Humongous book algebra problems by michael kelley

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Copyright © 2008 by W. Michael Kelley
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Contents
Introduction
Your one-stop shop for a review of numbers
Chapter 1: Arithmetic Fundamentals
1
Numbers fall into different groups
Number Classification .................................................................................................
2
subtract, multiply, and divide positive and negative numbers 5
Expressions Containing Signed Numbers ..Add,
......................................................................
When numbers band together, deal with them first
Grouping Symbols .......................................................................................................
8
Basic assumptions about algebra
Algebraic Properties ...................................................................................................
11

em
Chapter 2: Rational Numbers Understanding fractions sure beats being afraid of th 17
Proper and improper fractions, decimals, and mixed numbers 18
Rational Number Notation . .......................................................................................
Reducing fractions to lowest terms, like 1/2 instead of 5/10
Simplifying Fractions . ...............................................................................................
23
Add,
subtract,
multiply,
and
divide

fractions
Combining Fractions ................................................................................................. 26
37
Chapter 3: Basic Algebraic Expressions
Time for x to make its stunning debut
The alchemy of turning words into math
Translating Expressions . ...........................................................................................
38
Rules for simplifying expressions that contain powers
Exponential Expressions ............................................................................................
40
Multiply one thing by a bunch of things in parentheses
Distributive Property . ................................................................................................
45
My dear Aunt Sally is eternally excused
Order of Operations . .................................................................................................
48
Replace variables with numbers
Evaluating Expressions .............................................................................................
51
Chapter 4: Linear Equations in One Variable

How to solve basic equations

55

Add to/subtract from both sides
Adding and Subtracting to Solve an Equation ..............................................................
56
Multiply/divide

both
sides
Multiplying and Dividing to Solve an Equation . .......................................................... 59
Nothing new here, just more steps
Solving Equations Using Multiple Steps .......................................................................
61
Most
of
them
have
two
solutions
Absolute Value Equations . ......................................................................................... 70
Equations with TWO variables (like x and y) or more 73
Equations Containing Multiple Variables . ...................................................................

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iii


Table of Contents

Identify the points that make an e
quat
ion true
Chapter 5: Graphing Linear Equations in Two Variables
77
Which should you use to graph?

Number Lines and the Coordinate Plane ......................................................................
78
Plug
in
some
x’s,
plot
some
points,
call
it
a
day
Graphing with a Table of Values ................................................................................. 83

The easiest way to plot two points on a line quickly
Graphing Using Intercepts .........................................................................................
90
Figure
out
how
slanty
a
line
is
Calculating Slope of a Line ........................................................................................ 93
Don’t miss the point in these graphs (Get it?)
Graphing Absolute Value Equations ...........................................................................100
Chapter 6: Linear Equations in Two Variables

Generating equations of lines

105

Point +€slope€=€equation
Point-Slope Form of a Linear Equation .......................................................................106
Lines that look like€y€=€mx€+€b
Slope-Intercept Form of a Linear Equation ...................................................................110
Graphing equations that are solved for y
Graphing Lines in Slope-Intercept Form ......................................................................113
Write equations of lines in a uniform way
Standard Form of a Linear Equation ..........................................................................118
Practice all the skills from this chapter
Creating Linear Equations ........................................................................................121
Chapter 7: Linear Inequalities

127
They’re like equations without the equal sign
Dust off your equation-solving skills from Chapter 4
Inequalities in One Variable ......................................................................................128
Shoot arrows into number lines
Graphing Inequalities in One Variable . ......................................................................132
Two inequalities for the price of one
Compound Inequalities . ...........................................................................................135
Break these into two inequalities
Absolute Value Inequalities . ......................................................................................137
A fancy way to write solutions
Set Notation ............................................................................................................140
Lines that give off shade in the coordinate plane
Graphing Inequalities in Two Variables ......................................................................142

W ork wit
h more
than one equation at a time

147
Chapter 8: Systems of Linear Equations and Inequalities
Graph two lines at once
Graphing Linear Systems ..........................................................................................148
Solve one equation for a variable and plug it into the other
The Substitution Method . .........................................................................................153
Make one variable disappear and solve for the other one
Variable Elimination ................................................................................................162
The answer is where the shading overlaps
Systems of Inequalities ..............................................................................................168
Use the sharp points at the edge of a shaded region
Linear Programming ................................................................................................173
181
Chapter 9: Matrix Operations and Calculations
Numbers in rows and columns
The order of a matrix and identifying elements
Anatomy of a Matrix . ..............................................................................................182
Combine numbers in matching positions
Adding and Subtracting Matrices ..............................................................................183
Not as easy as adding or subtracting them
Multiplying Matrices . ..............................................................................................188
Values defined for square matrices only
Calculating Determinants .........................................................................................192
Double-decker matrices that solve systems
Cramer’s Rule .........................................................................................................200

