Algebra 1 by Fred Duckworth

For information: Fred W. Duckworth, Jr. c/o Jewels Educational Services 1560 East Vernon Avenue Los Angeles, CA 90011-3839 E-mail: admin@trinitytutors.com Website: www.trinitytutors.com

Copyright ÂŠ 2011 by Fred Duckworth. All rights reserved. This publication is copyrighted and may only be copied, distributed or displayed for personal use on an individual, one-time basis. Transmitting this work in any form by any means, electronic, mechanical or otherwise, without written permission from the publisher is expressly prohibited. All copyright notifications must be included and you may not alter them in any way. Moreover, you may not modify, transform, or build upon this work, nor use this work for commercial purposes.

Table of Contents – Unit 1 Introduction to Algebra………………...……….………………………..……4 Roots and Exponents…………………………………....……...……………17 Absolute Value………………………………….…..……………….…………18 Simplifying Expressions…………………..………….....……………..….…19 Multi-step Problems……………...……………………....……..…….………20 Graphing Linear Equations…………….…….……..…..………………….. 21 Equation of a Line………………………..……….……....…………………..22 Lines and Slopes…………………………….…………....……...……………23 Linear Equations in Two Variables………….…..………...…….…………24 Solving Polynomials…………………..………….....……...…………..….…25 Factoring…………………..……………….……….....……...….………..….…26 Fractions with Polynomials…...……..…..………………………...……….. 27 Solving Rational Expressions and Functions…………………….…….. 28 Completing the Square…………………......…………....………………….. 29 Solving Problems……………………....…………………………………….. 30 Relations and Functions……………..………….……....……….………….. 31 Domains and Ranges………………………………………………….…….. 32 Determining Functions……………………………………………….…….. 33 The Quadratic Formula………………………….…………………….…….. 34 Quadratic Functions………………………….…….………………….…….. 36 Reasoning………………………………...……….…………………….…..….. 39

Introduction / Part I WHAT IS ALGEBRA? When you studied arithmetic, you were learning all about numbers and the four ways in which they work together—the four operations: addition, subtraction, multiplication and division. You also learned most of the rules that govern or control such interactions, called properties. However, while algebra pretty much continues on that same track, it replaces some of the numbers with variables, which are symbols (usually letters from the alphabet printed in italics) substituting for various numerical values. Algebra does this in order to fulfill two of its primary functions: 1. First of all, algebra is designed to make it easier to study the properties of numbers. You see, while arithmetic is able to look at what particular numbers do under narrowly defined circumstances, using variables allows algebra to look at what entire categories of numbers do in general. Algebra can take into consideration a whole range of numbers all at once (along with the general properties of those numbers) without having to worry about their specific attributes. Again, this is because variables can assume any value in a problem. Algebra therefore makes it easier to see how certain properties act on (or affect) numbers and operations generally speaking. 2. Algebra’s second primary function is that of solving a variety of real-life problems. Some of these problems involve changing values—quantities that are not fixed or which vary over time—and it is only by using variables to represent these changing quantities that we are able to write, relatively easily, statements expressing their corresponding relationships. So then, stated as simply as possible: Algebra is the study of the general properties governing how numbers operate.

Introduction / Part II WHY STUDY ALGEBRA? As just stated, algebra is used to solve a variety of real-life problems, but in actuality, there are at least six good reasons for studying algebra: 1. First of all, algebra sharpens one’s mind in ways that enhance one’s ability to solve the afore mentioned problems 2. Second, it shows us surprising things about the world, and makes sense of things that are important to understand. 3. Also, algebra empowers us by giving us compact, general formulas that take the place of long lists of numbers. For example, if the telephone company tells us that a long-distance call cost $2.50 for the first three minutes and 70¢ per minute after that, rather than create a table showing the cost of a call for every length of time under the sun, we can determine the cost of a call of any linked by simply using a formula. 4. Fourth, once we have a general expression, we can manipulate it to find out things that we don’t know. For example, since distance equals rate times time (d = rt), if we know that we have to go 73 miles and will be traveling at 65 mph, we can determine how long it will take to get there by manipulating the expression (t = d/r). 5. Moreover, using algebra to solve abstract problems has led to many solutions to real-life problems, such as how to fly a man to the moon. 6. And finally, perhaps the most immediate reason for studying algebra, at least for students planning to attend college, is that algebra is the language used in advanced math classes, such as geometry, trigonometry and calculus. So, studying algebra will not only help you develop the ability to reason, but also enable you to understand the symbolic language of math and science. And as already mentioned twice, algebraic skills and concepts are used in a wide variety of problem-solving situations (which you can apply to your everyday life). So, let’s begin by defining the basic concepts behind all those symbols starting with the concept of numbers.

ARITHMETIC PROPERTIES

Number Sets

From all thatâ&#x20AC;&#x2122;s been said thus far, you may have gathered that a standard objective for students taking a course in basic algebra is to develop the ability to use properties of numbers to solve problems. However, there are different classifications of numbers, and how they behave depends on the particular group with which you are working at the moment. Accordingly, our first task will be to become familiar with the different kinds, types or categories of numbers that are out there.

Stated more formally, our first goal is for you to: Be able to identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable. So, we see that when it comes to the different classifications of numbers, there are at least four: integers, rational, irrational and real. Of these four number groups, real numbers are the most general, so we will begin there. SUBSETS OF THE NUMBER SYSTEM Weâ&#x20AC;&#x2122;ve already established that, at its core, mathematics is the study of how numbers behave; and that different types of numbers behave in different kinds of ways. Moreover, we stated that while arithmetic looks at the behavior of particular numbers in a given situation, algebra uses variables to analyze the behavior of entire groups of numbers without respect to their unique attributes.

Of course, if you’re unfamiliar with the different kinds of numbers that exist, you can’t help but be confused by references to their behavior. Therefore, this first chapter is designed to give you a working knowledge of the various number groups whose properties we will be studying. REAL NUMBERS The numbers we work with most often are called real numbers. The official definition is something like: Real numbers are numbers that can be represented by points on a number line Of course, that implies there are numbers that can’t be represented by points on a number line. In referring to “real” numbers, one suggests that there is such a thing as “fake” numbers. So…is there? IMAGINARY NUMBERS Well actually, numbers that aren’t real numbers are called imaginary numbers, which are a subset of a group called complex numbers. Imaginary numbers are easily identified because they’re always accompanied by the lowercase letter “i” printed in italics. Here are several examples of what imaginary numbers look like: 3½ i

πi

9.4 i

7i 2

However, at this time we will neither conduct an in-depth examination of, nor attempt to develop the concept of, imaginary numbers. We mentioned them here only to make it possible for us to provide the following informal definition for real numbers: REAL NUMBERS (continued) For our purposes (at least for the time being), we can define a real number as any number that is not accompanied by the lowercase letter “i” printed in italics. So, now that we have a working definition for real numbers, let’s consider some of its basic subsets.

By the way, the fact that we are talking about subsets begs the question: What is the definition of a set? (If you can answer this question intelligently, you can probably skip this next section.)

DEFINING SETS According to G. Cantor (1845-1918), developer of set theory… A set is a grouping together of single objects into a whole. Practically speaking, they give mathematicians a way to talk about collections of things (including numbers) in an abstract way. So then, a set is any collection of clearly specify things. Please note however that they do not have to be related in any special way. Anything may be lumped together and designated as a set. The things in the set are called the elements or members of the set. If each element of one set is also a member of another set, the first set is said to be a subset of the second. COUNTING NUMBERS So now that we’ve clarified our vocabulary, let’s get on with identifying the various subsets of real numbers, the most basic being the one people normally use when they want to count, called counting numbers (1, 2, 3... etc.). NATURAL NUMBERS Unfortunately, when referring to counting numbers, some people include zero, while others do not. Those that don’t say counting numbers plus zero are natural numbers (0, 1, 2… etc.). But believe it or not, there’s not solid agreement as to whether zero should be included among the natural numbers either. WHOLE NUMBERS And as if that isn’t confusing enough, while the category known as whole numbers are defined as positive, one finds the term “negative whole numbers” being used to describe another classification of numbers called integers, which are usually defined as counting numbers, their corresponding negatives, and zero.

MORE PRECISE VOCABULARY So then, depending on whom you’re talking to, counting numbers, natural numbers, and integers are all, at one time or another, referred to as whole numbers. With all of this confusion and lack of standardization, let’s just forget the whole convoluted bunch (with the exception of integers) and use more precise terminology instead. Rejecting the use of the vague and imprecise terms, we will instead use: • • • • •

integers non-negative integers positive integers negative integers zero

INTEGERS Integers include zero, all of the positive numbers representing one or more complete units, and all of their corresponding negatives, from negative infinity to infinity. . . . –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6 . . . NEGATIVE INTEGERS We refer to integers as either negative… . . . –5, –4, –3, –2, –1 NONNEGATIVE INTEGERS …nonnegative… 0, 1, 2, 3, 4, 5, 6 . . . ZERO …zero… 0 POSITIVE INTEGERS …or positive… 1, 2, 3, 4, 5, 6 . . .

And with that, we now have the required vocabulary to describe the remaining subsets of real numbers.

RATIONAL NUMBERS Typically, real numbers are divided into two major subsets. The first is called rational numbers. Rational numbers are all real numbers except those that have neverending non-repeating decimals.

IRRATIONAL NUMBERS The second subgroup of real number is called irrational numbers. If rational numbers are all real numbers except those that have neverending non-repeating decimals, then irrational numbers must be numbers that do have never-ending non-repeating decimals. It therefore follows that rational and irrational numbers are totally separate number groups. In other words, there is no overlap between them, even though they are both subsets of real numbers. RATIONAL vs. IRRATIONAL Rational numbers are technically regarded as ratios (divisions) of integers. To put it another way, a rational number is formed when you divide one integer by another integer. Consequently, they are sometimes referred to as fractional numbers. (Remember, you can turn any integer into a fraction by using the integer as the numerator and the numeral 1 as the denominator. Consequently, all integers are rational numbers.) But again, irrational numbers are numbers that have never-ending non-repeating decimals. Examples of irrational numbers include: pi (3.14159262â&#x20AC;Ś.), e (2.71828182845â&#x20AC;Ś) and the square root of two ( 2 ). Note that irrational numbers cannot be expressed as fractions. Note also that fractions (or rational numbers) can either be written as terminating (or ending) decimals (like 16.0, 0.5, or 0.76), or as repeating decimals (like 0.33333. . .).

ABBREVIATIONS Real numbers, rational numbers, and integers each have their own abbreviation, typically designated by capital letters printed in italics. •

Real numbers are usually denoted by: R.

•

Rational numbers by: Q (for quotient).

•

Integers by: Z (because the German word for integers is Zahlen).

If we wish to refer only to the positive members of each set, we use the following symbols… •

R+ is the set of all positive real numbers.

•

Q+ is this set of all positive rational numbers.

•

Z+ is this set of all positive integers.

If we wish to refer only to the nonzero members of each set, we use the symbols below… •

R* is the set of all nonzero real numbers.

•

Q* is this set of all nonzero rational numbers.

•

Z* is this set of all nonzero integers.

In some textbooks the set of integers may be denoted by “J.” Moreover, there are textbooks that may indicate that the symbol for the set of natural or counting numbers is “N.” However, due to a lack of uniformity with respect to those terms, we’ll always refer to “natural numbers” as “nonnegative integers.”

SET BUILDER NOTATION A lot of the work throughout his book is going to involve sets of numbers. Therefore, we had better learn their notation. We can designate or differentiate the different groups of numbers using set notation. The relationship of a member to its set is described by the symbol: ∈ , which means “is an element of.” (Recall, the things in a set are called the elements or members of the set.) A set may often be described by writing out the names of its elements inside braces, which is called the roster method. However, large sets are more frequently described using set-builder notation: Z+ = { x │ x > 0, x is an integer} The above example means “positive integers equal the set of all numbers x such that x is greater than 0.” Notice that the brace is read “the set of all” and the bar line is read “such that.”

