Humongous book algebra problems by michael kelley

73 Your one-stop shop for a review of numbers Numbers fall into different groups Add, subtract, multiply, and divide positive and negative numbers When numbers band together, deal with t

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Introduction

Number Classification 2

Expressions Containing Signed Numbers 5

Grouping Symbols 8

Algebraic Properties 11

Chapter 2: Rational Numbers 17 Rational Number Notation 18

Simplifying Fractions 23

Combining Fractions 26

Chapter 3: Basic Algebraic Expressions 37 Translating Expressions 38

Exponential Expressions 40

Distributive Property 45

Order of Operations 48

Evaluating Expressions 51

Chapter 4: Linear Equations in One Variable 55 Adding and Subtracting to Solve an Equation 56

Multiplying and Dividing to Solve an Equation 59

Solving Equations Using Multiple Steps 61

Absolute Value Equations 70

Equations Containing Multiple Variables 73

Your one-stop shop for a review of numbers Numbers fall into different groups

Add, subtract, multiply, and divide positive and negative numbers When numbers band together, deal with them first

Basic assumptions about algebra

U nderstandin g fr a c t i on s sure be ats being a fraid of t hem

Proper and improper fractions, decimals, and mixed numbers Add, subtract, multiply, and divide fractions

Reducing fractions to lowest terms, like 1/2 instead of 5/10

Time for x to make its stunning debut The alchemy of turning words into math Rules for simplifying expressions that contain powers Multiply one thing by a bunch of things in parentheses

My dear Aunt Sally is eternally excused Replace variables with numbers

How to solve basic equations Add to/subtract from both sides Multiply/divide both sides Nothing new here, just more steps Most of them have two solutions

Equations with TWO variables (like x and y) or more

Chapter 5: Graphing Linear Equations in Two Variables 77

Number Lines and the Coordinate Plane 78

Graphing with a Table of Values 83

Graphing Using Intercepts 90

Calculating Slope of a Line 93

Graphing Absolute Value Equations .100

Chapter 6: Linear Equations in Two Variables 105 Point-Slope Form of a Linear Equation .106

Slope-Intercept Form of a Linear Equation .110

Graphing Lines in Slope-Intercept Form .113

Standard Form of a Linear Equation .118

Creating Linear Equations .121

Chapter 7: Linear Inequalities 127 Inequalities in One Variable .128

Graphing Inequalities in One Variable .132

Compound Inequalities .135

Absolute Value Inequalities .137

Set Notation .140

Graphing Inequalities in Two Variables .142

Chapter 8: Systems of Linear Equations and Inequalities 147 Graphing Linear Systems .148

The Substitution Method .153

Variable Elimination .162

Systems of Inequalities .168

Linear Programming .173

Chapter 9: Matrix Operations and Calculations 181 Anatomy of a Matrix .182

Adding and Subtracting Matrices .183

Multiplying Matrices .188

Calculating Determinants .192

Cramer’s Rule 200

Which should you use to graph?

Plug in some x’s, plot some points, call it a day The easiest way to plot two points on a line quickly

Figure out how slanty a line is Don’t miss the point in these graphs (Get it?)

Point + slope = equation Lines that look like y = mx + b Graphing equations that are solved for y Write equations of lines in a uniform way Practice all the skills from this chapter

Generating equations of lines

They’re like equations without the equal sign Dust off your equation-solving skills from Chapter 4

Shoot arrows into number lines Two inequalities for the price of one Break these into two inequalities

A fancy way to write solutions

Lines that give off shade in the coordinate plane

Work with more than one equation at a time

Graph two lines at once Solve one equation for a variable and plug it into the other Make one variable disappear and solve for the other one

The answer is where the shading overlaps Use the sharp points at the edge of a shaded region