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Table of Contents

Advanced matrix stuff

Chapter 10: Applications of Matrix Algebra

207

Extra columns and lots of 0s and 1s
Augmented and Identity Matrices . .............................................................................208
Swap rows, add rows, or multiply by a number
Matrix Row Operations ............................................................................................211
More matrices full of 0s with a diagonal of 1s
Row and Reduced-Row Echelon Form .........................................................................216
Matrices that cancel other matrices out
Inverse Matrices ......................................................................................................228
Chapter 11: Polynomials

237
Clumps of numbers and variables raised to powers
Labeling them based on the exponent and total terms
Classifying Polynomials ............................................................................................238
Only works for like terms
Adding and Subtracting Polynomials .........................................................................239

FOIL and beyond
Multiplying Polynomials . .........................................................................................244
A lot like long dividing integers
Long Division of Polynomials ....................................................................................246
Divide using only the coefficients
Synthetic Division of Polynomials ...............................................................................251

Chapter 12: Factoring Polynomials

The opposite of multiplying polynomials

257

Largest factor that divides into everything evenly
Greatest Common Factors ..........................................................................................258
You can factor out binomials, too
Factoring by Grouping ..............................................................................................265
Difference of perfect squares/cubes, sum of perfect cubes
Common Factor Patterns ...........................................................................................267
Turn one trinomial into two binomials
Factoring Quadratic Trinomials . ...............................................................................270
Square r
oots, c
Chapter 13: Radical Expressions and Equations ube roots, and fractional exponents 275
Moving things out from under the radical
Simplifying Radical Expressions . ...............................................................................276
Fractional powers are radicals in disguise
Rational Exponents . ................................................................................................281
Add, subtract, multiply, and divide roots
Radical Operations ..................................................................................................283

Use exponents to cancel out radicals
Solving Radical Equations ........................................................................................288
Numbers that contain i, which equals √—
–1
Complex Numbers.....................................................................................................290
295
Chapter 14: Quadratic Equations and Inequalities Solve equations containing x2
Use techniques from Chapter 12 to solve equations
Solving Quadratics by Factoring ................................................................................296
Make a trinomial into a perfect square
Completing the Square ..............................................................................................300
Use an equation’s coefficients to calculate the solution
Quadratic Formula ..................................................................................................305
What b2€–€4ac tells you about an equation
Applying the Discriminant ........................................................................................312
Inequalities that contain x2
One-Variable Quadratic Inequalities ...........................................................................316

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Table of Contents

Chapter 15: Functions

323
Named expressions that give one output per input

What makes a function a function?
Relations and Functions ...........................................................................................324
+, –, ·, and ÷ functions
Operations on Functions ...........................................................................................326

Plug one function into another
Composition of Functions . ........................................................................................330
Functions that cancel each other out
Inverse Functions . ...................................................................................................335
Function rules that change based on the x-input
Piecewise-Defined Functions . .....................................................................................343
Chapter 16: Graphing Functions

347
Drawing graphs that aren’t lines
Plug in a bunch of things for x
Graphing with a Table of Values ................................................................................348
What can you plug in? What comes out?
Domain and Range of a Function ..............................................................................354
Pieces of a graph are reflections of each other
Symmetry ................................................................................................................360
The graphs you need to understand most
Fundamental Function Graphs...................................................................................365
Move, stretch, squish, and flip graphs
Graphing Functions Using Transformations ................................................................369
These graphs might have sharp points
Absolute Value Functions...........................................................................................374

Chapter 17: Calculating Roots of Functions

379
Roots€=€solutions€=€x-intercepts
Factoring polynomials given a head start
Identifying Rational Roots ........................................................................................380
The ends of a function describe the ends of its graph
Leading Coefficient Test ............................................................................................384
Sign changes help enumerate real roots
Descartes’ Rule of Signs ............................................................................................388
Find possible roots given nothing but a function
Rational Root Test ...................................................................................................390
Factoring big polynomials from the ground up
Synthesizing Root Identification Strategies ...................................................................394

399
Chapter 18: Logarithmic Functions Contains enough logs to build yourself a cabin
Given loga€b€=€c, find a, b, or c
Evaluating Logarithmic Expressions . .........................................................................400
All log functions have the same basic shape
Graphs of Logarithmic Functions ...............................................................................402
What the bases equal when no bases are written
Common and Natural Logarithms .............................................................................406
Calculate log values that have weird bases
Change of Base Formula ...........................................................................................409
Expanding, contracting, and simplifying log expressions
Logarithmic Properties ..............................................................................................412
417
Chapter 19: Exponential Functions Functions with a variable in the exponent
Graphs that start close to y€=€0 and climb fast
Graphing Exponential Functions ...............................................................................418
They cancel each other out

Composing Exponential and Logarithmic Functions .....................................................423
Cancel logs with exponentials and vice versa
Exponential and Logarithmic Equations .....................................................................426
Use f(t)€=€Nekt to measure things like population
Exponential Growth and Decay . ................................................................................433

