Table Of ContentMaster Essential Algebra Skills Practice Workbook with Answers
Chris McMullen, Ph.D.
Copyright © 2020 Chris McMullen, Ph.D.
www.improveyourmathfluency.com
www.monkeyphysicsblog.wordpress.com
www.chrismcmullen.com
All rights are reserved. However, educators or parents who purchase one copy of this
workbook (or who borrow one physical copy from a library) may make and distribute
photocopies of selected pages for instructional (non-commercial) purposes for their
own students or children only.
Zishka Publishing
ISBN: 978-1-941691-34-2
Mathematics > Algebra
Study Guides > Workbooks > Math
CONTENTS
Introduction iv
1 Getting Ready 5
2 Solving Linear Equations 41
3 Exponents 78
4 The Distributive Property 101
5 Factoring Expressions 126
6 Quadratic Equations 156
7 Variables in the Denominator 186
8 Systems of Equations 212
9 Inequalities 247
Answer Key 262
Glossary 364
Index 376
INTRODUCTION
The goal of this workbook is to help students master essential algebra skills through
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practice.
The first chapter is an essential preparatory chapter. It explains what algebra
is, defines key vocabulary terms, discusses the language of algebra, shows how
multiplication and division are expressed in algebra, and shows how to plug
numbers into an equation. It also reviews the order of operations, fractions, and
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negative numbers.
The remaining chapters cover essential algebra skills, such as combining like
terms, distributing, factoring, the FOIL method, variables in the denominator,
cross multiplying, ratios, rates, the quadratic formula, powers, roots, substitution,
simultaneous equations, rationalizing the denominator, inequalities, and word
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problems.
Each section concisely introduces the main ideas, explains essential concepts,
and provides representative examples to help serve as a guide. A full solution
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is given for every example.
Answer key. Practice makes permanent, but not necessarily perfect. Check the
answers at the back of the book and strive to learn from any mistakes. This will
help to ensure that practice makes perfect.
1 GETTING READY
1.1 What Is Algebra? 6
1.2 Essential Vocabulary 7
1.3 Multiplying and Dividing in Algebra 10
1.4 Algebra in English 13
1.5 Order of Operations 15
1.6 Algebra Operations 17
1.7 Fractions 20
1.8 Decimals 24
1.9 Percents 27
1.10 Negative Numbers 29
1.11 Powers and Roots 32
1.12 Number Classification 36
1.13 Evaluating Formulas 38
1 Getting Ready
1.1 What Is Algebra?
Algebra is a highly practical branch of mathematics for the following reason:
Algebra uses letters (like 𝑥𝑥, 𝑦𝑦, or 𝑡𝑡) to represent unknown quantities, and provides
a system of rules for determining the unknowns.
This system of rules makes algebra very useful for solving a wide variety of problems.
•
Following are a few examples where it is useful to represent numbers with letters.
If a car travels with constant speed, the distance traveled equals the speed of
the car times the elapsed time. By using the letter 𝑑𝑑 to represent the distance
traveled, the letter 𝑟𝑟 to represent the speed (which is a rate), and the letter 𝑡𝑡
to represent the elapsed time, we can express this relationship with the formula
𝑑𝑑 = 𝑟𝑟𝑡𝑡. If we know any two of these letters, the rules of algebra allow us to solve
for the unknown quantity. (Although this example is simple enough that you
could solve such problems without algebra, there are many formulas that would
2 2
𝑦𝑦 𝑥𝑥
• be very difficult to solve without using algebra, such as �𝑏𝑏2 −𝑎𝑎2 = 𝑐𝑐.)
Word problems can be solved systematically and efficiently by using letters to
represent unknowns and applying the rules of algebra. For example, if we know
that five times a number minus forty equals eighty, algebra lets us write the
equation 5𝑥𝑥−40 = 80 and provides a prescription for determining that 𝑥𝑥 is
equal to 24, which we will learn in Chapter 2. (Again, this simple example can
be solved without algebra, but for more challenging problems applying algebra
•
makes the solution much more straightforward and efficient.)