Numbers in rows and columns The order of a matrix and identifying elements

Combine numbers in matching positions Not as easy as adding or subtracting them Values defined for square matrices only Double-decker matrices that solve systems

entify the points that make a n e q ua t i o n tr

u

Chapter 10: Applications of Matrix Algebra 207

Augmented and Identity Matrices .208

Matrix Row Operations .211

Row and Reduced-Row Echelon Form .216

Inverse Matrices .228

Chapter 11: Polynomials 237 Classifying Polynomials .238

Adding and Subtracting Polynomials .239

Multiplying Polynomials .244

Long Division of Polynomials .246

Synthetic Division of Polynomials .251

Chapter 12: Factoring Polynomials 257 Greatest Common Factors .258

Factoring by Grouping .265

Common Factor Patterns .267

Factoring Quadratic Trinomials .270

Chapter 13: Radical Expressions and Equations 275 Simplifying Radical Expressions .276

Rational Exponents .281

Radical Operations .283

Solving Radical Equations .288

Complex Numbers 290

Chapter 14: Quadratic Equations and Inequalities 295 Solving Quadratics by Factoring .296

Completing the Square .300

Quadratic Formula .305

Applying the Discriminant .312

One-Variable Quadratic Inequalities .316

Extra columns and lots of 0s and 1s

Advanced matrix stuff

Swap rows, add rows, or multiply by a number More matrices full of 0s with a diagonal of 1s Matrices that cancel other matrices out

Clumps of numbers and variables raised to powers Labeling them based on the exponent and total terms

Only works for like terms FOIL and beyond

A lot like long dividing integers Divide using only the coefficients

S qua r e ro o ts, cube r oots, a nd fractional expo n en t s

Moving things out from under the radical Fractional powers are radicals in disguise Add, subtract, multiply, and divide roots

Use exponents to cancel out radicals Numbers that contain i, which equals √— –1

Solve equations containing x2

Use techniques from Chapter 12 to solve equations Make a trinomial into a perfect square

Use an equation’s coefficients to calculate the solution

What b2 – 4ac tells you about an equation

Inequalities that contain x2

Largest factor that divides into everything evenly The opposite of multiplying polynomials

You can factor out binomials, too Difference of perfect squares/cubes, sum of perfect cubes

Turn one trinomial into two binomials

Chapter 15: Functions 323

Relations and Functions .324

Operations on Functions .326

Composition of Functions .330

Inverse Functions .335

Piecewise-Defined Functions .343

Chapter 16: Graphing Functions 347 Graphing with a Table of Values .348

Domain and Range of a Function .354

Symmetry .360

Fundamental Function Graphs 365

Graphing Functions Using Transformations .369

Absolute Value Functions 374

Chapter 17: Calculating Roots of Functions 379 Identifying Rational Roots .380

Leading Coefficient Test .384

Descartes’ Rule of Signs .388

Rational Root Test .390

Synthesizing Root Identification Strategies .394

Chapter 18: Logarithmic Functions 399 Evaluating Logarithmic Expressions .400

Graphs of Logarithmic Functions .402

Common and Natural Logarithms .406

Change of Base Formula .409

Logarithmic Properties .412

Chapter 19: Exponential Functions 417 Graphing Exponential Functions .418

Composing Exponential and Logarithmic Functions .423

Exponential and Logarithmic Equations .426

Exponential Growth and Decay .433

Named expressions that give one output per input

What makes a function a function?

+, –, ·, and ÷ functions Plug one function into another Functions that cancel each other out Function rules that change based on the x-input

Drawing graphs that aren’t lines

Plug in a bunch of things for x What can you plug in? What comes out?