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Table of Contents

Chapter 20: Rational Expressions

439
Fractions with lots of variables in them
Reducing fractions by factoring
Simplifying Rational Expressions ...............................................................................440
Use common denominators
Adding and Subtracting Rational Expressions .............................................................444
denominators not necessary
Multiplying and Dividing Rational Expressions .Common
.........................................................452
Reduce fractions that contains fractions
Simplifying Complex Fractions ...................................................................................457
Rational functions have asymptotes
Graphing Rational Functions . ..................................................................................459

r 20
pte
a
h
Chapter 21: Rational Equations and Inequalities Solve equations using the skills from C
465
When two fractions are equal, “X” marks the solution
Proportions and Cross Multiplication .........................................................................466
Ditch the fractions or cross multiply to solve
Solving Rational Equations ......................................................................................470
Turn a word problem into a rational equation
Direct and Indirect Variation .....................................................................................475
Critical numbers, test points, and shading
Solving Rational Inequalities ....................................................................................479

Chapter 22: Conic Sections

Parabolas, Circles, Ellipses, and Hyperbolas

487

Vertex, axis of symmetry, focus, and directrix
Parabolas ...............................................................................................................488
Center, radius, and diameter
Circles ....................................................................................................................494
Major and minor axes, center, foci, and eccentricity
Ellipses ...................................................................................................................499
Transverse and conjugate axes, foci, vertices, and asymptotes
Hyperbolas ..............................................................................................................506
st is earned?

intere
h
c
u
two trains leave the station full of consecutive integers, how m
515
Chapter 23: Word Problems If

Integer and age problems
Determining Unknown Values ...................................................................................516
Simple, compound, and continuously compounding
Calculating Interest . ................................................................................................521
Area, volume, perimeter, and so on
Geometric Formulas . ................................................................................................525
Distance equals rate times time
Speed and Distance ..................................................................................................529
Measuring ingredients in a mixture
Mixture and Combination ........................................................................................534
How much time does it save to work together?
Work ......................................................................................................................538

Appendix A: Algebraic Properties

545

Appendix B: Important Graphs and Graph Transformations

547

Appendix C: Key Algebra Formulas

551

Index

555

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vii


Introduction
Are you in an algebra class? Yes? Then you NEED this book. Here’s why:
Fact #1: The best way to learn algebra is by working out algebra problems.
There’s no denying it. If you could figure this class out just by reading the
textbook or taking good notes in class, everybody would pass with flying colors.
Unfortunately, the harsh truth is that you have to buckle down and work
problems out until your fingers are numb.
Fact #2: Most textbooks only tell you WHAT the answers to their practice
problems are, but not HOW to do them! Sure, your textbook may have 175
problems for every topic, but most of them only give you the answers. That
means if you don’t get the answer right you’re totally out of luck! Knowing you’re
wrong is no help at all if you don’t know why you’re wrong. Math textbooks sit on a
huge throne like the Great and Terrible Oz and say, “Nope, try again,” and we
do. Over and over. And we keep getting the problem wrong. What a delightful
way to learn! (Let’s not even get into why they only tell you the answers to the
odd problems. Does that mean the book’s actual author didn’t even feel like
working out the even ones?)

Fact #3: Even when math books try to show you the steps for a problem, they
do a lousy job. Math people love to skip steps. You’ll be following along fine with
an explanation and then all of a sudden BAM, you’re lost. You’ll think to yourself,
“How did they do that?” or “Where the heck did that 42 come from? It wasn’t
there in the last step!” Why do almost all of these books assume that in order to
work out a problem on page 200, you’d better know pages 1 through 199 like the
back of your hand? You don’t want to spend the rest of your life on homework!
You just want to know why you keep getting a negative number when you’re
calculating the minimum cost of building a pool whose length is four times the
sum of its depth plus the rate at which the water is leaking out of a train that
left Chicago at 4:00 a.m. traveling due west at the same speed carbon decays.

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Introduction

Fact #4: Reading lists of facts is fun for a while, but then it gets
old. Let’s cut to the chase. Just about every single kind of
algebra problem you could possibly run into is in here—after all,
this book is HUMONGOUS! If a thousand problems aren’t enough,
then you’ve got some kind of crazy math hunger, my friend, and
I’d seek professional help. This practice book was good at first,
but to make it great, I went through and worked out all the
problems and took notes in the margins when I thought something
was confusing or needed a little more explanation. I also drew
little skulls next to the hardest problems, so you’d know not to

freak out if they were too challenging. After all, if you’re working
on a problem and you’re totally stumped, isn’t it better to know
that the problem is SUPPOSED to be hard? It’s reassuring, at
least for me.

All of my
notes are off to
the side like this and
point to the parts of
the book I’m trying to
explain.

I think you’ll be pleasantly surprised by how detailed the answer explanations
are, and I hope you’ll find my little notes helpful along the way. Call me crazy,
but I think that people who want to learn algebra and are willing to spend the
time drilling their way through practice problems should actually be able to
figure the problems out and learn as they go, but that’s just my two cents.
Good luck and make sure to come visit my website at www.calculus-help.com. If
you feel so inclined, drop me an email and give me your two cents. (Not literally,
though—real pennies clog up the Internet pipes.)