REVIEW •

The real numbers, abbreviated with R, are all numbers that actually exist. In plain terms, they are numbers that can be represented by points on the number line—from negative infinity to positive infinity— and every point in between. In even simpler language, a real number is any number that is not accompanied by the lower case letter “i” printed in italics (which identifies imaginary numbers).

•

The two main subsets into which real numbers are divided are rational numbers and irrational numbers, which are completely separate number types.

•

The rational numbers, which are abbreviated with Q, are regarded as ratios of integers, can be written as fractions, and can be expressed as terminating or repeating decimals.

•

On the other hand irrational numbers are numbers that have never-ending non-repeating decimals. They cannot be expressed as fractions.

•

The set of rational numbers includes integers, which are made up of zero, all the positive numbers that represent complete units, and all of their corresponding negatives.

•

Integers, which are abbreviated with Z, can be further divided into negative, nonnegative, zero, and positive.

So, now we have defined the different subsets of numbers and you’re ready to begin mastering the use of various arithmetic properties of real numbers and their subsets in solving all kinds of math problems.

OUTLINE OF SETS / SUBSETS OF THE NUMBER SYSTEM Complex numbers Imaginary numbers Real numbers Rational numbers • Rational numbers with terminating decimals • Rational numbers with repeating decimals • Integers ♦ Positive integers ♦ Nonnegative integers ♦ Zero ♦ Negative integers Irrational Numbers Directions: Be able to write the above outline by memory. Imaginary Numbers: If a ∈ R , a > 0, we define EXAMPLE:

−2 =i 2

− a = −1 a = i a

ARITHMETIC PROPERTIES

Properties

Now that we know a real number is any number that does not have a lowercase “i” accompanying it, let’s begin considering the properties of real numbers. The whole purpose of this Chapter is to deepen your understanding of the real number system by presenting (without all of the mathematical mumbojumbo) properties and laws of numbers that will equip you with the basic knowledge necessary to understand important algebraic concepts But, what do we mean by the term: properties?

Well, that’s just the fancy way of saying “rules.” The arithmetic properties are simply the laws that govern how real numbers behave with one another. In other words, when we say “arithmetic properties” we are simply referring to the rules that control how real numbers may be added, subtracted, multiplied or divided. You might say they are the rules that govern how real numbers work together, or how they operate, which is why addition, subtraction, multiplication and division are called: the four operations. However, the problem with basic arithmetic is that it can only look at what specific numbers do in specific situations. This is one of the main reasons we need algebra. You see, algebra has the ability to look at what entire categories of numbers do in general. Similarly, it can take into consideration a whole range of numbers all at once. This is because, in addition to numbers, the four operations and the rules that control how the numbers and operations interact—algebra also makes use of variables.

A variable is a symbol (usually a letter from the alphabet printed in italics) that can assume any value in a problem. Variables make it possible for algebra to consider the general properties of numbers without having to consider (worry about) their specific attributes. For example, as we learned in Chapter 1, the variable R is use to represent the entire category of real numbers. Moreover, the entire range of positive numbers can be represented by an expression using a variable, such as: x > 0. Algebra uses variables in other ways as well. For instance, when it comes to solving problems, there are often times when we donâ&#x20AC;&#x2122;t know all of the values involved and therefore need something else to stand for the values we donâ&#x20AC;&#x2122;t have. Supposing, for example, we have a friend named Roger, but we know nothing about his age, except that he is three years older than Darnell. If we know that Darnell is nine years old, then we can represent Rogerâ&#x20AC;&#x2122;s age with the equation: x = 9 + 3 Secondly, there are those cases when the solution to a problem has more than one value. For example, if we wanted to refer to every pair of numbers that have a sum greater than 45, we could do so using the following inequality: a + b > 45 And finally, there are other instances in which the value that satisfies a problem at any given time changes depending on a changing set of circumstances. Such a situation might be represented by an equation that looks something like this: y = x 2 + 6 So then, in situations where a value is unknown, nonspecific or unfixed, we find that a variable can come in quite handy. SITUATIONS WHERE VERIABLES ARE COMMONLY USED 1. To represent entire categories of numbers. 2. To represent a wide range of numbers. 3. To consider the general properties of numbers without respect to their specific attributes. 4. To represent unknown values when solving problems. 5. To represent the different answers to a problem whose requirements are satisfied by more than one solution. 6. To represent the values in a problem whose quantities and solution changes as the situation changes.

LIST OF PROPRETIES 1. Closure Under Both Addition and Multiplication 2. Identity Property for Both Addition and Multiplication 3. Inverse Property of Both Addition and Multiplication 4. Commutative Property of Both Addition and Multiplication 5. Associative Property of Both Addition and Multiplication 6. Multiplication Property of Zero 7. Multiplication Property of Negative One 8. Distributive Property of Multiplication Over Addition 9. Distributed Property of Multiplication Over Subtraction 10. Addition Property of Equality 11. Subtraction Property of Equality 12. Multiplication Property of Equality 13. Division Property of Equality 14. Addition Property of Inequality 15. Subtraction Property of Inequality 16. Multiplication Property of Inequality 17. Division Property of Inequality 18. Reflexive Property of Equality 19. Symmetric Property of Equality 20. Transitive Property of Equality 21. Transitive Property of Inequality As you know, the basic operations of numbers are addition, multiplication, subtraction, and division, which are called binary operations, because they’re defined in terms of combining numbers two at a time. (Of course, the prefix “bi-” means “two.”) And you’re undoubtedly familiar with the symbols used to represent these operations The symbols for plus (+); times (×) or ( · ); minus (–) and; divide by (÷). So, we’ll begin by looking at the properties (or laws) that apply to the set of nonnegative integers, which you may recall, are represented by this symbol you see here: Z* (the capital letter “Z” with an asterisk). Nonnegative integers are the numbers invented out of a need for counting. And a discussion of the basic operations on nonnegative integers will, in turn, show the need for expanding this group of numbers to the set of real numbers.

CLOSURE The first concept we’re going to explore with respect to the behavior of numbers is that of closure. In plain English, a particular set of numbers is said to be closed under a given operation if, whenever you compute two numbers from the set according to that operation, the result will also be a number from that same set. Closure Under Addition Consequently, the set of nonnegative integers is closed under addition, since that sum of any two nonnegative integers is always another uniquely determine nonnegative integer. The expression “uniquely determined” reflects the fact that, if you start with the same two numbers, the result of the operation will always be the same. So, since the some of any to nonnegative integers is always another uniquely determine nonnegative integer. nonnegative integers are said to be closed under addition (because whenever you add two non-negative integers you always get another non-negative integer). And one of the most important ideas with respect to the concept of closure is that of uniqueness. However, all that this means is, as long as you add the same two numbers, you’ll always get the same answer. By the way, you might want to note that the set of real numbers is also closed under addition. Summary The set of real numbers is closed under addition. The subset of nonnegative integers is closed under addition.

IDENTITY PROPERTY FOR ADDITION The identity property for addition is concerned with what number can be added to another number without changing the other number’s value? So, the identity element for addition is zero, because adding zero to a number won’t change the number’s value. For example, six plus 0 = 6. Again, the identity property for addition says, if you add zero to a number, that number’s value remains the same.

The Identity Property for Addition We are now going to look at a number of laws (or properties) that govern the addition of nonnegative integers, beginning with the identity property for addition, which states that… For every real number n, 1 · n + 0 = n EXAMPLE: 6 + 0 = 6 In other words, adding zero causes no change. So, zero is called the identity element for addition.

INVERSE PROPERTY OF ADDITION The inverse property of addition looks for the number that, when added to another number, gives you the identity element for addition. Since the identity element for addition is 0, we’re looking for a number that, when added to a nonnegative integer, gives us zero. That means, we’re going to have to add the number to its corresponding negative—so it’s now necessary for us to expand our scope beyond nonnegative integers to the set of real numbers. Consequently, the inverse property of addition says that for every real number there’s a number called the additive inverse that, if added to the first number, will return a value of zero. For example, eight plus negative eight equals zero.

Inverse Property of Addition The inverse property of addition states that… For every real number n, There is an additive inverse – n such that n + – n = 0 EXAMPLE: 8 + (– 8) = 0

Commutative Law of Addition To “commute” means to change for travel, and the commutative law of addition (in plain English) says you can switch the order of the numbers you are adding without it having any effect on their total value. The commutative law of addition states that… For every real number a and b, a + b = b + a EXAMPLE: 4 + 7 = 7 + 4

Associative Law of Addition To “associate” means to join together, connect or combine, and the associative law of addition says if you are adding three numbers with the use of grouping symbols, rearranging how the numbers are grouped will not change their total value. The associative law of addition states that… For every real number a, b and c, (a + b) + c = a + (b + c) EXAMPLE: (3 + 7) + 5 = 3 + (7 + 5) In other words: 10 + 5 = 3 + 12. In either case…you end up with 15

Multiplication We are now going to look at a number of laws (or properties) that govern the multiplication of nonnegative integers, which is written a · b, the meaning of which is… “a terms of b.” b + b + b + …+ b

Hence, you might say that the product of two nonnegative integers a and b is another name for sum, though it is actually defined to be the nonnegative integer a · b.

Closure Under Multiplication A set is said to be closed under multiplication if, every time you multiply two numbers from that set, you get an answer that is also an number from that set. A set is said to be closed under multiplication if, for all elements a, b of the set, the product ab is also an element of the set. Since that is the case with nonnegative integers, nonnegative integers are closed under multiplication. Now letâ&#x20AC;&#x2122;s look at multiplication by zero...

Multiplication by Zero For every real number n, n · 0 = 0 Example: 26 · 0 = 0 We saw that zero is the identity element for an addition. However, such is not the case when it comes to multiplication, though zero does still play a prominent role. The multiplication property of zero says that any real number multiplied times zero equals zero. Hmmm…That means we just expanded our discussion beyond nonnegative integers to include real numbers. Okay, fine…so for example…26 times zero equals zero.

Identity Property of Multiplication And have no fear… The fact that zero is not the identity element for multiplication, doesn’t mean that multiplication has no identity element. There’s an identity property of multiplication just as there was for addition. Except, the identity property for multiplication is concerned with what number can be MULTIPLIED times another number without changing the other number’s value? So, the identity element for multiplication is the number one, because multiplying one times a number won’t change that number’s value. For example, one times 35 equals 35.

For every real number n, 1 · n = n Example: 1 · 35 = 35 There is a unique number 1, called the multiplicative identity, such that for any nonnegative integer… n×1=1×n=n

Multiplication Property of Negative One We’ve already seen that multiplication has at least one property that addition doesn’t—the multiplication property of zero. Well, here’s another. The multiplication property of negative one says there’s a unique number –1, such that any nonnegative integer times –1 gives you that integer’s corresponding negative. For every real number n, – 1 · n = – n EXAMPLE: – 1 · 28 = – 28

Now, recall that the inverse property of addition looks for the number that, when added to another number, gives you the identity element for addition. And since the identity element for addition is 0, we looked for a number that, when added to a nonnegative integer, gave us zero. However, as you now know, the identity element for multiplication is the number 1. So, when it comes to the inverse property of MUTLIPLICATION, we need to find what we can multiply times any number that will give us the number 1, and the answer is, of course, the number’s reciprocal. In plain English, you write the number as a fraction…then flip it upside down so that the numerator becomes the denominator and the denominator becomes the numerator. That’s the number’s reciprocal. Multiply that times the number, and you get the number 1. For example, the reciprocal on four is one fourth, so if you multiply one fourth times four, you get the number 1.

For example, three times eight equals eight times three.

And as was the case with the commutative law, like addition, multiplication also has an associative law. Remember, to “associate” means to join together, connect or combine, so the associative law of multiplication says if you’re multiplying three numbers with the use of grouping symbols, rearranging how the numbers are grouped won’t change their final product. For example…in other words, 15 times 8 equals 5 times 24. In either case, you get 120.