Pieces of a graph are reflections of each other

The graphs you need to understand most Move, stretch, squish, and flip graphs These graphs might have sharp points

Roots = solutions = x-intercepts Factoring polynomials given a head start The ends of a function describe the ends of its graph Sign changes help enumerate real roots

Find possible roots given nothing but a function

Factoring big polynomials from the ground up

Contains enough logs to build yourself a cabin

Given loga b = c, find a, b, or c All log functions have the same basic shape What the bases equal when no bases are written Calculate log values that have weird bases

Expanding, contracting, and simplifying log expressions

Functions with a variable in the exponent Graphs that start close to y = 0 and climb fast

They cancel each other out Cancel logs with exponentials and vice versa Use f(t) = Nekt to measure things like population

Chapter 20: Rational Expressions 439

Simplifying Rational Expressions .440

Adding and Subtracting Rational Expressions .444

Multiplying and Dividing Rational Expressions .452

Simplifying Complex Fractions .457

Graphing Rational Functions .459

Chapter 21: Rational Equations and Inequalities 465 Proportions and Cross Multiplication .466

Solving Rational Equations .470

Direct and Indirect Variation .475

Solving Rational Inequalities .479

Chapter 22: Conic Sections 487 Parabolas .488

Circles .494

Ellipses .499

Hyperbolas .506

Chapter 23: Word Problems 515 Determining Unknown Values .516

Calculating Interest .521

Geometric Formulas .525

Speed and Distance .529

Mixture and Combination .534

Work .538

Fractions with lots of variables in them Reducing fractions by factoring Use common denominators Common denominators not necessary Reduce fractions that contains fractions Rational functions have asymptotes

Solve equations using the skills from Chapte

r 20

When two fractions are equal, “X” marks the solution Ditch the fractions or cross multiply to solve Turn a word problem into a rational equation Critical numbers, test points, and shading

Parabolas, Circles, Ellipses, and Hyperbolas Vertex, axis of symmetry, focus, and directrix

Center, radius, and diameter Major and minor axes, center, foci, and eccentricity Transverse and conjugate axes, foci, vertices, and asymptotes

If two trains leave the station full of consecutiveintegers, how muc interest

isearned?

Integer and age problems Simple, compound, and continuously compounding Area, volume, perimeter, and so on

Distance equals rate times time Measuring ingredients in a mixture How much time does it save to work together?

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.

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.

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

double-checked 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.

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

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.

Chapter 1

AlgebrAic FundAmentAls

Algebra, at its core, is a compendium of mathematical concepts, axioms, theorems, and algorithms rooted in abstraction Mathematics is most pow- erful 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 be- fore they can make the next logical step, representing concrete values with abstract notation.

This chapter ensures that you are thoroughly familiar with the most mon classifications used to describe numbers, provides an opportunity to manipulate signed numbers arithmetically, and investigates the founda- tional mathematical principles that govern algebra.

com-You might be anxious to dive into the nuts and bolts of algebra, but

don’t skip over the stuff in this chapter It’s full of key vocabulary words, such as “rational number” and “commutative property.” You also learn things like the difference between real and 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.

Your one - sto p shop f or a review of nu m bers

Number Classification

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. 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 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 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

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

8, that aren’t prime

because they are

divisible by too many

things, are called

“composite numbers.”

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

615384615384615384

1.6 The irrational mathematical constant p is sometimes approximated with the

fraction 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

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.

Complex numbers are discussed in more detail later

in the book, in Problems 13.37- 13.44.

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 by identifying the number sets to which it belongs.Because 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

According

to Problem 1.1,

the single element

that the whole

numbers contain and

the natural numbers

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

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

In this problem, the adjacent signs are different, “+ –,” so you must replace them

with a single negative sign: –

1.1 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

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 – 10Simplify the expression from left to right, beginning with 4 + 5 = 9

4 + 5 – 10 = 9 – 10

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.

Think of it this way If the two signs agree with each other (if they’re both positive or both negative), then that’s a good thing, a POSITIVE thing On the other hand, when two signs can’t agree with each other (one’s positive and one’s negative), then that’s

no good That’s NEGATIVE.

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.

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

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

You could

also write

, but you don’t HAVE

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.

1.17 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

Absolute value bars are the anti- depressants 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 mood- altering 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.

See? There’s the double sign

When turned into (+5), the negative sign

in front of the absolute values didn’t go away

In the next step, you eliminate the double sign “– +” to get –5 – 5.