Acknowledgments
Special thanks to the technical reviewer, Paula Perry, an expert who doublechecked the accuracy of what you’ll learn here. I met Paula when she was a
student teacher (and I had only a year or two under my belt at the time). She
is an extremely talented educator, and it’s almost a waste of her impressive
skill set to merely proofread this book, but I am appreciative nonetheless.

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ix


Trademarks
All terms mentioned in this book that are known to be or are suspected of being
trademarks or service marks have been appropriately capitalized. Alpha Books
and Penguin Group (USA) Inc. cannot attest to the accuracy of this information.
Use of a term in this book should not be regarded as affecting the validity of
any trademark or service mark.

Dedication
For my son Nick, the all-American boy who loves soccer, Legos, superheroes,
the Legend of Zelda, and pretending that he knows karate. You summarized
it best, kiddo, when you said, “You know why I love you so much, Dad? Because
we’re the same.”
For my little girls, Erin (who likes to hold my hand during dinner) and Sara (who
loves it when I tickle her until she can’t breathe). In a strange way, I am proud
that at three years old, you’ve both mastered the way to say “Daaadeeeee...,”
which suggests that I both amuse and greatly embarrass you at the same time.
Most of all, for my wife, Lisa, who cheers me on, pulls me through, picks me up, and
makes coming home the only reason I need to make it through every day.

˘

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Chapter 1

Algebraic Fundamentals

Your one-stop shop for a review of numbers
Algebra, at its core, is a compendium of mathematical concepts, axioms,
theorems, and algorithms rooted in abstraction. Mathematics is most powerful when it is not fettered by the limitations of the concrete, and the first
step toward shedding those restrictions is the introduction of the variable,
a structure into which any number of values may be substituted. However,
algebra students must first possess considerable knowledge of numbers before they can make the next logical step, representing concrete values with
abstract notation.
This chapter ensures that you are thoroughly familiar with the most common classifications used to describe numbers, provides an opportunity to
manipulate signed numbers arithmetically, and investigates the foundational mathematical principles that govern algebra.

You might be anxious to dive into the
nuts and bolts of algebra, but
don’t skip over the stuff in this chap
ter. It’s full of key vocabulary words
,
such as “rational number” and “comm
utative property.” You also learn th
ings
like the difference between real an
d complex numbers and whether 0
is odd or even. Some of the problems
might be easy, but you might be
surprised to learn something new.

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Chapter One — Algebraic Fundamentals

Number Classification
The natural
numbers are also
called the “counting
numbers,” because
when you read them,
it sounds like you’re
counting: 1, 2, 3, 4, 5,
and so on. Most people
don’t start counting
with 0.

Numbers fall into different groups
1.1

Describe the difference between the whole numbers and the natural numbers.
Number theory dictates that the set of whole numbers and the set of natural
numbers contain nearly all of the same members: {1, 2, 3, 4, 5, 6, …}. The
characteristic difference between the two is that the whole numbers also
include the number 0. Therefore, the set of natural numbers is equivalent to
the set of positive integers {1, 2, 3, 4, 5, …}, whereas the set of whole numbers is
equivalent to the set of nonnegative integers {0, 1, 2, 3, 4, 5, …}.

1.2

What set of numbers consists of integers that are not natural numbers? What
mathematical term best describes that set?
The integers are numbers that contain no explicit fraction or decimal.
Therefore, numbers such as 5, 0, and –6 are integers but 4.3 and

So you
get the
whole numbers
by taking the
natural numbers
and sticking 0
in there.

are not.

Thus, all integers belong to the set {…, –3, –2, –1, 0, 1, 2, 3, …}. According to
Problem 1.1, the set of natural numbers is {1, 2, 3, 4, 5, …}. Remove the natural
numbers from the set of integers to create the set described in this problem:
{…, –4, –3, –2, –1, 0}. This set, which contains all of the negative integers and
the number 0, is described as the “nonpositive numbers.”

1.3

Is the number 0 even or odd? Positive or negative? Justify your answers.
By definition, a number is even if there is no remainder when you divide it by 2.
To determine whether 0 is an even number, divide it by 2:
. (Note that 0
divided by any real number—except for 0—is equal to 0.) The result, 0, has no
remainder, so 0 is an even number.
However, 0 is neither positive nor negative. Positive numbers are defined as
the real numbers greater than (but not equal to) 0, and negative numbers are
defined as real numbers less than (but not equal to) 0, so 0 can be classified
only as “nonpositive” or “nonnegative.”

1.4

Numbers, like
8, that aren’t prim
because they are e
divisible by too m
a
things, are called ny
“composite numbe
rs.”



Identify the smallest positive prime number and justify your answer.
A number is described as “prime” if it cannot be evenly divided by any number
other than the number itself and 1. According to this definition, the number 8
is not prime, because the numbers 2 and 4 both divide evenly into 8. However,
the numbers 2, 3, 5, 7, and 11 are prime, because none of those numbers is
evenly divisible by a value other than the number itself and 1. Note that the
number 1 is conspicuously absent from this list and is not a prime number.
By definition, a prime number must be divisible by exactly two unique values,
the number itself and the number 1. In the case of 1, those two values are equal
and, therefore, not unique. Although this might seem a technicality, it excludes
1 from the set of prime numbers, so the smallest positive prime number is 2.