Now, the distributive law of multiplication over addition, in plain English, states that… …when you have two terms that are being added inside of parentheses… …multiplied times one term that is outside of the parentheses, before adding, distribute the tern that’s on the outside… …to each of the terms that are on the inside, multiplying them individually. Then remove the parentheses… …and add the resulting products. For example …

The properties that remain are, relatively speaking, much easier to understand than the ones we dealt with first. So, Iâ&#x20AC;&#x2122;ll go ahead and state them using standard mathematical language, beginning with the distributive property over subtractionâ&#x20AC;Ś Which is virtually the same as the distributed property over additionâ&#x20AC;Ś

Which, to me, seems rather self-evident. (He says, “Hey, I can see myself!” You know…refection? Get it?

(Itâ&#x20AC;&#x2122;s symmetrical)

The relationship carries across, or carries through.

They’re not all that hard to remember when you organize ‘em that way. And with that, we’re ready to deepen our understanding of important algebraic concepts.

Standard 1.0 California requires students to know and apply the arithmetic properties of subsets of integers, rational numbers, irrational numbers and real numbers; including closure properties for the four operations where applicable. Consequently, you’ll need to first be able to identify the various subsets of numbers. This, in turn, calls for the ability to use set builder notation, so let’s begin there…

There are two ways to define a set. The first is to list and separate by commas all of the objects that make up the set. We write the defining list within braces. This is called the roster method.

The Roster Method A set may be described by writing out the names of all its elements inside braces: { } The braces are read as “the set of all elements.” Specifying a set by writing out a list is called the roster method because it consists of simply creating a roster naming all the elements of the set. However, this becomes impractical when it comes to sets with a large number of elements. In such cases, it is better to use the second way of defining a set— what is known as: set-builder notation. This method requires us to give a rule that identifies the elements of the set. In this case, the defining rule is written within braces.

Set Builder Notation Set-builder notation is generally used to represent a group of real numbers. It stipulates that sets be written in the following format… { x : x has property Y } …which is read as “the set of all elements x such that x has the property Y. (The colon means “such that.”) Using this notation, a set is often defined as the collection of real numbers that belong to either an open, closed, half-open, or infinite interval (of real numbers). OPEN INTERVAL An open interval is a set of real numbers represented by a line segment on the real number line, whose endpoints are not included in the interval. This concept is made clear by the following definition: ( a, b ) = { x : a < x < b } where a < b Again, the endpoints of an open interval are NOT part of the interval. CLOSED INTERVAL

In contrast, a closed interval is a set of real numbers represented by a line segment of the real number line, whose endpoints are included in the interval. Its definition is as follows: [ a, b ] = { x : a ≤ x ≤ b } where a < b Again, the endpoints of a closed interval ARE part of the interval. HALF-OPEN INTERVAL

A half-open interval is also a set of real numbers represented by a line segment on the real number line, but with one endpoint included in the interval, and the other endpoint not included in the interval. They are defined as follows: [ a, b ) = { x : a ≤ x < b } where a < b ( a, b ] = { x : a < x ≤ b } where a < b HALF-OPEN INTERVAL

Similar definitions apply when the infinite interval is open at one end. REVIEW Application Standard 1.1 1) Is the equation 3(2x – 4) = –18 equivalent to the equation 6x – 12 = –18? a) Yes, the equations are equivalent by the Associative Property of Multiplication. b) Yes, the equations are equivalent by the Commutative Property of Multiplication. c) Yes, the equations are equivalent by the Distributive Property of Multiplication over Addition. d) No, the equations are not equivalent. 2) Which statement is false? a) The order in which two whole numbers are subtracted does not affect the difference. b) The order in which two whole numbers are added to does not affect the sum. c) The order in which to rational numbers are added to does not affect some. d) The order in which to rational numbers are multiplied is not affect the product.

THE ASSOCIATIVE PROPERTIES (APPLICATION?) Generally speaking, the associative property states that if you change the grouping of numbers upon which some operation is being performed, the result remains the same. (Neither the commutative nor the associative properties work with subtraction or division.) Let's look at the associative property of addition next: a + (b + c) = (a + b) + c Again, the two sides of the equation are equivalent to one another. PROBLEM: Use the associative property to write an equivalent expression to (a + 7b) + 3c. SOLUTION: Using the associative property of addition (where changing the grouping of the addends does not change the value of the sum) we get (a + 7b) + 3c = a + (7b + 3c). a(bc) = (ab)c Now let's turn to the associative property of multiplication: a(bc) = (ab)c Of course, both sides are equivalent. PROBLEM: Use the associative property to write an equivalent expression to (3 路 n)x. SOLUTION: Using the associative property of multiplication (where changing the grouping of the factors does not change the value of the product) we get (3 路 n)x = 3(nx) THE DISTRIBUTIVE PROPERTY The distributive property applies when you are multiplying some term that is outside a set of parentheses times two or more other terms that are being added or subtracted inside of the parentheses ( ). The

distributive property states that you multiply the outside term times every term on the inside. (The terms on the inside must be separated by + or –.) EXAMPLE: a(b + c – d) = ab + ac – ad

PROBLEM: Use the distributive property to write 2(x - y) without parenthesis.. SOLUTION: Distribute the 2 to each term on the inside the parentheses, or in other words, multiply every term inside of ( ) by 2, which give you . . . 2(x – y) = 2(x) – 2(y) = 2x – 2y PROBLEM: Use the distributive property to write –(2x + 5) without parenthesis. SOLUTION: Basically, when you have a negative sign in front of a ( ), like this example, you can think of it as taking a -1 times the ( ). What you end up doing in the end is taking the opposite of every term in the ( ). –(2x + 5) = (–1)(2x + 5) = (–1)(2x) + (–1)(5) = (–2x – 5) the negative sign outside the parentheses is equivalent to (–1) so . . . distributing the (-1) to each term inside the parentheses gives us . . . and multiplying we get . . . for our solution

PROBLEM: Use the distributive property to find the product of 4(2a + b + 3c). SOLUTION: As mentioned above, you can extend the distributive property to as many terms as are inside the parentheses. The basic idea is that you multiply the outside term times each term on the inside. 4(2a + b + 3c) = 4(2a) + 4(b) + 4(3c) = 8a + 4b + 12c =

distributing the 4 to each term inside the parentheses gives us . . . and multiplying we get . . . for our solution

THE IDENTITY AND INVERSE PROPERTIES The additive identity is 0, meaning that when you add 0 to any number, you end up with that number as a result. The additive identity is 0 a + 0 = a (or 0 + a = a)

The multiplication identity is 1, meaning that when you multiply any number by 1, you wind up with that number as your answer. a(1) = a (or 1(a) = a)

The additive inverse (or negative) states that for each real number a, there is a unique real number, denoted –a, such that a + (–a) = 0. In other words, when you add a number to its additive inverse, the result is zero. Other terms that are synonymous with additive inverse are negative and opposite. PROBLEM: Write the opposite (or additive inverse) of -3. SOLUTION: The opposite of -3 is 3, since -3 + 3 = 0. The multiplicative inverse (or reciprocal) states that for each real number a, except 0, there is a unique real number such that a · ½ = 1 (or ½ · a = 1). In other words, when you multiply a number by its multiplicative inverse the result is one. A more common term used to indicate a multiplicative inverse is the reciprocal. A multiplicative inverse or reciprocal of a real number a (except 0) is found by "flip-flopping" it. The numerator of a becomes the denominator of the reciprocal of a

and the denominator of a becomes the numerator of the reciprocal of a. PROBLEM: Write the reciprocal (or multiplicative inverse) of -3. SOLUTION: The reciprocal of -3 is -1/3, since -3(-1/3) = 1. When you take the reciprocal, the sign of the original number stays intact. Remember that you need a number that when you multiply times the given number you get 1. If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no. These additive and multiplicative inverses will come in handy big time when you go to solve equations later on, so keep them in your memory bank until that time. PROBLEM: Example 9: Write the opposite (or additive inverse) of ⅛. SOLUTION: The opposite of ⅛ is -⅛, since ⅛ + (-⅛) = 0. PROBLEM: Example 9: Write the reciprocal (or multiplicative inverse) of ⅛. SOLUTION: The reciprocal of ⅛ is 8, since 8(⅛) = 1 PRACTICE PROBLEMS

These problems are from West Texas A&M University's beginning algebra website. They will allow you to check your understanding of these properties. To get the most out of them, you should work the problem out on your own and then check your answer by clicking on the link to West Texas A&M University (www.wtamu.edu) for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer. 1. Use a commutative property to write an expression equivalent to: xy

2. Use a commutative property to write an expression equivalent to: 1 + 3x 3. Use an associative property to write an expression equivalent to: (a + b) + 1.5 4. Use an associative property to write an expression equivalent to: 5(xy) 5. Use the distributive property to find the product: -2(x - 5) 6. Use the distributive property to find the product: 7(5a + 4b + 3c). 7. Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of -7. 8. Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of 3/5

Why

Introduction

This Book’s Function… The Algebra I standards adopted by the California State Board of Education indicate that students are to understand and use the rules of exponents, including such operations as raising to a fractional power. HOW COME THIS PUBLICATION EXISTS? However, I do not find the organizational structure of most algebra books conducive to this task in the sense that relevant information and related concepts are typically scattered helter-skelter throughout the breadth of such publications. Also, while most math textbooks typically require students to trudge their way through an endless array of problems, my experience has convinced me that this practice of expecting students to complete a plethora of tedious exercises by mimicking examples provided at the beginning of each new topic only ensures they finish the course unable to articulate virtually anything they supposedly “learned” over the duration of the semester. So, rather than require students to generate pages upon pages of homework, I prefer instead to emphasize the acquisition and retention of knowledge—giving relevant concepts sufficient time to “sink in” by allowing students to mull over them again and again, or better yet, to become adept at expressing them in their own words through whatever means proves most effective in helping learners make the skills and concepts they encounter truly “theirs.” You see, I’m not all that impressed with high test scores if students cannot recall the same material a week or two later. Hence, I’ve taken the time to gather all of the pertinent information here in one place where students can easily review as needed, thereby facilitating the process of mastering the laws of exponents here and now so that there will be no future need to purchase books by Kaplan, Cliff or Barron in order to review material that was never truly mastered in the first place. However, before actually plunging into the rules themselves, I think it wise to begin by providing a little background information…

What

Background

Recognizing Exponents The Purpose of this document is to help students develop an understanding of exponents, including… • • •

The laws of exponents Negative exponents Fractional exponents

REPRESENTING REPEATED MULTIPLICATION An exponent can be an efficient way to express repeated multiplication. For example, instead of going through all the trouble of writing: 5×5×5×5×5×5×5 Exponents allow us to express the same idea relatively quickly with: 57 In this case, 5 is called the base, while 7 is called the exponent.

The entire expression is called a power: 57

The exponent tells us how many times to use the base as factors in a multiplication problem. Powers can be multiplied times one another in accordance with the power rules. We’ll first consider the case of two powers that have the same base…

Background

Rules are Rules! The Purpose of this document is to help students develop their understanding of exponents, including… 1. The laws of exponents 2. Negative exponents 3. Fractional exponents REPEATED MULTIPLICATION (CONTINUED) From what we know so far, we can determine that… 2 2 × 2 4 = (2 × 2) × (2 × 2 × 2 × 2) = 2 6 Notice that if we are multiplying powers that have the same base, their product will also be a power having that same base, with an exponent equivalent to the sum of the exponents from the original powers. So, the general rule can be stated as follows:

If a is a real number and n and m are nonnegative integers, then an × am = an + m

EXAMPLE

82 × 83 = 82 + 3 = 85

Since most problems involving exponents can be simplified within five steps if the exponents are dealt with in a particular order, let’s look at the remaining rules of exponents in that very same order.