Using letters to represent numbers allows us to express mathematical rules in
a general form. For example, note that 5×(6+4) = 5×10 = 50 has the same
answer as 5×6+5×4 = 30+20 = 50. Similarly, 7×(5+3) = 7×8 = 56
has the same answer as 7×5+7×3 = 35+21 = 56. Using letters, we can
express this rule in the general form 𝑎𝑎(𝑏𝑏+𝑐𝑐) = 𝑎𝑎𝑏𝑏+𝑎𝑎𝑐𝑐 (where 𝑎𝑎𝑏𝑏 means 𝑎𝑎
times 𝑏𝑏). This is known as the distributive property.
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Master Essential Algebra Skills Practice Workbook with Answers
1.2 Essential Vocabulary
In order to learn algebra, you will first need to understand some important words. It
wouldn’t be helpful to tell you, “The next step is to divide by the coefficient of the
variable,” if you have no idea what the words “coefficient” and “variable” mean. If you
want to understand what is going on when we discuss algebra, you need to study the
following words and definitions. The sooner you can remember these definitions, the
better. If you come across a mathematical word in this book that you don’t understand,
you can look it up in the handy glossary at the back of the book.
An unknown refers to a letter, like 𝑥𝑥 or 𝑦𝑦, that you are trying to solve for in a problem.
A variable refers to a letter, like 𝑥𝑥 or 𝑦𝑦. We call it a “variable” because it doesn’t have
the same value for different problems. For example, you might find that 𝑥𝑥 equals 3 for
one problem, but that 𝑥𝑥 equals 12 in another problem. The value of 𝑥𝑥 “varies” from one
problem to another.
The terms “unknown” and “variable” both refer to letters like 𝑥𝑥, 𝑦𝑦, 𝑡𝑡, etc. that we don’t
know the values for (until we solve for them).
3
A constant has a fixed value. All real numbers, like 5, −2, 418.27, and even 2√3 are
constants. Where it can get confusing is when we use letters to represent constants
1 2
as well as variables. For example, in the formula ℎ = 2𝑔𝑔𝑡𝑡 , we consider ℎ and 𝑡𝑡 to be
variables, but consider 𝑔𝑔 to be a constant. Why? Because near the surface of the earth,
2
𝑔𝑔 has a constant value (with a magnitude of 9.8 m/s ).
A coefficient is a number that multiplies a variable. For example, in 6𝑥𝑥 the coefficient
4
is the number 6, while in 9𝑦𝑦 the coefficient is 9.
The terms “coefficient” and “constant” aren’t interchangeable. Although a coefficient
may be constant, a coefficient has a very specific role: it must multiply a variable. For
7
1 Getting Ready
2
example, in 4𝑥𝑥 = 36, the coefficient is 4, while 4 and 36 are both constants (but we
would call 36 the constant “term”; see below for the definition of “term”).
An equation is easy to spot because it has an equal (=) sign. For example, 7𝑥𝑥+2 = 30
is an equation. If it doesn’t have an equal sign, like 3𝑥𝑥−8, it isn’t an equation.
An expression doesn’t have an equal sign (=) or inequality (like < or >). For example,
3𝑥𝑥−8 is an expression.
You can solve an equation. For example, 𝑥𝑥 = 4 solves the equation 7𝑥𝑥+2 = 30 since
7(4)+2 = 28+2 = 30. (If you didn’t follow the math in this paragraph, don’t worry.
We will learn this in Sec. 1.13.)
You can simplify an expression (but you cans’to lve it). For example, 5𝑥𝑥−4+3𝑥𝑥−2
simplifies to 8𝑥𝑥−6. (We’ll understand why in Chapter 2. For now, you should be able
to see that 8𝑥𝑥−6 is indeed simpler than 5𝑥𝑥−4+3𝑥𝑥−2.) To simplify an expression
means to find an equivalent expression that has a simpler form. (When we learn how
to simplify expressions, you’ll see concrete examples of what this means.)