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Chapter One — Algebraic Fundamentals

1.5

List the two characteristics most commonly associated with a rational number.
The fundamental characteristic of a rational number is that it can be expressed
as a fraction, a quotient of two integers. Therefore,

and

are examples

of rational numbers. Rational numbers expressed in decimal form feature
either a terminating decimal (a finite, rather than infinite, number of values
after the decimal point) or a repeating decimal (a pattern of digits that repeats
infinitely). Consider the following decimal representations of rational numbers
to better understand the concepts of terminating and repeating decimals.

615384615384

1.6

615384

The irrational mathematical constant p is sometimes approximated with the
fraction

Little bars
like this are
used to indicate
which digits of a

repeating decimal
actually repeat.
Sometimes, a few digits
in front won’t repeat,
but the number is still
rational. For example,
is a rational number.

. Explain why that approximation cannot be the exact value of p.

When expanded to millions, billions, and even trillions of decimal places,
the digits in the decimal representation of p do not repeat in a discernable
pattern. Because p is equal to a nonterminating, nonrepeating decimal, p is an
irrational number, and irrational numbers cannot be expressed as fractions.

1.7

Which is larger, the set of real numbers or the set of complex numbers?
Explain your answer.
Combining the set of rational numbers together with the set of irrational
numbers produces the set of real numbers. In other words, every real number
must be either rational or irrational. The set of complex numbers is far larger
than the set of real numbers, and the reasoning is simple: All real numbers are
complex numbers as well. The set of complex numbers is larger than the set of
real numbers in the same way that the set of human beings on Earth is larger
than the set of men on Earth. All men are humans, but not all humans are
necessarily men. Similarly, all real numbers are complex, but not all complex
numbers are real.

Complex

numbers are
discussed in
more detail late
r
in the book, in
Problems 13.3713.44.

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Chapter One — Algebraic Fundamentals

According
to Problem 1.1,
the single element
that the whole
numbers contain and
the natural numbers
exclude is the
number 0.
Any infinitely
long decimal
that has no
pattern of repeating
digits represents an
irrational number. On
the other hand, rational

decimals either have
to repeat or terminate.
Because there are a
lot more ways to write
irrational numbers as
decimals than there
are to write rational
numbers as decimals,
there are a lot more
irrational numbers
than rational
numbers.

1.8

List the following sets of numbers in order from smallest to largest: complex
numbers, integers, irrational numbers, natural numbers, rational numbers,
real numbers, and whole numbers.
Although each of these sets is infinitely large, they are not the same size. The
smallest set is the natural numbers, followed by the whole numbers, which is
exactly one element larger than the natural numbers. Appending the negative
integers to the whole numbers results in the next largest set, the integers. The
set of rational numbers is significantly larger than the integers, and the set
of irrational numbers is significantly larger than the set of rational numbers.
The real numbers must be larger than the irrational numbers, because all
irrational numbers are real numbers. The complex numbers are larger than
the real numbers, as explained in Problem 1.7. Therefore, this is the correct
order: natural numbers, whole numbers, integers, rational numbers, irrational
numbers, real numbers, and complex numbers.

1.9

Describe the number 13 by identifying the number sets to which it belongs.
Because 13 has no explicit decimal or fraction, it is an integer. All positive
integers are also natural numbers and whole numbers. It is not evenly divisible
by 2, so 13 is an odd number. In fact, 13 is not evenly divisible by any number
other than 1 and 13, so it is a prime number. You can express 13 as a fraction
, so 13 is a rational number. It follows, therefore, that 13 is also a real
number and a complex number. In conclusion, 13 is odd, prime, a natural
number, a whole number, an integer, a rational number, a real number, and a
complex number.

1.10 Describe the number
Because

by identifying the number sets to which it belongs.

is less than 0 (i.e., to the left of 0 on a number line), it is a negative

number. It is a fraction, so by definition it is a rational number and, therefore, it
is a real number and a complex number as well.

Any number
divided by itself
is 1, so
.

˘

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Chapter One — Algebraic Fundamentals

Expressions Containing Signed Numbers

Add, subtract, multiply, and divide positive and negative numbers
1.11 Simplify the expression: 16 + (–9).
This expression contains adjacent or “double” signs, two signs next to one
another. To simplify this expression, you must convert the double sign into a
single sign. The method is simple: If the two signs in question are different,
replace them with a single negative sign; if the signs are the same (whether both
positive or both negative), replace them with a single positive sign.

Some algebra
books write positive
and negative signs
higher and smaller, like
this: 16 + –9. I’m sorry, but
that’s just weird. It’s
perfectly fine to turn
that teeny floating
sign into a regular
sign: 16 + –9.

In this problem, the adjacent signs are different, “+ –,” so you must replace them
with a single negative sign: –.