0 Rule #1

Zero Base and Zero Exponent Rule Where we begin is not all that difficult to remember in that we start with zero as both the base and the exponent.

THE ZERO BASE AND ZERO EXPONENT RULE This is the easiest exponent to deal with because, like division by 0, it’s undefined. (It kind of like…doesn’t really exist.) In other words, it has no answer—there is no solution.

0 Rule #2

Zero Exponent Rule Where we go next is not all that difficult to remember either, since we are still working with zero as the exponent.

THE ZERO EXPONENT RULE Staying with zero a bit longer, we next move to powers where zero is only the exponent. Any number raised to the power of zero equals one. EXAMPLE

50 = 1

So, why is that? Well, if we’re to remain consistent with the information on page five, a 0 = 1 is the only possible definition. Recall that… an × am = an + m

Replacing a with 2, n with 0, and m with 4, we now have: 20 × 24 = 20 + 4 This can, of course, be rewritten as… 20 × 24 = 24 And as you know, the only way that the above can be true is if 2 0 = 1 since by the Identity Element for Multiplication: 1 × 2 4 = 2 4 Hence, any number raised to the power of zero must equal one (1).

1 Rule #3

One Exponent Rule After zero comes the numeral one. So, letâ&#x20AC;&#x2122;s now use that number as the exponent.

THE ONE EXPONENT RULE Since the exponent tells us how many times to use the base as factors in a multiplication problem, any number raised to a power of one equals itself. EXAMPLE

51 = 5

3 2

(5 )

Rule #4

Power Base Rule What do you do when the base of a power is itself a power?

THE POWER BASE RULE When you have a power as the base of a power—a power within a power, sort of speaking—remove the parentheses and multiply the exponents. (a m) n = a m n EXAMPLE

(5 3) 2= (5 × 5 × 5) × (5 × 5 × 5) = 5 3 × 2 = 5 6

(3·5)

2 Rule #5

Product Base Rule What do you do when the base of a power is the product of two or more numbers?

THE PRODUCT BASE RULE If a power has a base that consists of multiple values which need to be multiplied, remove the parentheses and distribute the exponent to each of the values

(ab) m = a m b m EXAMPLE 2

× 5 2 = 9 × 25 = 225

(3·5) 2 = (15) 2 = 225 (3·5) 2 = 3

(⅜)

2 Rule #6

Quotient Base Rule What do you do when the base of a power is some value divided by another value?

THE QUOTIENT BASE RULE If a power has a base that consists of a fraction, you again remove the parentheses and distribute the exponent to each variable.

EXAMPLE

32 9 3   = 2 = 64 8 8

8 ×8

3 Rule #7

Product Rule What do you do when you have to multiply powers that have the same base?

THE PRODUCT RULE We saw rule seven explained back on page five, so you’re already familiar with the fact that, when you’re multiplying powers that have the same base, you copy the base once and simply add the exponents together. an × am = an + m

EXAMPLE

82 × 83 = 82 + 3 = 85

Rule #8

Quotient Rule What do you do when you have to divide powers that have the same base?

THE QUOTIENT RULE To obtain the quotient of two powers that have the same base, you once again copy the base once. But, instead of adding the exponents, this time you subtract the exponent in the denominator from the exponent in the numerator.

EXAMPLE

53 = 5 3â&#x2C6;&#x2019; 2 = 51 = 5 2 5

Background

Divide and Conquer The Purpose of this document is to help students develop their understanding of exponents, including… • • •

The laws of exponents Negative exponents Fractional exponents

REPRESENTING DIVISION In addition to being used to express repeated multiplication, exponents can also be used to represent division. This is accomplished by using negative integers as exponents.

Note: Because a negative exponent indicates division, in such cases the base must not be zero. Again, the definition of negative exponents must also be consistent with all of the properties (or laws) of exponents. With this in mind, – let’s see how negative exponents should be defined, using 4 1 as an example. According to the product rule: –

4 1 × 4 1 should equal 4 equals 1.

–1+1

which is of course equal to 4 0 which

We can replace 4 1 with 4, which means that 4

–1

× 4 should equal 1.

However, the only way this could be true is if 4 Ergo: 4

–1

must equal ¼.

–1

= ¼.

Âź can also be written as

1 41

which, trust me, is how it must be

expressed if itâ&#x20AC;&#x2122;s to remain consistent with all of the other properties of exponents as well. Therefore, when the base of some power is a real number (other than zero), and the exponent is a negative value, the power can be represented as a fraction that has one (1) as the numerator, and a denominator identical to the original power, except that the negative sign is omitted!

EXAMPLE 8 â&#x2C6;&#x2019; 2 =

1 82

–2 Rule #9

Negative Exponent Rule When you have a number raised to a negative power, change the power to a fraction that has the numeral one as its numerator, and the very same power as the denominator, except that you omit the negative sign

THE NEGATIVE EXPONENT RULE

EXAMPLE 2 − 4 =

1 1 = 4 16 2

(It ain’t very mathematical, but I simply visualize myself as taking the negative sign and stretching it out over the base and the exponent. I then write the numeral one above the whole thing and voilà—there it is!)

Roots Background

Let’s Get Radical The Purpose of this document is to help students develop their understanding of exponents, including… 1. The laws of exponents 2. Negative exponents 3. Fractional exponents

REPRESENTING ROOTS We first used exponents to represent repeated multiplication. We then used negative exponents to represent division. Now we’ll see how to use fractional exponents to represent roots. We’ll start with a generic value (n), which we would like to equal its square root ( n ) when it is raised to some unknown power (n x). So, we want n x to equal n . Our question is, what must x equal to make this so? Now, according to the definition of square roots:

( n)

= n.

We could also write it this way without changing any of the values:

( n)

= n1

In that our goal is for n x to equal

( )

above equation: n x

n we replace

n with n x in the

= n1

Now, according to the law of exponents that I call The Power Base Rule, we’re supposed to multiply x × 2 and end up with 1. Now, in order for that to happen x must equal ½. So, if we want n x to equal

n then x must = ½.

This suggests that if the base of a power is a real number, and its exponent is a fraction that has 1 as the numerator and some positive integer as the denominator, that power can be expressed as a radical by using the power’s base as the radicand and the fraction’s denominator as the index.

EXAMPLE: 4 5 = 5 4

For the sake of simplicity (meaning avoiding radicals with negative radicands) we won’t define fractional exponents that use a negative base. And thankfully, there’s only one additional situation we’ve not yet covered. Doing so will allow us to extend our definition of exponents to include all positive rational numbers.

Roots

continued

Fractional Exponents

Being Rational The Purpose of this document is to help students develop their understanding of exponents, including… • • •

The laws of exponents Negative exponents Fractional exponents

REPRESENTING ROOTS

( )

To express powers such as 4 7 using just one exponent, we need

( ) =7 1

an expression that is equivalent to 7 4 3

1 ×3 4

Of course, that expression is 7 4 suggesting that if the base of a power is a real number and its exponent is a fraction, the expression may also be written as a power whose base is a radical by using the original expression’s base as the radicand, the original exponent’s denominator as the index, and the original exponent’s numerator as the power to which the radical is raised

Âž Rule #10

Fractional Exponent Rule What do you do when the base of a power is some value divided by another value?

THE FRACTIONAL EXPONENT RULE When you have a number raised to a fraction, place the base under a radical sign, use the denominator as the index, and raise the resulting radical to a power equal to the fractionâ&#x20AC;&#x2122;s numerator.

EXAMPLE: 16 4 =

( 16 ) = 2 4

http://www.purplemath.com/modules/exponent.htm You already know of one relationship between exponents and radicals: the appropriate radical will "undo" an exponent, and the right power will "undo" a root. For example:

But there is another relationship (which, by the way, can make computations like those above much simpler). For the square (or "second") root, we can write it as a power, like this:

...or:

The cube (or "third") root is also the one-third power:

The fourth root is also the one-fourth power:

The fifth root is also the one-fifth power; and so on. Looking at the first examples, we can re-write them like this:

You can enter fractional exponents on your calculator for evaluation, but you must remember to use parentheses. If you are trying to evaluate, for (4/5) instance, 15 , you must put parentheses around the "4/5", because 4 otherwise your calculator will think you mean "(15 ) Ăˇ 5". Fractional exponents allow greater flexibility (you'll see this a lot in calculus), are often easier to write than the equivalent radical format, and permit you to do calculations that you couldn't before. For instance:

Whenever you see a fractional exponent, remember that the top number is the power, and the lower number is the root (if you're converting back to the radical format). For instance:

By the way, some decimal powers can be written as fractional exponents, 5.5 11 too. If you are given something like "3 ", recall that 5.5 = /2, so:

35.5 = 311/2 Generally, though, when you get a decimal power (something other than a fraction or a whole number), you should just leave it as it is, or, if necessary, pi evaluate it in your calculator. For instance, 3 cannot be simplified or rearranged as a radical.

There is one fussy point here. When you are dealing with these exponents with variables, you might have to take account of the fact that you are sometimes taking even roots. Think about it: Suppose you start with â&#x20AC;&#x201C;2. Then:

In other words, you put in a negative number, and got out a positive number! This is the official definition of absolute value:

(Yeah, I know: they never told you this, but they expect you to know, so I'm 3/6 telling you now.) So if they give you, say, x , then x had better not be 3 negative, because x would still be negative, and you would be trying to take 4/6 the sixth root of a negative number. If they give you x , then a negative x becomes positive (because of the fourth power) and is then sixth-rooted, so 2/3 it becomes | x | (by reducing the fractional power). On the other hand, if 4/5 they give you something like x , then you don't care whether x is positive or negative, because a fifth-root doesn't have any problem with negatives. (By the way, these considerations are irrelevant if your book specifies that you should "assume all variables are non-negative".)

Review We’ve learned ten power rules, also referred to as the laws of exponents. If performed in a particular order, they’ll allow us to simplify most problems involving exponents within five steps. Consequently, that’s the order in which we learned (and hopefully memorized) them. First we had zero as both the base and the exponent, which is undefined. In other words, it doesn’t exist. I call this the zero base and exponent rule. Second was when you have any other real number with zero as the exponent, which always equals one. I call this the zero exponent rule. Third are powers that have one as an exponent, which will always equal the base. This I call the one exponent rule. Fourth is what I call the power base rule. When you have a power within a power, in which case, you remove the parentheses and multiply the exponents. Fifth is the product base rule. When the base of a power consists of numbers that are to be multiplied times one another. In this case, you first remove the parentheses and then, distribute the exponent to each of the values. After that, you multiply. Sixth is the quotient base rule. When the base of a power consists of numbers to be divided (in other words…a fraction), again, first you remove the parentheses and distribute the exponent to the numerator and the denominator—then you divide. Seventh is the product rule. When multiplying powers that have the same base, copy the base once, and then add the exponents. Eighth is the quotient rule. When dividing powers that have the same base, again you copy the base once and subtract the top exponent (in the numerator) minus the bottom one (in the denominator). Ninth is the negative exponent rule. When the exponent is negative, change the power to a fraction that has one as the numerator and the exact same power as the denominator—except you remove the negative sign.

And finally there is the fractional exponent rule. When the exponent is a fraction, rewrite the power as a radical, using the base as the radicand and the fractionâ&#x20AC;&#x2122;s denominator as the index. Then place the radical in parentheses, and raise it to a power equivalent to the fractionâ&#x20AC;&#x2122;s numerator. FRACTIONAL EXPONENTS

Chapter 2 The state of California expects you to understand and use such operations as taking the opposite; finding the reciprocal; taking a root; and raising to a (fractional) power (understand and use the rules of exponents).

EXPONENTS (Purple book, p. 129) And exponent is a convenient way to express repeated multiplication. An exponent is an easy way to represent the value of a number that is multiplied times itself repeatedly. It has two components: the base and the exponent. Together, they are called a power. The base tells you what number to multiply times itself and the exponent telss you how many time to do so.

