . . 16
#1926. . . . . . . . . . . . . . . . . 16
#4518. . . . . . . . . . . . . . . . . 18
#4466. . . . . . . . . . . . . . . . . 19
#6738. . . . . . . . . . . . . . . . . 32
quadratic equations
#5236. . . . . . . . . . . . . . . . . 44
quadratic equations solved by factoring
#4913. . . . . . . . . . . . . . . . . 42
#4829. . . . . . . . . . . . . . . . . 42
#4603. . . . . . . . . . . . . . . . . 42
#4699. . . . . . . . . . . . . . . . . 42
#5001. . . . . . . . . . . . . . . . . 42
#4127. . . . . . . . . . . . . . . . . 43
#4020. . . . . . . . . . . . . . . . . 43
#6019. . . . . . . . . . . . . . . . . 43
#5800. . . . . . . . . . . . . . . . . 43
#5597. . . . . . . . . . . . . . . . . 43
#5609. . . . . . . . . . . . . . . . . 43
#5790. . . . . . . . . . . . . . . . . 43
#5101. . . . . . . . . . . . . . . . . 44
#1524. . . . . . . . . . . . . . . . . 58
quadratic equations—completing the
square
#1683. . . . . . . . . . . . . . . . . 55
#1903. . . . . . . . . . . . . . . . . 55
#1942. . . . . . . . . . . . . . . . . 55
#5020. . . . . . . . . . . . . . . . . 55
#2340. . . . . . . . . . . . . . . . . 55
#2410. . . . . . . . . . . . . . . . . 55
#2575. . . . . . . . . . . . . . . . . . 55
#2615. . . . . . . . . . . . . . . . . . 55
#4977. . . . . . . . . . . . . . . . . . 55
#1945. . . . . . . . . . . . . . . . . . 56
#2630. . . . . . . . . . . . . . . . . . 58
#596. . . . . . . . . . . . . . . . . . . 61
#5812. . . . . . . . . . . . . . . . . . 66
quadratic equations— quadratic formula
#1009. . . . . . . . . . . . . . . . . . 55
#1010. . . . . . . . . . . . . . . . . . 55
#1044. . . . . . . . . . . . . . . . . . 55
#594. . . . . . . . . . . . . . . . . . . 55
#1252. . . . . . . . . . . . . . . . . . 55
#495. . . . . . . . . . . . . . . . . . . 55
#1378. . . . . . . . . . . . . . . . . . 56
#5050. . . . . . . . . . . . . . . . . . 56
#5611. . . . . . . . . . . . . . . . . . 56
#6115. . . . . . . . . . . . . . . . . . 56
#6325. . . . . . . . . . . . . . . . . . 56
#1208. . . . . . . . . . . . . . . . . . 61
#2302. . . . . . . . . . . . . . . . . . 62
#6511. . . . . . . . . . . . . . . . . . 67
quadratic equations— solving pure
quadratics
#2611. . . . . . . . . . . . . . . . . . 50
#1900. . . . . . . . . . . . . . . . . . 50
#2060. . . . . . . . . . . . . . . . . . 50
#2323. . . . . . . . . . . . . . . . . . 50
#2407. . . . . . . . . . . . . . . . . . 50
#2570. . . . . . . . . . . . . . . . . . 50
#4141. . . . . . . . . . . . . . . . . . 50
#3620. . . . . . . . . . . . . . . . . . 50
#3512. . . . . . . . . . . . . . . . . . 53
#6304. . . . . . . . . . . . . . . . . . 53
#4300. . . . . . . . . . . . . . . . . . 53
#3020. . . . . . . . . . . . . . . . . . 57
#1024. . . . . . . . . . . . . . . . . . 62
#5114. . . . . . . . . . . . . . . . . . 63
radical equations
#4747. . . . . . . . . . . . . . . . . . 52
#5010. . . . . . . . . . . . . . . . . . 52
#4940. . . . . . . . . . . . . . . . . . 52
#3720. . . . . . . . . . . . . . . . . . 52
#3979. . . . . . . . . . . . . . . . . . 52
#7425. . . . . . . . . . . . . . . . . . 52
#4912. .
#2301. .
#277. . .
#3030. .
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53
57
62
67
radius
#5566. . . . . . . . . . . .
#6320. . . . . . . . . . . .
ratio problems
#154. . . . . . . . . . . . .
#718. . . . . . . . . . . . .
#619. . . . . . . . . . . . .
#511. . . . . . . . . . . . .
#852. . . . . . . . . . . . .