1.12 Simplify the expression: –5 – (+6).

This expression contains the adjacent signs “– +.” As explained in Problem 1.11,
the double sign must be rewritten as a single sign. Because the adjacent signs
are different, they must be replaced with a single negative sign.
–5 – (+6) = –5 – 6
To simplify the expression –5 – 6, or in fact any expression that contains
signed numbers, think in terms of payments and debts. Every negative number
represents money you owe, and every positive number represents money you’ve
earned. In this analogy, –5 – 6 would be interpreted as a debt of $5 followed by
a debt of $6, as both numbers are negative. Therefore, –5 – 6 = –11, a total debt
of $11.

Think of it
this way. If the
two signs agree w
it
each other (if th h
ey’re
both positive or bo
negative), then th th
a
good thing, a POS t’s a
IT
thing. On the othe IV E
r hand,
when two signs ca
n’t
agree with each
other
(one’s positive and
one’

negative), then th s
at’s
no good. That’s
NEGATIV E.

1.13 Simplify the expression: 4 – (–5) – (+10).
This expression contains two sets of adjacent or “double” signs: “– –” between
the numbers 4 and 5 and “– +” between the numbers 5 and 10. Replace like
signs with a single + and unlike signs with a single –.
4 – (–5) – (+10) = 4 + 5 – 10
Simplify the expression from left to right, beginning with 4 + 5 = 9.
4 + 5 – 10 = 9 – 10

There’s one other technique you can use to add and subtract
signed numbers. If two numbers have different signs (like 9 and –10), then subtract
them (10 – 9 = 1) and use the sign from the bigger number (10 > 9, so use the negative sign
attached to the 10 to get –1 instead of 1). If the signs on the numbers are the same, then
add the numbers together and use the shared sign. In other words, to simplify –12
– 4, add 12 and 4 to get 16 and then stick the shared negative sign
out front: –16.

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Chapter One — Algebraic Fundamentals
To simplify 9 – 10 using the payments and debts analogy from Problem 1.12, 9
represents $9 in cash and –10 represents $10 in debts. The net result would be a

debt of $1, so 9 – 10 = –1.

1.14 Simplify the expression:

.

Choosing the sign to use when you multiply and divide numbers works very
similarly to the method described in Problem 1.11 to eliminate double signs.
When two numbers of the same sign are multiplied, the result is always positive.
If, however, you multiply two numbers with different signs, the result is always
negative.
In this case, you are asked to multiply the numbers 6 and –3. Because one is
positive and one is negative (that is, their signs are different), the result must be
negative.

You could
also write

,
but you don’t HAVE
to write a + sign in
front of a positive
number. If a number
has no sign in
front of it, that
means it’s
positive.

There’s no
multiplication

sign written
between (3) and
(–3), so how did
you know to multiply
them together? It’s
an “unwritten rule”
of algebra. When two
quantities are written
next to one another
and no sign separates
them, multiplication is
implied. That means
things like 4(9), 10y,
and xy are all
multiplication
problems.

˘

1.15 Simplify the expression:

.

When signed numbers are divided, the sign of the result once again depends
upon the signs of the numbers involved. If the numbers have the same sign, the
result will be positive, and if the numbers have different signs, the result will be
negative. In this case, both of the numbers in the expression, –16 and –2, have
the same sign, so the result is positive:
.

1.16 Simplify the expression: (3)( –3)(4)(–4).
Multiply the signed numbers in this expression together working from left to
right. In this way, because you are multiplying only two numbers at a time, you
can apply the technique described in Problem 1.14 to determine the sign of
each result. The leftmost two numbers are 3 and –3; they have different signs, so
multiplying them together results in a negative number: (3)( –3) = –9.
(3)( –3)(4)( –4) = (–9)(4)( –4)
Again multiply the two leftmost numbers. The signs of –9 and 4 are different, so
the result is negative: (–9)(4) = –36.
(–9)(4)( –4) = (–36)( –4)
The remaining signed numbers are both negative; because the signs match,
multiplying them together results in a positive number.
(–36)( –4) = 144

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Chapter One — Algebraic Fundamentals

1.17 Simplify the expression:

.

The straight lines surrounding 9 in this expression represent an absolute value.
Evaluating the absolute value of a signed number is a trivial matter—simply
make the signed number within the absolute value bars positive and then
remove the bars from the expression. In this case, the number within the
absolute value notation is already positive, so it remains unchanged.

You are left with two signed numbers to combine: +4 and –9. According to the
technique described in Problem 1.11, combining $4 in assets with $9 in debt has
a net result of $5 in debt: 4 – 9 = –5.

1.18 Simplify the expression:

.

The absolute value of a negative number, in this case –10, is the opposite of the
negative number:
.

1.19 Simplify the expression:

.

If this problem had no absolute value bars and used parentheses instead, your
approach would be entirely different. The expression –(5) – (–5) has the
double sign “– –,” which should be eliminated using the technique described in
Problems 1.11–1.13. However, absolute value bars are treated differently than
parentheses, so this expression technically does not contain double signs. Begin
by evaluating the absolute values:
and
.