EXAMPLE: 52 = 5 × 5 Here 5 is called the base and 2 is the exponent. The entire expression is called a power. An exponent may be any natural number and the base maybe any real number. (What is a “natural” number?) What is two multiplied by itself three times?

(2)(2)(2) = ? A number that is repeatedly multiplied by itself can be represented in exponential form.

(2)(2)(2) = 23 In exponential form, the number that is multiply repeatedly by it self is called the base. In exponential form, the number that represents how many times a number is multiplied by itself is called the exponent.

The number two is multiplied by it self three times.

DIRECTIONS: Respond to each item below. the prompts 3

The term 4 indicates that you should that the number four is multiplied how many times? _____

Given the term x how many times is the veritable x multiplied by itself? _____

Given the term 5 which number is the base?

_____ 4.

Given the term 10 which number is the exponent? ? _____

Write 8 × 8 × 8 × 8 × 8 in exponential form. _____

THE POWER RULES 00

Zero raised to the zero power is undefined. 00 =

Any other number raised to the zero power equals one. x1 = 1

x–a When you have a number raised to a negative power, change the negative sign into a fraction line, then write the number along with its exponent under the number one. 1 x–a = a x

xb When you have a number raised to a fraction, change the fraction line to a radical sign, place the number and the numerator under a radical sign, and the denominator outside the radical sign. b

When you multiply terms, add their exponents.

xa · x b = x a + b

When you divide terms, write the number one time, then change the faction line to a subtraction sign and subtract the exponent in the numerator minus the exponent in the denominator.

xa = xa –b b x

THE POWER RULES CONTINUED;

You multiply the powers here… (x a) b = x ab

Here you distribute the powers to each variable… (xyz) a = x a y a z a

Here you distribute the powers to each variable… a

x xa   = a y  y

Here you multiply it out the ling way. (x + y) 2 = (x + y)(x + y)

OPERATIONS AND EXPONENTS Use operations such as taking the root and raising to a fractional power, as well as the rules of exponents. 16 + 3 8 = Simplify each expression.

x4 x3 x5 x3 x7 x3 x9 x3

Rules of Exponents 1.

ZERO BASE AND EXPONENT RULE The power in which zero is both the base and the exponent is undefined. In other words, it doesn't exist.

ZERO EXPONENT RULE Any other real number with an exponent of zero is equivalent to one.

ONE EXPONENT RULE Powers that have one as their exponent are always equal the base.

POWER BASE RULE When you have a power within a power, you remove the parentheses and multiply the exponents.

PRODUCT BASE RULE When the base of a power consists of numbers that are to be multiplied times one another, you first remove the parentheses and then, distribute the exponent to each of the values. After that, you multiply.

QUOTIENT BASE RULE When the base of a power consists of numbers to be divided (in other wordsâ&#x20AC;Śa fraction), again, you first remove the parentheses and then distribute the exponent to the numerator and the denominator. After that, you divide.

PRODUCT RULE When multiplying powers that have the same base, copy the base once and then add the exponents

QUOTIENT RULE When dividing powers that have the same base, again you copy the base once and subtract the top exponent (in the numerator) minus the bottom exponent (in the denominator).

NEGATIVE EXPONENT RULE When the exponent is negative, change the power to a fraction that has one as the numerator and the exact same power as the denominatorâ&#x20AC;&#x201D;except that you remove the negative sign.

10. FRACTIONAL EXPONENT RULE

When the exponent is a fraction, rewrite the power as a radical, using the base as the radicand and the fractionâ&#x20AC;&#x2122;s denominator as the index. Then place the radical in parentheses, and raise it to a power equivalent to the fractionâ&#x20AC;&#x2122;s numerator.

OPPOSITES AND RECIPROCALS To find the reciprocal of a number, reverse its sign from positive to negative (or from negative to positive). To find the opposite of a variable, reverse its sign from positive to negative, or from negative to positive. To find the opposite of a number or variable, either change the sign to its opposite or multiply both sides by negative one. A number or variable and its opposite added together equals zero. The opposite of a number or veritable is called its additive inverse. To change in expression to its un-simplified opposite, multiply or divide the entire expression by negative one or change the sign of the entire expression.

A negative in front of parentheses is applied into each term just like negative one.

A negative in front of the parentheses changes the sign of every term inside the parentheses when it is distributed.

To simplify in expression in parentheses, change every term inside the parentheses by the number and/or sign outside.

What is the opposite of (x + 3)? –x – 3

What is the opposite of (x – 5)? –x + 5

Another way to change in expression to its opposite is to change the sign of every term in the expression to its opposite.

What is the opposite of 2x – 3? –2x + 3 What is the reciprocal of ⅓ ? The reciprocal of any whole number is one over the original number. The product of a number end its reciprocal is one.

What is the reciprocal of 1/x? What is the reciprocal of x2?

What is the reciprocal of a/b? What is the reciprocal of zero? There is not a reciprocal for zero.

Lets see how variables are used, along with some basic arithmetic properties, to solve some elementary math problems. Kenyan had a little get-together with four of his friends. At 5:00 P.M. he invited some more of his friends by telephone, and at the end of the

night a total of five people were at his little get-together. How many more people came after 5:00 P.M.? This situation can be represented by… 5+n=5 It’s obvious that no one else came to Kenyan’s get-together after 5:00. In other words,

n = 0. But on a test you might be asked to justify your answer. You would need to know that this is an example of the…

(x + 7)(x – 4 ) = 0

Inverse Property of Addition For every real number n, there is an additive inverse -n such that n + (-n) = 0.

Multiplication Property of Zero For every real number n, n · 0 = 0.

Identity Property of Addition …which says that: For every real number n, n + 0 = n. In other words, if you add nothing to a number, you end up with the same number with which you started. So in plain language…the Identity Property of Addition says that if you add zero to a number, that number’s value remains the same.

Here is another situation…

n · 60 = 60 Joe was traveling at 60 miles per hours. At the end of the trip he traveled a distance of 60 miles. How long was the trip?

Comes in hand wihen you want to multipliy a quantity time a number without changing the number’s value.

Identity Property of Multiplication For every real number n, 1 · n = n.

As you progress in your study of algebra, you will encounter references to different kinds of numbers. If you do not know what each type of number it is, you will not understand what your textbook is talking about. We will therefore take if you moments to have you learned the names of the major number groups or categories of numbers, as well as the characteristics of their members. We will start with the most general category and move down to the more specific. This means that, initially at least, you'll be memorizing a bunch of meaningless words. Nonetheless this task will be very useful in that the activity will provide you with a hook on which to hang the information that will immediately follow, and make sense of all the words you went through the trouble of memorizing. In

Twice as many stray dogs were picked up in February as were in January, and 10 more stray dogs were picked up in March than were in February. A total of 300 stray dogs were picked up over the three months. How many stray dogs were picked up in February? Last year, Tom drank 58 more than three times as many glasses of water as Pedro drank. If the two boys drank a total of 870 glasses of water, how many glasses did Tom drink? Sylvia makes three times as much money per month as her best friend, LaToya. The sum of their monthly incomes is at most $970. What is the greatest amount of monthly income that LaToya could earn?

14. A 120-foot-long rope is cut into 3 pieces. The first piece of rope is

twice as long as the second piece of rope. The third piece of rope is three times as long as the second piece of rope. What is the length of the longest piece of rope?

The cost to rent a construction crane is $750 per day plus $250 per hour of use. What is the maximum number of hours the crane can be used each day if the rental cost is not to exceed $2500 per day? The lengths of the sides of a triangle are y, y +1, and 7 centimeters. If the perimeter is 56 centimeters, what is the value of y? Beth is two years older than Julio. Gerald is twice as old as Beth. Debra is twice as old as Gerald. The sum of their ages is 38. How old is Beth? Marcy has a total of 100 dimes and quarters. If the total value of the coins is $14.05, how many quarters does she have?

Members of a senior class held a car wash to raise funds for their senior prom. They charged $3 to wash a car and $5 to wash a pick-up truck or a sport utility vehicle. If they earned a total of $275 by washing a total of 75 vehicles, how many cars did they wash?

all thatâ&#x20AC;&#x2122;s been said thus far, you may have gathered that a standard objective for students taking a course in basic algebra is to develop the ability to

Studying algebra will help you develop the ability to reason and killing with symbols as a step in the process developing an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are also used in a wide variety of problem-solving situations. So, let's begin by defining the basic concepts behind all those symbols starting with the concept of variables.

Algebra teaches us how to reason and calculate with symbols, an important step in developing an understanding of this embolic language of mathematics and science, which makes out about totally indispensable for anyone planning to (or who might just possibly) look for a job in either of these fields. For those of you quite sure that your future endeavors will never lead you in such a direction, understand that out abroad can also sharpen one's mind in ways that prove quite helpful when faced with the need to solve many different types of problems. In fact, algebra was invented as a tool to solve real-life problems in real-world situations. But though it has applications to virtually every human endeavor—from art to zoology—it is unfortunately all too often treated as nothing more than… And asked if the above reasons for studying algebra were not enough, there's also the fact that algebra shows us surprising… So let's begin by defining the basic concepts behind all those symbols. And since the State of California expects students to be able to identify and use the arithmetic properties of number sets, let's start there. In stating these properties, we use lower case letters to take the place of numbers. Such letters are called variables. (A variable is a letter that can assume various values in a problem.) It is usual to that capital letters signify the sets themselves. (By the way, sets are a means by which mathematicians talk of collections of things [such as numbers] in an abstract way.)

There are two ways... (page 3 – rainbow book)

Introduction to Algebra WHY STUDY ALGEBRA? Studying algebra will help you develop the ability to reason and killing with symbols as a step in the process developing an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are also used in a wide variety of problem-solving situations. So, let's begin by defining the basic concepts behind all those symbols starting with the concept of variables.

It is usual to that capital letters signify the sets themselves. (By the way, sets are a means by which mathematicians talk of collections of things [such as numbers] in an abstract way.)

There are two ways... (page 3 â&#x20AC;&#x201C; rainbow book)

ARITHMETIC PROPERTIES

Number Sets

sets well.

Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable: According to G. Cantor (18451918), developer of set theory, a set is a grouping together of single objects into a whole. They are a way mathematicians can talk of collections of things (including numbers) in an abstract way. A lot of the work throughout his book is going to involve sets of numbers. Therefore, we had better learn the notation for these

There are two ways to define a set. The first is to list and separate by commas all of the objects that make up the set. We write the defining list within braces. This is called the roster method.

The Roster Method

A set may be described by writing out the names of all its elements inside braces: { } The braces are read as “the set of all elements.” Specifying a set by writing out a list is called the roster method because it consists of simply creating a roster naming all the elements of the set. However, this becomes impractical when it comes to sets with a large number of elements. In such cases, it is better to use the second way of defining a set— what is known as: set-builder notation. This method requires us to give a rule that identifies the elements of the set. In this case, the defining rule is written within braces.

OPEN INTERVAL An open interval is a set of real numbers represented by a line segment on the real number line, whose endpoints are not included in the interval. This concept is made clear by the following definition: ( a, b ) = { x : a < x < b } where a < b Again, the endpoints of an open interval are NOT part of the interval.

CLOSED INTERVAL

An infinite interval is a set of real numbers, but it is represented by a ray or line on the real number line. As befits the name, an infinite interval does not have an endpoint in one or both directions. They are defined as follows: (-∞, a ] = { x : x ≤ a } [ a, -∞) = { x : x ≥ a } (-∞,∞) = { x : x is a real number } Similar definitions apply when the infinite interval is open at one end. REVIEW Directions: Do not go on to the next page until you can verbalize the information below completely unassisted. 1. Those with the four operations, numbers, variables that often take the place of numbers, and the properties that govern them all. 2. The reason for having variable substitute for numbers is that they make it easier to see how certain properties act on (control, direct or affect) numbers and operations. They also substitute for changing values. 3. A variable is a letter that takes the place of a number.