#326. . . . . . . . . . . . .
#39. . . . . . . . . . . . . .
#467. . . . . . . . . . . . .
#845. . . . . . . . . . . . .
#1690. . . . . . . . . . . .
#2930. . . . . . . . . . . .
#6308. . . . . . . . . . . .
#5795. . . . . . . . . . . .
#205. . . . . . . . . . . . .
#5017. . . . . . . . . . . .
#2634. . . . . . . . . . . .
rational numbers
#790. . . . . . . . . . . . .
#711. . . . . . . . . . . . .
#629. . . . . . . . . . . . .
#1380. . . . . . . . . . . .
#6129. . . . . . . . . . . .
#3037. . . . . . . . . . . .
rationalizing the denominator
#1007. . . . . . . . . . . .
#1069. . . . . . . . . . . .
#6509. . . . . . . . . . . .
#248. . . . . . . . . . . . .
#6825. . . . . . . . . . . .
#4860. . . . . . . . . . . .
#2066. . . . . . . . . . . .
#1212. . . . . . . . . . . .
#1220. . . . . . . . . . . .
. . . . . 18
. . . . . 38
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13
13
13
13
13
13
13
13
14
14
14
32
38
49
54
61
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22
22
22
22
24
38
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52
52
52
52
52
53
57
62
67
263
real numbers and irrational numbers
#6290. . . . . . . . . . . . . . . . . .
#6025. . . . . . . . . . . . . . . . . .
#6234. . . . . . . . . . . . . . . . . .
#5799. . . . . . . . . . . . . . . . . .
#597. . . . . . . . . . . . . . . . . . .
reflexive property of equality. . . . . . .
#20. . . . . . . . . . . . . . . . . . . .
#1613. . . . . . . . . . . . . . . . . .
#3927. . . . . . . . . . . . . . . . . .
#1681. . . . . . . . . . . . . . . . . .
repeating decimals
#2888. . . . . . . . . . . . . . . . . .
#4330. . . . . . . . . . . . . . . . . .
#1904. . . . . . . . . . . . . . . . . .
#3555. . . . . . . . . . . . . . . . . .
#8952. . . . . . . . . . . . . . . . . .
set builder notation
#6685. . . . . . . . . . . . . . . . . .
#7105. . . . . . . . . . . . . . . . . .
#7792. . . . . . . . . . . . . . . . . .
#8895. . . . . . . . . . . . . . . . . .
#8973. . . . . . . . . . . . . . . . . .
#941. . . . . . . . . . . . . . . . . . .
#948. . . . . . . . . . . . . . . . . . .
sets braces
#492. . . . . . . . . . . . . . . . . . .
sets and braces
#221. . . . . . . . . . . . . . . . . . .
#317. . . . . . . . . . . . . . . . . . .
#45. . . . . . . . . . . . . . . . . . . .
#923. . . . . . . . . . . . . . . . . . .
#699. . . . . . . . . . . . . . . . . . .
#900. . . . . . . . . . . . . . . . . . .
#667. . . . . . . . . . . . . . . . . . .
#930. . . . . . . . . . . . . . . . . . .
#9130. . . . . . . . . . . . . . . . . .
sets— empty set
#37. . . . . . . . . . . . . . . . . . . .
sets— equality of
#500. . . . . . . . . . . . . . . . . . .
#71. . . . . . . . . . . . . . . . . . . .
#888. . . . . . . . . . . . . . . . . . .
#401. . . . . . . . . . . . . . . . . . .
#53. . . . . . . . . . . . . . . . . . . .
264
51
51
51
51
61
21
21
24
43
58
54
54
57
61
66
21
21
21
21
21
24
24
11
11
11
11
11
11
11
14
14
66
11
11
11
11
14
49
#598. . . . . . . . . . . . . . .
sets—finite and infinite
#123. . . . . . . . . . . . . . .
#560. . . . . . . . . . . . . . .
#635. . . . . . . . . . . . . . .
sets—is a member of
#5580. . . . . . . . . . . . . .
simplifying fractions
#3114. . . . . . . . . . . . . .
#3262. . . . . . . . . . . . . .
#3508. . . . . . . . . . . . . .
#3705. . . . . . . . . . . . . .
#933. . . . . . . . . . . . . . .
#2950. . . . . . . . . . . . . .
#3858. . . . . . . . . . . . . .
slope
#1258. . . . . . . . . . . . . .
#2894. . . . . . . . . . . . . .
#5740. . . . . . . . . . . . . .
#2511. . . . . . . . . . . . . .