The terms of an equation, expression, or inequality are separated by + signs, − signs,
2
= signs, or inequal signs (like < or >). For example, 2𝑥𝑥 +9𝑥𝑥+4 has 3 terms (which
2
are 2𝑥𝑥 , 9𝑥𝑥, and 4) and 7𝑥𝑥+5 = 3𝑥𝑥+25 has 4 terms (which are 7𝑥𝑥, 5, 3𝑥𝑥, and 25).
2
Example 1. Is (𝑥𝑥−2) +5 an expression or an equation?
It is an expression because it doesn’t contain an equal (=) sign.
24
Example 2. Is 𝑥𝑥 = 6 an expression or an equation?
It is an equation because it contains an equal (=) sign.
3
Example 3. What are the terms of 𝑥𝑥 −2 = 6?
3
There are three terms: 𝑥𝑥 , 2, and 8. Terms are separated by +, −, and = signs.
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Master Essential Algebra Skills Practice Workbook with Answers
2
Example 4. For 5𝑥𝑥 −4 = 3𝑦𝑦, what are the variables and what are the coefficients?
The variables are 𝑥𝑥 and 𝑦𝑦. The coefficients are 5 and 3. Coefficients multiply variables.
Exercise Set 1.2
Directions: Apply the definitions from this section to answer the following questions.
2
1) Is (𝑥𝑥−1) = 16 an expression or an equation?
3 4 1
2
2) Is 𝑥𝑥 −𝑥𝑥+8 an expression or an equation?
3 2
3) What are the terms of 𝑥𝑥 +8𝑥𝑥 −3𝑥𝑥+6?
4) What are the terms of 9−𝑥𝑥 = 4?
2 3
5) What are the terms of 5𝑥𝑥𝑦𝑦 −7𝑦𝑦 +3?
6) For 3𝑥𝑥−8 = 7, what are the variables and what are the coefficients?
2 2
7) For 5𝑥𝑥 −4+2𝑦𝑦 , what are the variables and what are the coefficients?
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1 Getting Ready
1.3 Multiplying and Dividing in Algebra
We almost never use the standard times symbol (×) in algebra. Why not? It’s because
𝑥𝑥 is the most commonly used variable in algebra. If you wrote down an equation using
both the variable 𝑥𝑥 and the times symbol ×, these could easily be confused (especially
when writing by hand). It is a good habit to avoid using the times symbol (×). When
you read algebra, you need to be aware of the different ways that multiplication may
be represented.
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A common way to multiply numbers is to use parentheses like one of these examples:
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(3)(4) means 3 times 4.
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3(4) also means 3 times 4.
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(3)4 is less common, but still means 3 times 4.
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(3)(4)(5) means 3 times 4 times 5.
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3(4)(5) also means 3 times 4 times 5.
(3)4(5) and (3)(4)5 are less common, but still mean 3 times 4 times 5.
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An alternative is to use a middle dot (·):
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3·4 means 3 times 4.
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3·4·5 means 3 times 4 times 5.
(3)∙(4) unnecessarily uses both parentheses and a middle dot, but it still means
3 times 4. We recommend avoiding this, but beware that you may encounter it.
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When a variable multiplies another quantity, no multiplication symbol is used:
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5𝑥𝑥 means 5 times 𝑥𝑥.
•
𝑥𝑥𝑦𝑦𝑥𝑥 means 𝑥𝑥 times 𝑦𝑦 times 𝑥𝑥.
• 2 2
4𝑥𝑥 𝑦𝑦 means 4 times 𝑥𝑥 times 𝑦𝑦.
2(𝑥𝑥−3) means 2 times the quantity 𝑥𝑥−3.
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You should avoid the following (but beware that you may encounter them):
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3∙𝑥𝑥 should instead be written as 3𝑥𝑥.
2(𝑥𝑥) should instead be written as 2𝑥𝑥. Compare these unnecessary parentheses
to the example above where parentheses are needed (with 𝑥𝑥−3).
10