Types of Numbers

The elements of the sets whose notation you will be learning are, of course, numbers. As you progress in your study of algebra, you will encounter references to different kinds of numbers. If you do not know what each type of number is, you will not understand what your textbook is talking about. We will therefore take this time to have you learn the names of the major number groups (or categories of numbers), as well as the characteristics of their members. We will start with the most general category and move to the more specific. This means that, initially at least, you will be memorizing a meaningless collection of words. Nonetheless, this task will be very

useful in that the activity will provide you with “hooks” on which to “hang” the information that will immediately follow, which will, in turn, make sense of all the words that you went through the trouble of memorizing.

Number Groups The most basic group is the one that people traditionally used when they wanted to count: counting numbers (1, 2, 3…). Some people include zero. Others don't. They say that counting numbers plus zero are natural numbers (0, 1, 2…). However, there is not solid agreement as to whether zero should or shouldn’t be included among the natural numbers either. And while whole numbers are defined as positive, one finds the term “negative whole numbers” used to describe integers. With all this confusion and lack of standardization in terminology, lets just forget all of the above (with the single exception of integers) and use the following more precise terminology instead… By the way, depending on whom you’re talking to, counting numbers, natural numbers, and integers, at one time or another, are all be referred to as whole numbers. Moreover, counting numbers are sometimes referred to as natural numbers. To avoid such confusion, algebra rejects the use of such vague and imprecise terms, employee and instead only the terms mentioned above: integers, negative integers, non-negative integers, zero, and positive integers.

CHAPTER 1.0 Subsets of Integers Integers include zero, all of the positive numbers representing one or more complete units, and all of their corresponding negatives. . . . –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6 . . . We refer to integers as either negative: . . . –6, –5, –4, –3, –2, –1 As nonnegative: 0, 1, 2, 3, 4, 5, 6 . . . As zero: 0 Or as positive: 1, 2, 3, 4, 5, 6 . . .

TYPES OF NUMBERS The state of California expects you to be familiar with the behavior of numerical subsets such as integers, rational numbers, irrational numbers and the like. Of course, that requires that you first know what these entities are, so let’s begin with the most general (as we said we would) and provide a definition for real numbers.

Real numbers are all numbers that actually exist. In plain terms, real numbers are all of the points on the number line— from negative infinity to positive infinity, and every point in between.

Then, we have rational numbers.

Rational numbers are all numbers except those with never-ending non-repeating de

You probably guessed that irrational numbers are next. Examples include pi (3.14159262….), e (2.71828182845…) and the square root of two ( 2 2 ).

Irrational areofnumbers never-ending decima Integers are thenumbers last group numbersthat wehave will define (for thenon-repeating time being).

Integers are the last group of numbers we will define (for the time being).

The integers are made up of zero, along with all of the positive numbers that repres complete (or whole) units, and all of the negative numbers that represent whole un

sson 4

More About Rational, Irrational and Real Numbers Rational numbers are technically regarded as ratios (divisions) of integers. In other words, a rational number is formed by dividing one integer by another integer. Consequently, they are sometimes referred to as fractional numbers. It follows from this definition that all fractions in which both of the numerator and denominator are integers are irrational numbers. And remember, you can turn any integer into a fraction by placing it over a fraction line and the number 1, so obviously, all integers are rational numbers! On the other hand, irrational numbers cannot be written as fractions. Remember that fractions (rational numbers) can be written as terminating (ending) or repeating decimals, such as 16.0, 0.5, 0.76, or 0.333333 . . . Irrational numbers like pi (π = 3.14159262….) and the square root of two ( 2 2 = 1.414213562…) cannot be expressed as fractions.

Rational and irrational numbers are totally separate number types—there is no overlap. Putting these two major classifications together gives you the real numbers, which make up the real number system. They are numbers that are represented by points on the number line. A real number is also defined as any number that can be written as a fraction. I thought you just said that irrational numbers cannot be written as a fraction. “But why,” you ask, “are they called ‘real’ numbers? Are there ‘pretend’ numbers?” Well yes, there are. They’re actually called “imaginary” numbers. They are used to make complex numbers and always include the lower-case letter "i". (If a number does not have an “i” in it, it is a real number.)

Set Notation and Integers We introduced this chapter by saying that we were going to make sure we began learning the notation for the number subsets right away, and since we are starting with integers, we had better not delay learning its notation any longer. It is usual to let capital letters signify sets, and the letter we use for integers is a capital Z. (But don't ask us why! If it helps, you might want to focus on the fact that the term integers ends with the sound /zzzz/.) There are two ways to define a set. The first is to list them objects that compose the set, separating them with commas. We write the defining list within braces. For example, integers are defined as follows:

Z = {. . . –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6 . .

The three dots (ellipses) indicate that we continue in the same manner The second way is to give the rule that identifies the elements of the set. Again, we write the defining rule within braces…

Z = {all positive numbers ≥ 1 whole unit, all of their corresponding negatives, and zero} Hi, My name is Mr. Will and today I will be presenting a lesson on the arithmetic properties. But, what do we mean by arithmetic…and what do we mean by properties? Well, of course, arithmetic has to do with arithmetic; and arithmetic involves three basic concepts. 1. Arithmetic deals with numbers. 2. Arithmetic looks at how those numbers work or interact together. In other words, if it looks at how they operate. And how do they operate? They do it in four ways: through at

addition, subtraction, multiplication, and division; which is why we call these the four operations. 3. And finally, arithmetic looks at the rules that govern or control numbers and their operations. However, a mathematics, but don't call the rules—we call them properties. Is so, in a nutshell, arithmetic looks at numbers, the four operations through which members interact with one another, and the rules (or properties) that govern those interactions. Now, in most math books, if you look for the arithmetic properties they are hard to find. There are usually interspersed all throughout the book, which I find frustrating and irritating. So, I've listed them all here for your convenience—so that you can see them all at a glance. I’ve grouped them, or organized them, in such a way as to make them easier to remember. In stating these properties, we use letters, called variables, to stand for numbers. We have four properties, six laws, two identities and multiplication by zero.

The Arithmetic Properties Our seven properties of equality.

1. THE ADDITION PROPERTY OF EQUALITY 2. THE SUBTRACTION PROPERTY OF EQUALITY 3. THE MULTIPLICATION PROPERTY OF EQUALITY 4. THE DIVISION PROPERTY OF EQUALITY 5. REFLEXIVE PROPERTY OF EQUALITY 6. SYMMETRIC PROPERTY OF EQUALITY 7. TRANSITIVE PROPERTY OF EQUALITY Also five properties of inequality: 8. THE ADDITION PROPERTY OF INEQUALITY 9. THE SUBTRACTION PROPERTY OF INEQUALITY 10. THE MULTIPLICATION PROPERTY OF INEQUALITY

11. THE DIVISION PROPERTY OF INEQUALITY 12. TRANSITIVE PROPERTIES OF INEQUALITIES We have one law…

13. THE TRICHOTOMY LAW Five properties of movement… 14. THE ASSOCIATIVE PROPERTY OF ADDITION 15. THE ASSOCIATIVE PROPERTY OF MULTIPLICATION 16. THE COMMUTATIVE PROPERTY OF ADDITION 17. THE COMMUTATIVE PROPERTY OF MULTIPLICATION 18. THE DISTRIBUTIVE PROPERTY 19. THE DISTRIBUTIVE LAW OF MULTIPLICATION OVER ADDITION Then the two identities…

20. THE IDENTITY PROPERTY OF ADDITION 21. THE IDENTITY PROPERTY OF MULTIPLICATION One inverse property identities…

22. INVERSE PROPERTY OF ADDITION And finally, two properties associated with numbers…

23. MULTIPLICATION PROPERTY OF ZERO 24. MULTIPLICATION PROPERTY OF NEGATIVE ONE

Closure Properties

Application Standard 1.1

3) Is the equation 3(2x – 4) = –18 equivalent to the equation 6x – 12 = –18? a) Yes, the equations are equivalent by the Associative Property of Multiplication. b) Yes, the equations are equivalent by the Commutative Property of Multiplication. c) Yes, the equations are equivalent by the Distributive Property of Multiplication over Addition. d) No, the equations are not equivalent.

4) Which statement is false? a) The order in which two whole numbers are subtracted does not affect the difference. b) The order in which two whole numbers are added to does not affect the sum. c) The order in which to rational numbers are added to does not affect some. d) The order in which to rational numbers are multiplied is not affect the product.

Chapter 2 THE RULES OF ROOTS AND EXPONENTS • • • • •

Taking the opposite Finding the reciprocal Taking a root Raising to a fractional power Rules of exponents

Taking the Opposite An equation

Finding the Reciprocal Division is related to multiplication in the sense that it is made possible by the existence of multiplicative inverses, also known as reciprocals. In other words, when you divide by a number, what you are doing is the same thing as multiplying by its reciprocal. Reciprocals, or multiplicative inverses, are pairs of numbers that have one as their product. For example, 5 is the reciprocal of 15 , and because 5 ⋅ 15 = 1 and

1 5

is the reciprocal of 5,

⋅5 = 1.

If you are a glutton for punishment and want to see this concept written mathematically, here you go... For each nonzero real number a, there is a unique real number 1/a such that a · (1/a) = 1. Because (1/a) · a = 1, we have the law…

The reciprocal of 1/a is a. Reciprocals are two numbers whose product equals 1. In other words, a pair of numbers are reciprocals of each other if their product equals 1. So, the reciprocal of 3 is

1 3

because 3 × 13 = 1

Another name for the reciprocal of a number is multiplicative inverse, and all nonzero real numbers have one. So then, for every nonzero real number a, there exists a unique real number 1a such that a ⋅ 1a = 1 . More generally, we have the following law: The reciprocal of

1 a

is a.

Prove the zero factor law (pg 32)

Division by a Nonzero Number Division by a nonzero number means multiplication by its reciprocal, according to the following definition: If a and b are real numbers, and if b ≠ 0, then a/b = a · (1/b).

Division by Zero Because zero has no reciprocal, division by zero is not defined.

When Minus Signs Appear in a Fraction When minus signs appear in a fraction, the fraction can often be written more simply according to the following rules: If a and b are real numbers with a ≠ b, then (–a)/b = –(a/b) a/(–b) = –( a/b) (–a)(–b) = a/b In other words, if you have a fraction in which the liberator or the denominator is negative, you can just make the whole fraction native. And if you have a fraction where the numerator and the denominator are negative, you can just make the whole fraction positive.

How to Divide a Number Appearing as a Fraction

To divide by a number appearing as a fraction, you multiply by its reciprocal (multiplicative inverse), which is derived by inverting the fraction—by turning it upside-down. Here is how the concept is written mathematically: If a and b are each a nonzero real number, then the multiplicative inverse of real a/b is b/a. So, to divide by a given fraction, we must multiply it by that fraction's multiplicative inverse (reciprocal). That's the general law may be stated as follows: If a, b, c, and d are real numbers with b, c, and d nonzero, then a/b ÷ c/d = (a/b) · (d/c).

Cancellation Law for Multiplication Division of the house us to the ride of the following law: If a, b, and c are real numbers with a ≠ 0, and if ab = ac, the b = c. An important consequence of the cancellation law is the zero factor law: If a and b are real numbers such that ab = 0, then either a = 0 or the b = 0 (or both).

Personally, when I went to come up the reciprocal (or multiplicative inverse) of a number, I simply write the number has a fraction, and then switch the numerator with the denominator.