#2072. . . . . . . . . . . . . .
#1049. . . . . . . . . . . . . .
slope-intercept form of the line
#1205. . . . . . . . . . . . . .
#994. . . . . . . . . . . . . . .
#645. . . . . . . . . . . . . . .
#1022. . . . . . . . . . . . . .
#1391. . . . . . . . . . . . . .
#1260. . . . . . . . . . . . . .
solving equations
#740. . . . . . . . . . . . . . .
#5607. . . . . . . . . . . . . .
#1265. . . . . . . . . . . . . .
#5816. . . . . . . . . . . . . .
square roots— simplifying
#523. . . . . . . . . . . . . . .
#3012. . . . . . . . . . . . . .
#3121. . . . . . . . . . . . . .
#3716. . . . . . . . . . . . . .
#4321. . . . . . . . . . . . . .
#4150. . . . . . . . . . . . . .
#4838. . . . . . . . . . . . . .
#1040. . . . . . . . . . . . . .
#2522. . . . . . . . . . . . . .
#8009. . . . . . . . . . . . . .
. . . 62
. . . 11
. . . 11
. . . 11
. . . 19
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45
45
45
45
45
45
48
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60
60
60
63
63
63
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60
60
60
60
61
61
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24
54
62
66
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50
50
50
50
50
50
53
57
62
65
symmetric property of equality
#204. . . . . . . . . . . . . . . . . . .
#5266. . . . . . . . . . . . . . . . . .
#1125. . . . . . . . . . . . . . . . . .
transposing
#1930. . . . . . . . . . . . . . . . . .
#2294. . . . . . . . . . . . . . . . . .
#2901. . . . . . . . . . . . . . . . . .
trapezoid— area of
#8950. . . . . . . . . . . . . . . . . .
#9122. . . . . . . . . . . . . . . . . .
#4123. . . . . . . . . . . . . . . . . .
two equations solved by elimination
#4010. . . . . . . . . . . . . . . . . .
#1891. . . . . . . . . . . . . . . . . .
#2292. . . . . . . . . . . . . . . . . .
#2904. . . . . . . . . . . . . . . . . .
#4505. . . . . . . . . . . . . . . . . .
#4519. . . . . . . . . . . . . . . . . .
#2935. . . . . . . . . . . . . . . . . .
#6414. . . . . . . . . . . . . . . . . .
#4700. . . . . . . . . . . . . . . . . .
#4284. . . . . . . . . . . . . . . . . .
#4295. . . . . . . . . . . . . . . . . .
#1685. . . . . . . . . . . . . . . . . .
#5104. . . . . . . . . . . . . . . . . .
two equations solved by graphing
#2300. . . . . . . . . . . . . . . . . .
#2909. . . . . . . . . . . . . . . . . .
#4818. . . . . . . . . . . . . . . . . .
two equations solved by substitution
#5588. . . . . . . . . . . . . . . . . .
#6315. . . . . . . . . . . . . . . . . .
#6902. . . . . . . . . . . . . . . . . .
#4909. . . . . . . . . . . . . . . . . .
union of sets
#924. . . . . . . . . . . . . . . . . . .
#3328. . . . . . . . . . . . . . . . . .
#4280. . . . . . . . . . . . . . . . . .
#4500. . . . . . . . . . . . . . . . . .
#800. . . . . . . . . . . . . . . . . . .
#600. . . . . . . . . . . . . . . . . . .
#1893. . . . . . . . . . . . . . . . . .
#950. . . . . . . . . . . . . . . . . . .
#3005. . . . . . . . . . . . . . . . . .
#5111. . . . . . . . . . . . . . . . . 44
22
54
62
30
30
30
20
20
24
29
29
29
29
29
29
32
32
37
37
43
58
66
volume
#2717. . . . . . . . . . . . . . .
#2788. . . . . . . . . . . . . . .
#2943. . . . . . . . . . . . . . .
#3366. . . . . . . . . . . . . . .
#4450. . . . . . . . . . . . . . .
#4820. . . . . . . . . . . . . . .
#7790. . . . . . . . . . . . . . .
whole numbers
#347. . . . . . . . . . . . . . . .
#5590. . . . . . . . . . . . . . .
word problems in two unknowns
#6207. . . . . . . . . . . . . . .
#6305. . . . . . . . . . . . . . .
#8207. . . . . . . . . . . . . . .
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36
36
36
36
36
37
38
. . 12
. . 37
. . 31
. . 31
. . 32
33
33
37
34
34
34
37
29
29
29
29
29
29
29
32
37
265