1.0 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

1.1 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

1. 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:

1.3 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

–5 + 5 = 0

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.

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

1.4 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

1.5 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

1.6 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

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?

If you’re not sure how turned into , you divide the numbers in the top and bottom

of the fraction by 2: and That process

is called “simplifying” or “reducing” the fraction and is

explained in Problems 2.11–2.17.

1.7 Simplify the expression: 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

1.8 Simplify the expression: 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

1.9 Simplify the expression:

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

The “numerator”

is the top part of the

fraction and the

“denominator” is

the bottom part.

According to

the end of Problem

1.27, when you divide

a number and its

opposite (like 7 and

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 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

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.

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

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

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

There’s also an

associ-ative

property for

mul-tiplication, which

says that you can

regroup numbers that

are multiplied together

and it won’t change

the answer Here’s an

example:

The rule

stating that you

should multiply a string

of numbers from left to

right is part of the order

of operations Problems

3.30–3.39 cover this in

more detail.

“Product” is

a fancy word for

“what you get when

you multiply things,” like

“sum” is a fancy way to

say “what you get when

you add things.”

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:

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

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 Apply the distributive property to simplify the

expression 3(2 – 7)

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

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

Don’t over think this one—it’s nothing you don’t already know If you multiply a number by

1 or add 0 to it, the number’s IDENTITY 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.

1.39 Simplify the expression 3(2 – 7) (from Problem 1.38) once again, this time calculating the sum within the grouping symbols first Verify that the result matches the answer produced by the distributive property in Problem 1.38.Although the distributive property applies to this expression (as explained in Problem 1.38), simplifying the expression inside the grouping symbols first makes the problem significantly easier.

The expressions in Problems 1.38 and 1.39, though simplified differently, have the same value

1.40 Explain the additive inverse property and demonstrate it mathematically using

a real number

The additive inverse property states that adding any real number to its opposite results in the additive identity, 0 Consider the number 6; the sum of 6 and its opposite, –6, is 0: 6 + (–6) = 0 The property applies to negative numbers as well Adding –3 to its opposite, +3, also results in 0: –3 + 3 = 0

1.41 Explain the multiplicative inverse property and demonstrate it mathematically using a real number

The multiplicative inverse property states that multiplying a number by its reciprocal results in the multiplicative identity, 1 For instance, if you multiply

2 by its reciprocal , the product is 1

1.4 Complete the following statement and explain your answer:

According to the transitive property, if a = b and b = c, then _.

The transitive property describes the relative equality of three quantities Here,

the quantity a is equal to the quantity b In turn, b is equal to a third quantity, c

If b is equal to both a and c, it follows logically that a and c must also be equal Therefore, the equation a = c correctly completes the statement.

fraction is the fraction

you get by reversing

the numerator and

denominator (So the

you nauseated, don’t

worry You’ll find

lots of practice in

Problems 2.33–

2.37

1.43 Identify the mathematical property that justifies the following statement and

explain your answer:

Both sides of the equation contain the same numbers in the same order The

only difference between the sides of the equation is the way the numbers are

grouped by the parentheses Therefore, this statement is true by the associative

property of multiplication, which states that the way a set of real numbers is

grouped does not affect its product

1.44 Identify the mathematical properties that justify the following statement and

explain your answer:

If 3 = x and x = y, then y = 3.

According to the transitive property, if 3 = x and x = y, then 3 = y To rewrite

3 = y as y = 3 to match the given statement, you must apply the symmetric

property

If the numbers had been in a different order

on either side

of the equal sign, the commutative property would have come into play The commutative property says “order doesn’t matter,” and the associative property says “it doesn’t matter where you stick the parentheses.”

See Problem 1.42 for more information about the transitive property and Problem 1.37 for more info about the symmetric property.