Taking a Root As you probably already know, exponents can be used to indicate repeated multiplication, which can also be represented by printing a number under a radical sign, as in the case of 9 . The above expression is called a radical. The number (9) which appears under the radical sign ( ) is called the radicand. This concept is expressed by the formal definition of square roots:

So then, 3 = 9 because 3 2 = 9. Also, 0 = 0 , because 0 2 = 0 However, while the definition of square roots involves an equation using the exponent 2, there are times when equations use higher exponents and therefore need to be defined using higher roots. For example, the equation 81 = 3 4 indicate that three has been raised to the fourth power. We may express this by saying that 3 is in a fourth root of 81. In symbols, this would be written... 3 = 4 81 Here 4 is the index of the radical. (Note that when the index is not written in a radical symbol, it is understood that the radical refers to a square root.

Principal Values Recall that according to our definition of square roots…

So, even though both –3 2 and 3 2 are equal to 9 , because –3 is not a nonnegative real number, only the positive square, 3, is used as the square root of 9 . In other words, 9 ≠ −3 To put it another way, positive square roots have been arbitrarily singled out and called the principal values of square roots, and whenever we use the radical symbol, we always mean the principal value of the square root.

EXAMPLES: 3

− 64 = – 4 because – 64 = (– 4) 3

− 32 = – 2 because – 32 = (– 2) 5

Irrational Numbers Roots give rise to important examples of irrational numbers. Let us see what we can say about 2 .

Raising to a Fractional Power (p 267) If a is a positive real number and n is a positive integer, 1

then a n defined by a n = n a Page 273

Rules for Exponents (page 129) An exponent is a convenient way to express repeated multiplication.

EXAMPLE: 52 = 5 × 5. Here 5 is called the base, and 2 is the exponent. The entire expression is called a power. If most problems involving exponents can be simplified within five steps—provided you deal with the different types of exponents in a particular order—wouldn’t it make sense to learn the rules for exponents in that same order? The following explanations will therefore define the different types of exponents in that very order. It's easy enough to remember where you start, since it almost seems logical—you start with zero.

1. Zero Base and Exponent 00 Zero raised to the zero power is undefined. (It doesn't exist.)

So, you started with two zeros with zero as the base and zero as the exponent. Sticking with our “aero” theme, we move from two zeros to one zero, with zero as the exponent.

2. Zero Exponent a0 = 1 Any number raised to the zero power equals one. And where are you going to go from zero if not to one?

3. One Exponent a1 = a Any number raised to a power of one equals itself. Those are pretty simple. Now, let’s get a little bit more complicated and move on to the power.

4. Power Rules (am)n = amn Remove the parentheses and multiplied the exponents. (ab)m = bmam Remove the parentheses and distribute the exponent to each variable. m

a a   = m b b Remove the disease and distribute the exponent to each variable.

5. Product Rule am · an = am + n When multiplying terms that have the same base, copy the base one time and add their exponents. You multiply powers that have identical bases by copy the base once and using the sum of their powers as the exponent.

6. Quotient Rule am = a m −n n a When you divide terms that have the same base, you copy the base one time, changing the fraction line into a subtraction sign and subtracting the exponent in the denominator from the exponent in the numerator. 7. Negative Exponent 1 a −m = −m a

When you have a number raised to a negative power, change the negative sign into a fraction line and write the base along with its exponent under the numeral 1.

8. Radical Exponent a

x b = b xa When you have a number raised to a fraction, place the number under a radical sign. Then keep the denominator as the number’s exponent and make the numerator the radicand. Don’t be fooled! (x + y)2 = (x + y)(x + y) Here you multiply out the long way.

The state of California expects you to be able to apply the rules of exponents and using operations such as taking the root and raising to a fractional power An exponent is a convenient way to express repeated multiplication. For example, another way to represent two multiplied by itself three times… (2)(2)(2) = ___ …is to write it in exponential form: (2)(2)(2) = 23 So then, 23 means that the number two is multiplied by it self three times. In exponential form, the number that is multiplied repeatedly times itself is called the base, and the number that represents how many times a number is multiplied by itself is called the exponent. (The entire expression is called a power.)

Application

16 + 3 8 = ?

a) 4 b) 6 c) 9 d) 10

6) Which expression is equivalent to x6 x2? a) x4 x3 b) x5 x3 c) x7 x3 d) x9 x3

7) Which expression does not have a reciprocal? a) –1 b) 0 c)

1 1000

d) 3

8) What is the multiplicative inverse of a) –2

1 ? 2

b) – ½ c) ½ d) 2

So then, 3 = 9 because 3 2 = 9.

Also, 0 = 0 , because 0 2 = 0 However, while the definition of square roots involves an equation using the exponent 2, there are times when equations use higher exponents and therefore need to be defined using higher roots. For example, the equation 81 = 3 4 indicate that three has been raised to the fourth power. We may express this by saying that 3 is in a fourth root of 81. In symbols, this would be written... 3 = 4 81 Here 4 is the index of the radical. (Note that when the index is not written in a radical symbol, it is understood that the radical refers to a square root.

Principal Values Recall that according to our definition of square roots…

EXAMPLES: 3

− 64 = – 4 because – 64 = (– 4) 3

− 32 = – 2 because – 32 = (– 2) 5

Irrational Numbers Roots give rise to important examples of irrational numbers. Let us see what we can say about 2 .

Name __________________________________________________________

Date ______/______/______

Score _______

Roots What is two multiplied by itself three times?

(2)(2)(2) = ? Of course, the answer is eight, but that’s not really our main concern here. What we really want to focus on is the fact that a number that is repeatedly multiplied times itself can be represented in another way, called exponential form.

(2)(2)(2) = 23 In exponential form, the number that is multiply repeatedly by itself is called the base. In exponential form, the number that represents how many times a number is multiplied by itself is called 3 the exponent. In the case of 2 the number two is multiplied by itself three times.

DIRECTIONS: Respond to the prompts below. each item below. the prompts 3

The term 4 indicates that the number four is multiplied how many times?

Given the term x how many times is the veritable x multiplied by itself?

Given the term 5 which number is the base?

Given the term 10 which number is the exponent?

_____

Write 8 × 8 × 8 × 8 × 8 in exponential form.

_____

2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents

DIRECTIONS: Write the reciprocal of each number. A

â&#x20AC;&#x201C;1

1 1000

What is the multiplicative inverse of

1 ? 2

ROOTS RULES 0 =0

1 =1 (In basic calculus you canâ&#x20AC;&#x2122;t have a negative number under an even number root?) When you are dividing quantities under separate radicals that have the same roots, you can divide them under a single radical.

a b

a b n n

a n a = b b

a b

You can convert any route to a power and solve it using the power rules. To convert a route to a power, remove the radical sign and write the root as a power under the 1. (What?) When you are multiplying quantities under separate radicals that have the same roots, you can multiply them under a single radical. 3

a ⋅ b = a⋅b n

a ⋅3 b = 3 a⋅b

a ⋅ n b = n a ⋅b

Chapter 3 EQUATIONS, INEQUALITIES & ABSOLUTE VALUE CHAPTER 3 Prove equations and inequalities involving absolute values.

DIRECTIONS: Solve. │2x – 3│= 5 What is the solution set of this inequality? 5 –│x + 4│≤ –3

Solving Equations The first step to solving equations and inequalities in one variable is to use elementary transformations chosen so that all the terms involving the variable are on one side and all the numerical terms are on the other. However, in changing an equation by an elementary transformation, you must ensure that the old and new equations are equivalent. Then simplify the new equation (or inequality) by collecting like terms. Is an expression representing the same real number is added or subtracted to each site of an equation, then the old and new equations are equivalent. Also, if each side of in equation is multiplied or divided by the same expression representing a nonzero real number, then the old and new equations are equivalent.

Solving Inequalities The

Absolute Value The absolute value of a number is the distance between that number and the origin on the number line. Absolute value is indicated by enclosing the number in vertical | |.

Because absolute value is a distance, it is never negative. | 0 | = 0, but if a ≠ 0, then | a | > 0. The formal definition of absolute value is as follows: If a is a real number, then | a | = a if a ≥ 0 | a | = –a if a < 0 The distance between any two real numbers can be expressed in terms of absolute value, according to the following: If a and b are in any two real numbers, then | a – b | is equal to the distance a and b on the number line.

Solving Equations and Inequalities Involving Absolute Value In mathematics, the absolute value of a given real number is defined as that number's distance from zero. In other words, absolute value is the number's numerical value without regard to its sign. For example, both 4 and −4 have an absolute value of 4 since, were you to plot each of them on a number line, both of them would be four units away from zero.

Of course, that means the absolute value of zero is always zero, since there is no distance between zero and itself. Consequently, absolute values are always equal to or greater than zero—but never less than zero. To designate the absolute value of a number or expression, we write that number or expression between “absolute brackets.” For example, “the absolute value of −2” would be written like this: | −2 |

EXERCISE 1: Find the value(s) for x when | 4 − 2x | = −7. Since absolute values are never less than zero, the answer is: Ø. (The solution is a null, or empty, set.)

EXERCISE 2: Solve the equation | x − 4 | = 5. Normally, when we have the absolute value of a quantity involving a variable such as | x − 4 |, in order to find the solution we have to express that quantity as both a positive and negative value, and then apply the basic algebraic operations. Therefore, we must express x − 4 as both greater than or equal to zero and less than zero.

If x − 4 ≥ 0, then x ≥ 4 And the equation is changed to...

x–4 x–4+4 x x

=5 =5+4 =5+4 =9

So in this first case, the answer is the intersection of the solution sets of...

x ≥ 4 and x = 9 Hence, the solution set is {9}.

On the other hand, if x − 4 < 0, then x < 4 And the equation is changed to...

x–4 x–4+4 x x

=–5 =–5+4 =–5+4 =–1

So in this second case, the answer is the intersection of the solution sets of...

x < 4 and x = −1

Hence, the solution set is {−1}. The solution set of | x − 4 | = 5 is the union of the solution sets in both of the two cases.

Therefore, the solution set is {−1, 9}.

EXERCISE 3: Find the solution set of | 2 − 3x | = 7. Remember that we have to express | 2 − 3x | as both greater than or equal to zero and less than zero.

If 2 − 3x ≥ 0, then x ≤ ⅔ And the equation is changed to...

2 – 3x – 3x – 3x x

=7 =7–2 =5 = – 5/3

So in this first case, the answer is the intersection of the solution sets of...

x ≤ ⅔ and x = −5/3 Hence, the solution set is {−5/3}.

On the other hand, if 2 − 3x < 0, then x > ⅔ And the equation is changed to...

2 – 3x = – 7 – 3x = – 7 – 2

– 3x = – 9 x =3 So in this second case, the answer is the intersection of the solution sets of...

x > ⅔ and x = 3 Hence, the solution set is {3}.

The solution set of | 2 − 3x | = 7 is the union of the solution sets in both cases.

Therefore, the solution set is {−5/3, 3}.

EXERCISE 4: Find the solution set of | 2x − 3 | = x + 5. Remember that we have to express | 2x − 3 | as both greater than or equal to zero and less than zero.

If 2x − 3 ≥ 0, then x ≥ 3/2 And the equation is changed to...

2x – 3 2x – x – 3 x–3 x–3+3 x

=x+5 =x–x–5 =5 =5+3 =8

So in this first case, the answer is the intersection of the solution sets of...

x ≥ 3/2 and x = 8 Hence, the solution set is {8}.

On the other hand, if 2x − 3 < 0, then x < 3/2 And the equation is changed to...

2x – 3 2x – 3 2x + x – 3 3x – 3 3x – 3 + 3 x

= – (x + 5) =–x–5 =–x+x–5 =–5 =–5+3 = – 2/3

So in this second case, the answer is the intersection of the solution sets of...

x < 3/2 and x = −2/3 Hence, the solution set is {−⅔}.

The solution set of | 2 − 3x | = 7 is the union of the solution sets in both cases.

Therefore, the solution set is {−2/3, 8}. EXERCISE 5: Find the solution set of | 3x + 1 | = 4x + 3. Remember that we have to express | 3x + 1 | as both greater than or equal to zero and less than zero.