See? There’s
the double sign.
When
turned in
(+5), the negative to

si
in front of the a gn
bsolute
values didn’t go
awa
In the next step, y.
you
eliminate the dou
ble
sign “– +” to get
–5 – 5.

Absolute
value bars
are the antidepressants of the
mathematical world.
They make everything
inside positive. To say
that more precisely, they
take away the negative
of the number inside.
That means
.
However, the moodaltering lines have no
effect on positive
numbers:
.

Absolute
values are

simple when
there’s only one
number inside. If
the number inside
is negative, make it
positive and drop the
absolute value bars.
If the number’s
already positive,
leave it alone
and just drop
the bars.

Well, it
doesn’t contain
double signs YET.
It will in just a
moment.

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Chapter One — Algebraic Fundamentals

1.20 Simplify the expression:

.

Do not eliminate double signs in this expression until you have first addressed
the absolute values.

Combine the signed numbers two at a time, working from left to right. Begin
with 2 – 7 = –5.

–5 + 5 = 0

1.21 Simplify the expression:

.

This problem contains the absolute value of an entire expression, not just a single
number. In these cases, you cannot simply remove the negative signs from each
term of the expression, but rather simplify the expression first and then take the
absolute value of the result.
To simplify the expression 3 + (–16) – (–9), you must eliminate the double signs
and them combine the numbers one at a time, from left to right.

Grouping Symbols

When numbers band together, deal with them first
For now, the
parentheses and
other grouping
symbols will tell
you what pieces of
a problem to simplify
first. When parentheses

aren’t there to help, you
have to apply something
called the “order of
operations,” which
is covered in
Problems 3.30–
3.39.

˘

1.22 Simplify the expression:

.

When portions of an expression are contained within grouping symbols—like
parentheses (), brackets [], and braces {}—simplify those portions of the
expression first, no matter where in the expression it occurs. In this expression,
is contained within parentheses, so multiply those numbers:
.

1.23 Simplify the expression:

.

The only difference between this expression and Problem 1.22 is the placement
of the parentheses. This time, the expression 7 + 10 is surrounded by grouping
symbols and must be simplified first.

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Chapter One — Algebraic Fundamentals
By comparing this solution to the solution for Problem 1.22, it is clear that the
placement of the parentheses in the expression had a significant impact on the
solution.

1.24 Simplify the expression:

.

Although this expression contains parentheses and brackets, the brackets are
technically the only grouping symbols present; the parentheses surrounding –11
are there for notation purposes only. Simplify the expression inside the brackets
first.

1.25 Simplify the expression:

.

This expression contains two sets of nested grouping symbols, brackets and
parentheses. When one grouped expression is contained inside another,
always simplify the innermost expression first and work outward from there.
In this case, the parenthetical expression
should be simplified first.

A grouped expression still remains in the expression, so it must be simplified
next.

1.26 Simplify the expression:

.

Grouping symbols are not limited to parentheses, brackets, and braces. Though
it contains none of the aforementioned elements, this fraction consists of two
grouped expressions. Treat the numerator (6 + 10) and the denominator (14
– 8) as individual expressions and simplify them separately.

If you’re not sure how
turned
into , you divide the numbers in the top and bottom
and
. That process
of the fraction by 2:
is called “simplifying” or “reducing” the fraction and is
explained in Problems 2.11–2.17.

Double signs, like
in the expression
19 + (–11), are ugly
enough, but it’s just
too ugly to write the
signs right next to each
other like this:
19 + – 11. If you look
back at Problems
1.11–1.13, you’ll notice
that the second
signed number is
always encased

in parentheses if
leaving them out
would mean two
signs are
touching.

“Nested”
means that
one expression
is inside the
other one. In this
case,
is
nested inside the
bracketed expression
because
the expression inside
parentheses is also
inside the brackets.
Nested expressions are
like those egg-shaped
Russian nesting dolls.
You know the ones?
When you open
one of the dolls,
there’s another,
smaller one
inside?

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Chapter One — Algebraic Fundamentals

1.27 Simplify the expression:

The “numerator”
is the top part of the
fraction and the
“denominator” is
the bottom part.

.

Like Problem 1.26, this fractional expression has, by definition, two implicit
groups, the numerator and the denominator. However, it contains a second
grouping symbol as well, absolute value bars. The absolute value expression is
nested within the denominator, so simplify the innermost expression, first.

Now simplify the numerator and denominator separately.

Any number divided by itself equals 1, so

= 1, but note that the numerator

is negative. According to Problem 1.15, when numbers with different signs are
divided, the result is negative.

1.28 Simplify the expression:
According to
the end of Proble
m
1. 27, when you d
ivide
a number and it
s
opposite (like 7 a
nd
–7), you get –1.

.

This expression consists of two separate absolute value expressions that are
subtracted. The left fractional expression requires the most attention, so begin
by simplifying it.

Now that the fraction is in a more manageable form, determine both of the
absolute values in the expression.

1.29 Simplify the expression:
Three if you
don’t count
as a group
(because it has
only one number
inside). Four if
you do count

it.