Chapter 

rAtionAl numbers

Chapter 1 introduced the concept of rational numbers, real numbers that can be expressed as a fraction, a terminating decimal, or a repeating deci- mal Rational numbers are truly Gestalt, which is to say they are greater than the sum of their parts By nature they are merely quotients of integers, but their complexity requires a unique set of concepts (such as the least common denominator) and procedures (such as reducing to lowest terms and transforming rational numbers from fractions to decimals and vice ver- sa) Through the study of rational numbers, and the careful, often rigorous, techniques that surround them, unique and otherwise obfuscated proper- ties of integers are uncovered.

When you divide two integers, you get a fraction The fancy name

for “fraction” is “rational number,” and that’s what you deal with in

this chapter Fractions aren’t as hard to handle as most people think— you just need to know that when fractions are in the mix, you need

to follow specific rules For instance, you can only add and subtract

fractions if they have the same denominator Simple enough However,

if fractions have DIFFERENT denominators, you CAN multiply and divide them

Spend some time getting familiar with fractions in this chapter When you finish, you’ll be able to change decimals into fractions; change

fractions into decimals; simplify fractions; identify a least common

denominator; and add, subtract, multiply, and divide fractions to your heart’s content

Un de rst a nding f rac t i ons su re be a ts b e ing a f raid o f th em

Rational Number Notation

Proper and improper fractions, decimals, and mixed numbers

.1 Express 0.013 as a percentage

Decimal numbers, like percentages, express a value in terms of a whole

This whole value can be expressed in decimal form, as 1 (or 1.0), and as a percentage, 100% Transforming a decimal into a percent is as simple as moving the decimal point exactly two digits to the right Here, 0.013 = 1.3%

. Express 0.25% as a decimal

According to Problem 2.1, converting from a decimal to a percentage requires you to move the decimal point two digits to the right It comes as no surprise, then, that performing the opposite conversion, from percentage to decimal, requires you to move the decimal exactly two digits to the left

In this problem, only one digit, 0, appears to the left of the decimal The second, unwritten, digit is also 0 Therefore, 0.25% = 00.25% = 0.0025%

Note: Problems 2.3–2.4 refer to the rational number

.3 Express the fraction as a decimal

The rational number represents the quotient To express the fraction as

a decimal, use long division to divide 4 by 1 Set up the long division problem, writing an additional zero at the end of the dividend Copy the decimal point above the division symbol

For the moment, ignore the decimal point within the dividend and imagine that 1.0 is equal to 10 Because 4 divides into 10 two times, place a 2 above

the rightmost digit of 10 Because 4 does not divide evenly into 10, a remainder

mean the same thing

You can also add

zeroes at the end of

a decimal: 1.5, 1.50,

1.500, and so on.

The number

you’re dividing BY

is called the “divisor”

and the number you’re

dividing INTO is called

the “dividend.” The

answer you get

once you’re done

dividing is called

the “quotient.”

You can

add as many

zeroes as you want,

and you can do it at any

time during the problem

Here’s your goal: you want

the answer to have either

terminated or begun to

repeat If it hasn’t done

either, pop some more

zeroes up there and

keep going.

Subtract 8 from 10 and write the result below the horizontal line.

The difference (10 – 8 = 2) is not 0, so the quotient has not yet terminated Place

another 0 on the end of the dividend and on the end of the number below the

horizontal line

This time, dividing the bottommost number by the divisor produces no

remainder; 4 divides evenly into 20 Write the result above the division symbol

next to the 2 already there

Multiply the newest digit in the quotient (5) by the divisor (4) and write the

result beneath 20

Subtract the bottom two numbers

Because the remainder is 0, the division problem is complete:

Rational numbers either repeat or terminate When you get a remainder of 0, you stop dividing, and the decimal terminates

Note: Problems 2.3–2.4 refer to the rational number

.4 Express the fraction as a percentage

According to Problem 2.3, To transform a decimal into a percentage, move the decimal point two digits to the right: 0.25 = 25.0%, or simply 25%

Note: Problems 2.5–2.7 refer to the rational number

.5 Express the fraction as a decimal

Use the method described in Problem 2.3 to rewrite the fraction as a long vision problem Here, however, there is no immediate need to place a 0 in the dividend No decimal point is written explicitly, so write one at the end of the dividend and copy it above the division symbol

di-Six divides into eleven one time, so write 1 above the rightmost digit of 11

Multiply the divisor by the digit just written above the dividend , write the result below the dividend and subtract

The divisor (6) cannot divide into the result (5) a whole number of times As Problem 2.3 directed, change 5 into 50 and also add a zero to the end of the dividend

Six divides into 50 eight times, so place 8 at the right end of the quotient Multiply the new digit by the divisor (6), write the result below 50, and subtract

In Problem

2.3, you had

to add a zero

because 4 divides

into 10 but it really

doesn’t divide into 1

very well In Problem

dividing INTO has to

be bigger than what

you’re dividing BY If

it’s not, add a zero

to the dividend

and the bottom

number.

Six does not divide into 2 a whole number of times, so once again insert zeroes

after 2 and 11.0

Six divides into 20 three times Take the appropriate actions in the long division

problem

Once again, 2 is the bottommost number in the long division problem If you

place zeros after it and after the dividend, it produces another 3 in the quotient

Once again, 2 is the bottommost number Repeating this process is futile—each

time a zero is added, it produces another 3 in the quotient and a difference of

2, the same number with which you started Because the division problem has

turned into an infinite loop producing the same pattern of digits, you can

conclude that

Put a 3 above the division symbol, multiply it by the divisor (6), write that multiplication result (18) below

20, and then subtract it from 20.

If the decimal form of

a rational number doesn’t terminate, one or more digits will repeat infinitely Write the decimal with a little bar over the

3 to indicate that it’s an infinitely repeating digit in the decimal.

Note: Problems 2.5–2.7 refer to the rational number

.6 Express the fraction as a percentage

According to problem 2.5, Move the decimal two places to the right to convert the decimal into a percentage:

Note: Problems 2.5–2.7 refer to the rational number

.7 Express the fraction as a mixed number

The fraction is considered improper because its numerator is greater than its denominator To express it as a mixed number, divide 11 by 6 There is no need to use long division—it is sufficient to conclude that 6 divides into 11 one time with a remainder of 5 Thus The whole number portion of the mixed number is the number of times the divisor divides into the dividend; the remainder is the numerator of the fraction The denominator of the fraction matches the denominator of the original fraction

In other words, given the improper fraction , if x divides into y a total of w

times with a remainder of r, then

.8 Express the improper fraction as a mixed number

Four divides into 65 a total of 16 times with a remainder of 1 Therefore

The fractional part of the mixed number consists of the remainder divided by the original denominator

.9 Express as an improper fraction

To convert a mixed number into a fraction, multiply the denominator and the whole number and then add the numerator Divide by the denominator

of the mixed number:

.10 Express as an improper fraction

According to Problem 2.9, Substitute a = 4, b = 5, and c = 12 into

the formula

Improper

doesn’t mean

unacceptable

It’s fine to have

fractions with bigger

numbers on top than

on bottom, and most

figure this out

Either long divide

or use a

calculator You get

16.25 The number left

of the decimal (16) is

the nonfraction part

of the mixed number

To figure out the

remainder, multiply

that whole number

by the denominator

and

subtract what you

get from the

numerator:

65 – 64 = 1.

Simplifying Fractions

Reducing fractions to lowest terms, like instead of

.11 Explain the process used to reduce a fraction to its lowest terms

A fraction is in lowest terms only when its numerator and denominator no

longer share any common factors In other words, the fraction is reduced if no

number divides evenly into both the numerator and denominator One effective

way to reduce a fraction is to identify the greatest common factor (GCF) of both

numbers and then divide each by that GCF

Note: Problems 2.12–2.13 refer to the rational number

.1 Identify the greatest common factor of the numerator and denominator

List all the factors of the numerator (numbers that divide evenly into 8) and the

denominator (numbers that divide evenly into 24)