10

.

This problem contains numerous nested expressions—braces that contain
brackets that, in turn, contain parentheses that include an absolute value.
Begin with the innermost of these, the absolute value expression.

The innermost expression surrounded by grouping symbols is now (3 + 1), so
simplify it next.

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Chapter One — Algebraic Fundamentals

The bracketed expression is now the innermost group; simplify it next.

1.30 Simplify the expression:

.

The numerator and denominator both contain double signs within their
innermost nested expressions. Begin simplifying there, and carefully work your
way outward. For the moment, ignore the absolute value signs surrounding the
entire fraction.

Evaluate

and

Leave the
big, outside
absolute value bars
until the very end,
after you have a single
number on the top
and bottom of the
fraction.

to continue simplifying.

Now that the numerator and denominator each contain a single real number
value, take the absolute value of the fraction that remains.

Algebraic Properties

Basic assumptions about algebra
1.31 Simplify the expressions on each side of the following equation to verify that
the sides of the equation are, in fact, equal.

(3 + 9) + 10 = 3 + (9 + 10)
Each side of the equation contains a pair of terms added within grouping
symbols. According to Problem 1.22, those expressions should be simplified
first.

Both sides of the equation have a value of 22 and are, therefore, equal.

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Chapter One — Algebraic Fundamentals

1.32 What algebraic property guarantees that the equation (3 + 9) + 10 = 3 +
(9 + 10) from Problem 1.31 is true?

There’s also an
associ-ative
property for multiplication, which
says that you can
regroup numbers that
are multiplied together
and it won’t change
the answer. Here’s an
example:

The only difference between the sides of the equation is the placement of
the parentheses. According to the associative property of addition, if a set of
numbers is added together, the manner in which they are grouped will not
affect the total sum.

1.33 Simplify the expressions on each side of the following equation to verify that
the sides of the equation are, in fact, equal:

There are no grouping symbols present to indicate the order in which you
should multiply the numbers on each side of the equation. Therefore, you
should multiply the numbers from left to right, starting with
on the left
side of the equation and
on the right.

The rule
stating that you
should multiply a
of numbers from string
le
right is part of th ft to
e order
of operations. Prob
lem
3.30 –3.39 cover th s
is in
more detail.

1.34 What algebraic property guarantees that the equation in Problem 1.31 is true?
The sides of the equation in Problem 1.33 contain the same values; however,
they are listed in a different order. The commutative property of multiplication
states that re-ordering a set of real numbers multiplied together will not affect
the product.

Just like the
associative property, the
commutative property works for both
addition and multiplication. If you’ve got a

big list of numbers added together, you can add
them in any order you want, and you’ll get the same
thing. In case you’d like to see visual evidence, here’s
Exhibit A:

“Product” is
a fancy word fo
r
“what you get w
hen
you multiply thin
gs,” like
“sum” is a fancy
wa
say “what you ge y to
t when
you add things.”

12

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Chapter One — Algebraic Fundamentals

1.35 According to the associative properties of addition and multiplication,

the manner in which values are grouped does not affect the value of the
expression. However, Problems 1.22 and 1.23, which contain only addition and

multiplication, prove that
.

How is it possible that grouping the expressions differently changed their
values, despite the guarantees of the associative properties?
The associative properties of addition and multiplication are separate and
cannot be combined. In other words, you can apply the associative property of
addition only when addition is the sole operation present, and you can apply
the associative property of multiplication only when the numbers involved
are multiplied. Neither associative property can be applied to the expression
because it contains both addition and multiplication.

1.36 Describe the identity properties of addition and multiplication, including the
role of the additive and multiplicative identities.

According to the identity property of addition, adding 0 (the additive identity)
to any real number will not change the value of that number. Similarly,
the multiplicative identity states that multiplying a real number by 1 (the
multiplicative identity) doesn’t change the value either.

1.37 Complete the following statement and explain your answer:

According to the ______________ property, if a = b, then b = a.

The symmetric property guarantees that two equal quantities are still equal
if written on opposite sides of the equal sign. In other words, if x = 5, then it
is equally correct to state that 5 = x.

1.38 According to the distributive property, if a, b, and c are real numbers,
then

expression 3(2 – 7).

. Apply the distributive property to simplify the

The distributive property applies to expressions within grouping symbols that
are multiplied by another term. Here, the entire expression (2 – 7) is multiplied
by 3. The distributive property allows you to multiply each term within the
parentheses by 3.

Don’t over
think this one—it
’s
nothing you don’t
already know. If
yo
multiply a numbe u
r by
1 or add 0 to it,
the
number’s IDENTIT
Y
doesn’t change:
5 + 0 = 5 and
.

So if you’re
as old as I am,
then I am as old as
you are. Hmmmm.
Not very shocking

or particularly
groundbreaking.

Multiply 3(2) and 3(–7) before adding the terms together. According to the
algebraic order of operations, multiplication within an expression should be
completed before addition. For a more thorough investigation of this topic, see
Problems 3.30–3.39.

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13