The largest factor common to both lists is 8, so the greatest common factor of 8

and 24 is 8

Note: Problems 2.12–2.13 refer to the rational number

.13 Reduce the fraction to lowest terms

According to Problem 2.12, the greatest common factor of 8 and 24 is 8, so

divide both the numerator and the denominator by 8 to reduce the fraction

Though and are different fractions, they have equivalent values:

Note: Problems 2.14–2.15 refer to the rational number

.14 Identify the greatest common factor of the numerator and denominator

List the factors of the numerator and denominator

The largest factor common to both lists is 9, so the greatest common factor of

27 and 63 is 9

Except 1, because 1 can divide evenly into anything.

If you have two dozen eggs, most people would say, “I have one-third

of the eggs” rather than “I have 8 of the

24 eggs.” Both are correct, but the first is easier to visualize.

There’s no magic trick for generating the list of factors Start by trying to divide the number

by 2, then by 3, and

so on, until you identify all the numbers that divide in evenly Here’s one tip: All the numbers

in the list are paired up For instance, after you figure out that 3 is a factor of 63, then

also has

to be in the list

of factors.

Note: Problems 2.14–2.15 refer to the rational number

.15 Express the fraction in lowest terms

According to Problem 2.14, the greatest common factor of 27 and 63 is 9, so divide the numerator and denominator by 9 to reduce the fraction

.16 Reduce the fraction to lowest terms

List the factors of the numerator and denominator Although this fraction is negative, the technique you use to reduce it to lowest terms remains unchanged

Divide the numerator and denominator by 4, the greatest common factor

.17 Reduce the fraction to lowest terms and identify the greatest common factor of 2,024 and 8,448

The technique demonstrated in Problems 2.11–2.16 is not convenient when the numbers in the numerator and denominator are very large Because 2,024 and 8,448 each have many factors, rather than list all of them, identify one common factor (it does not matter which) and use it to reduce the fraction In this case, both numbers are even, so you can divide each by 2

The fraction is not yet reduced to lowest terms Notice that the numerator and denominator of the resulting fraction are again even, so divide both by 2

Once again the fraction consists of even numbers Divide by 2

If it

would take

too long to write

all the factors,

find something that

divides evenly into the

top and the bottom

of the fraction and

reduce it Keep doing

that until you can’t

find any common

factors.

Continue to look for common factors Notice that 253 and 1,056 are evenly

divisible by 11

Now that the fraction is reduced to lowest terms, calculate the greatest common

factor by multiplying each of the numbers that were eliminated from the

fraction

.18 Express 0.45 as a fraction in lowest terms

The decimal 0.45 is read “forty-five hundredths,” because it extends two digits

to the right of the decimal point Forty-five hundredths literally translates into

the fraction Reduce the fraction to lowest terms.

.19 Express 1.843 as an improper fraction

The decimal 1.843 is read “one and eight hundred forty-three thousandths.”

Therefore, you should divide 843 by one thousand to convert the decimal into

a mixed number: The number left of the decimal, 1, becomes the

whole part of the mixed number That fraction cannot be reduced, because the

greatest common factor of 843 and 1,000 is 1

Use the method described in Problem 2.7 to convert the mixed number into an

improper fraction

.0 Express 0.5 as a fraction

The decimal 0.5 repeats infinitely (0.5 0.5555555 = ); therefore, you cannot

use the method described in Problems 2.18–2.19 to convert this number into a

fraction If the digits of a repeating decimal begin repeating immediately after

the decimal point (that is, the repeated string begins in the tenths place of the

decimal), then you can apply a shortcut to rewrite the decimal as a fraction:

Divide the repeated string by as many 9s as there are digits in the repeated

string

When you’re looking for factors, don’t stop checking at 10, or you’ll miss this one You don’t have to try every number in the world, though—it depends on the smaller

of the two numbers

in the numerator and denominator If that number is less than 225, you can stop looking for factors at 15 If it’s less than 400, you can stop looking for factors

at 20 Basically, if the smaller number

is less than a number N, then stop checking

at

0.5 is five tenths, 0.05 is five hundredths, 0.005

is five thousandths, 0.0005 is five ten thousandths, etc.

Each digit right of the decimal point is a power of 10 Because 0.45 ends two digits right of the decimal, you divide it by 102 To make 0.12986 into a fraction, you divide

it by 105, because it has five digits left

of the decimal

In Problems 2.18–2.19, the number of d igits after the decimal told you what power of 10 t decimal never ends, so you can’t get t o divide by However, this he power of 10 you need.

In this case, the repeated string is consists of one digit (5) To convert into a fraction, divide the repeated string by 9.

The rationale behind this shortcut is omitted here, as it is based on skills not discussed until Chapter 4 This technique, in its more rigorous form, is explained in greater detail in Problems 4.26–4.28

Combining Fractions

Add, subtract, multiply, and divide fractions

. Explain what is meant by a least common denominator

Equivalent fractions might have different denominators For instance, Problem 2.13 demonstrated that and have the same value, as is expressed

in lowest terms It is often useful to rewrite one or more fractions so that their denominators are equal Usually, there are numerous options from which you can choose a common denominator, and the least common denominator is the smallest of those options

Note: Problems 2.23–2.25 refer to the fractions and

.3 Identify the least common denominator of the fractions

Begin by identifying the largest of the given denominators; here, the largest

denominator is 10 Because the other denominator (2) is a factor of 10,

then 10 is the least common denominator (LCD) The LCD is never smaller than the largest denominator

Note: Problems 2.23–2.25 refer to the fractions and

.4 Generate equivalent fractions using the least common denominator

To rewrite using the least common denominator, divide the LCD by the

current denominator: Multiply the numerator and denominator of

by that result.

Because already contains the least common denominator, it does not need

to be rewritten

Note: Problems 2.23–2.25 refer to the fractions and

.5 Calculate the sum of the fractions

To calculate the sum or difference of fractions, those fractions must have a

common denominator According to Problem 2.24,

Add the numerators of the fractions, but not the denominators

Unless otherwise directed, you should always reduce answers to lowest terms

Note: Problems 2.26–2.27 refer to the fractions , , and

.6 Identify the least common denominator

The largest denominator of the three fractions is 9 However, both of the

remaining denominators are not factors of 9, so 9 is not the LCD To identify

another potential LCD candidate, multiply the largest denominator by 2:

All the denominators (3, 6, and 9) are factors of 18, so it is the LCD

Multiplying the top and bottom of the fraction by 5 is like multiplying the entire fraction by You’re allowed

to do that because

, and multiplying any number

by 1 doesn’t change

it, according to the multiplicative identity property

in Problem 1.36.

In other words, if you want to ADD

or SUBTRACT fractions ….

If 18 didn’t work, you’d test

to see if were the LCD If

27 didn’t work, you’d multiply 9 by 4, then

5, then 6, and so on, until finally all the denominators divided in evenly.

Note: Problems 2.26–2.27 refer to the fractions , , and

.7 Simplify the expression:

To rewrite the fraction using the least common denominator, divide each denominator into 18: , , and Multiply the numerator and denominator of each fraction by the corresponding result In other words, multiply the first fraction by , the second fraction by , and the third fraction by

Combine the numerators into a single numerator divided by the least common denominator and simplify

The result is already in lowest terms:

.8 Simplify the expression: The largest denominator (16) is not evenly divisible by both of the other denominators (3 and 12) so multiply it by 2: However, 32 is not divisible by all of the denominators either, so multiply the largest denominator

by 3: Because 48 is divisible by 3, 12, and 16, it is the least common denominator

Rewrite the expression using the method described by Problem 2.27: Divide each denominator into the LCD and multiply the numerator and denominator

of each fraction by the corresponding result

Don’t simplify

these fractions,

or you’ll end up with

what you started

with and destroy

the common

denominators.