If 3x + 1 ≥ 0, then x ≥ −1/3 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of...

x ≥ −⅓ and x = −⅔ However, since −⅔ is neither greater than nor equal to −⅓, in this first case, the answer is a null set: {Ø}.

On the other hand, if 3x + 1 < 0, then x < −1/3

And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of...

x < −1/3 and x = −4/7 Hence, the solution set is {−4/7}. The solution set of | 2 − 3x | = 7 is the union of the solution sets in both cases.

Therefore, the solution set is {−4/7}. EXERCISE 6: Find the solution set of | 2x − 2 | = 2 − 2x Remember that we have to express | 2x − 2 | as both greater than or equal to zero and less than zero.

If 2x − 2 ≥ 0, then x ≥ 1 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of...

x ≥ 1 and x = 1 Hence, the solution set is {1}.

On the other hand, if 2x − 2 < 0, then x < 1

And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of...

x < 1 and 0x = 0 Hence, the solution set is { x | x < 1}

The solution set of | 2 − 3x | = 7 is the union of the solution sets in both cases.

Therefore, the solution set is {x | x ≤ 1}. EXERCISE 7: Find the solution set of | 3x − 7 | = | x + 3 | Remember that we have to express | 5x − 1 | as both greater than or equal to zero and less than zero.

If 3x − 7 ≥ 0, then x ≥ 7/3 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of...

x ≥ 7/3 and x = 5 Hence, the solution set is {5}.

On the other hand, if 3x − 7 < 0, then x < 7/3 And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of...

x < 7/3 and x = 1 Hence, the solution set is {1}.

The solution set of | 2 − 3x | = 7 is the union of the solution sets in both cases.

Therefore, the solution set is {1, 5}. Application 9) What is the solution for this equation? | 2 − 3x | = 5 a) x = −4 or x = 4 b) x = −4 or x = 3 c) x = −1 or x = 4 d) x = −1 or x = 3

10) What is the solution set of the inequality below? 5 – | x + 4| ≤ – 3 a) –2 ≤ x ≤ 6 b) x ≤ –2 or x ≤ 6 c) –12 ≤ x ≤ 4

d) x ≤ –12 or x ≤ 4

Chapter 3

Absolute Value Solving Equations and Inequalities Involving Absolute Value The state of California expects you to be able to solve equations and inequalities involving absolute values. It would therefore probably be a good idea to make sure you have a clear understanding of exactly what absolute value is. In mathematics, the absolute value of a given real number is defined as that number's distance from zero. For example, both 4 and −4 have an absolute value of 4 because if you plotted each of them on a number line, both of them would be four units away from zero.

In other words, the absolute value of a number is that number’s value without any negative sign. Consequently, absolute values are never less than zero. They are always equal to or greater than zero. (The absolute value of zero is always zero, since there is no distance between zero and itself.) To designate the absolute value of a number or expression, we write that number or expression between "absolute brackets." For example, the absolute value of −2 would be written like this: |−2|

Now that you have a clearer understanding of what absolute value is, let's find out how we go about solving equations and inequalities when absolute value is involved.

EXERCISE 1: Solve the equation |x − 4|= 5. Normally, when we have the absolute value of a quantity involving a variable such as |x − 4|, in order to find the solution we have to express that quantity as both a positive and negative value, and then apply the basic algebraic operations. Therefore, we must express x − 4 as both greater than or equal to zero and less than zero. If x − 4 ≥ 0, then x ≥ 4 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of... x ≥ 4 and x = 9 Hence, the solution set is {9}.

On the other hand, if x − 4 < 0, then x < 4 And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of... x < 4 and x = −1

Hence, the solution set is {−1}. The solution set of |x − 4|= 5 is the union of the solution sets in both of the two cases. Therefore, the solution set is {−1, 9}. EXERCISE 2: Find the solution set of |2 − 3x|= 7. Remember that we have to express |2 − 3x| as both greater than or equal to zero and less than zero. If 2 − 3x ≥ 0, then x ≤ ⅔ And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of... x ≤ ⅔ and x = −5/3 Hence, the solution set is {−5/3}.

On the other hand, if 2 − 3x < 0, then x > ⅔ And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of... x > ⅔ and x = 3 Hence, the solution set is {3}.

The solution set of |2 − 3x|= 7 is the union of the solution sets in both cases. Therefore, the solution set is {−5/3, 3}. EXERCISE 3: Find the solution set of |2x − 3|= x + 5. Remember that we have to express |2x − 3| as both greater than or equal to zero and less than zero. If 2x − 3 ≥ 0, then x ≥ 3/2 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of... x ≥ 3/2 and x = 8 Hence, the solution set is {8}. On the other hand, if 2x − 3 < 0, then x < 3/2 And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of... x < 3/2 and x = −2/3 Hence, the solution set is {−⅔}.

The solution set of |2 − 3x|= 7 is the union of the solution sets in both cases. Therefore, the solution set is {−2/3, 8}. EXERCISE 4: Find the solution set of |3x + 1|= 4x + 3. Remember that we have to express |3x + 1| as both greater than or equal to zero and less than zero. If 3x + 1 ≥ 0, then x ≥ −1/3 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of... x ≥ −⅓ and x = −⅔ However, since −⅔ is neither greater than nor equal to −⅓, in this first case, the answer is a null set: {Ø}.

On the other hand, if 3x + 1 < 0, then x < −1/3 And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of...

x < −1/3 and x = −4/7 Hence, the solution set is {−4/7}. The solution set of |2 − 3x|= 7 is the union of the solution sets in both cases. Therefore, the solution set is {−4/7}. EXERCISE 5: Find the solution set of |2x − 2|= 2 − 2x Remember that we have to express |2x − 2| as both greater than or equal to zero and less than zero. If 2x − 2 ≥ 0, then x ≥ 1 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of... x ≥ 1 and x = 1 Hence, the solution set is {1}.

On the other hand, if 2x − 2 < 0, then x < 1 And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of...

x < 1 and 0x = 0 Hence, the solution set is {x|x < 1}. The solution set of |2 − 3x|= 7 is the union of the solution sets in both cases. Therefore, the solution set is {x|x ≤ 1}. EXERCISE 6: Find the solution set of |3x − 7|= |x + 3| Remember that we have to express |5x − 1| as both greater than or equal to zero and less than zero. If 3x − 7 ≥ 0, then x ≥ 7/3 And the equation is changed to...

So in this first case, the answer is the intersection of the solution sets of... x ≥ 7/3 and x = 5 Hence, the solution set is {5}.

On the other hand, if 3x − 7 < 0, then x < 7/3 And the equation is changed to...

So in this second case, the answer is the intersection of the solution sets of... x < 7/3 and x = 1

Hence, the solution set is {1}. The solution set of |2 − 3x|= 7 is the union of the solution sets in both cases. Therefore, the solution set is {1, 5}. EXERCISE 7: Find the value(s) for x when |4 − 2x|= −7. Since absolute values are never less than zero, the answer is the null (or empty) set, which is represented by the symbol: Ø. In other words, the answer to this problem does NOT exist.

Chapter 4 SIMPLIFYING EXPRESSIONS Is expected that after you have completed this course, you will be able to solve multi-step problems involving linear equations and inequalities in one variable (including word problems) providing a justification for each step.

Solving Linear Equations An equation is linear if its variables have no exponents and none of its terms have more than one variable (as a factor).

CHAPTER 4 Simplifying expressions before solving linear equations and inequalities in one variable

DIRECTIONS: Simplify 3(2x – 5) + 4(x – 2) = 12. 5x – 2(7x + 1) = 14 x

EXTRA PRACTICE Using the Addition Property of Equality to Solve One-Step Equations Containing One Variable EXAMPLE: Solve

x–3 =–8

x–3+3 =–8+3 x=–5

Check

x–3 = –8 –5 – 3 = – 8

Add 3 to each side to get the variable alone on one side of the equal sign. Simplify.

Check your solution in the original equation. Substitute – 5 for x.

–8 = –8

Addition Property of Equality For every real number a, b, and c, if a = b, then a + c = b + c.

Lesson 1-1 Solve. 1. x – 2 = 90 2. x – 22 = 3 3. x – 14 = 9 4. x – 7 = 12 5. x – 9 = 0 6. x – 8 = 80 7. x – 31 = 8 8. x – 5 = 8 9. x – 1 = –1 10. x – 4 = –6 11. x – 2 = –4 12. x – 3 = 11

Application 11) Which equation is equivalent to 5 x − 2(7 x + 1) = 14x? a) −9 x − 2 = 14 x b) −9 x + 1 = 14 x c) −9 x + 2 = 14 x d) 12x −1 = 14 x

12) Which equation is equivalent to 4(2 – 5 x) = 6 − 3(1 – 3 x)? a) 8 x = 5 b) 8 x = 17 c) 29 x = 5 d) 29 x = 17

13) Which equation is equivalent to 3[4(7 x − 4( x − 3)] + 1 = 16? a) 9 x – 2 = 16 b) 9 x + 37 = 16 c) 17 x – 2 = 16 d) 17 x + 13 = 16

Chapter 5 SOLVING MULTISTEP PROBLEMS IN ONE VARIABLE At 14) Which equation is equivalent to

3[4(7 x − 4( x − 3)] + 1 = 16? a) 9 x – 2 = 16 b) 9 x + 37 = 16 c) 17 x – 2 = 16 d) 17 x + 13 = 16

CHAPTER 5 MULTI-STEP PROBLEMS Solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

The total cost (c) in dollars of renting a sailboat for n days is given by the equation

c = 120 + 60n If the total cost was $360, for how many days was the sailboat rented?

3(x + 5) 3x + 15 x + 15 x

= 2x + 35 = 2x + 35 = 35 = 20

A 120-foot-long rope is cut into 3 pieces. The first piece of rope is twice as long as the second piece of rope. The third piece of rope is

three times as long as the second piece of rope. What is the length of the longest piece of rope? The cost to rent a construction crane is $750 per day plus $250 per hour of use. What is the maximum number of hours to crane can be used each day if the rental cost is not to exceed $2500 per day?

DIRECTIONS: Solve. x â&#x20AC;&#x201C; 5 > 14? The length of the sides of a triangle are y, y + 1, and 7 centimeters. If the perimeter is 56 cm, what is the value of y?

Chapter 6 GRAPHING LINEAR EQUATIONS (PAGE 67)

Chapter 6 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). What is the y-intercept of the graph of 4x + 2y = 12

Relations In mathematics, relationships often exist between two different quantities. One of the ways such relationships can represented is through the use of equations containing two variables, one for each of the quantities. There are four different ways this situation may be represented: 1. 2. 3. 4.

in words in a table by a formula with a graph

Conditional Equations Formulas used to represent such relationships are often part of a conditional equation, which is an equation that contains one or more variables, for which some values make the equation true, while other values make the equation false. For example, if the replacement set for the equation 3x + 1 = 13 is the set of real numbers, then its solution set is {4} because it is the only real number value for x that makes the equation true. Similarly, given the equation y = 3x + 2, for some values of x and y, this equation will be true, while for other values it will be false. Consequently, it too is a conditional equation. Note however that it is impossible to decide whether the equation is true or false unless we are given values for both of the variables, x and y.

Graphing Relations As you can imagine, there are a countless number of solutions to y = 3x + 2. There are so many, in fact, that presenting them in words or in a table might be considered too impractical or inefficient. And while the equation itself does encompass all of the potential solutions to the relationship between x and y, in cases such as this (and many others like it) it is a picture (or graph) that is the key to thoroughly understanding the information represented.

Of course, determining which values make the above conditional equation true involves selecting a pair of numbers and specifying which of those numbers is to be associated with which of the variables. This is done through a concept referred to as: ordered pairs. An ordered pair consists of two numbers in some definite order. An ordered pair consisting of the two numbers a and b, with a first and b second, is denoted by (a, b). In the ordered a pair (a, b), a is called the first component, or entry, and b is called the second component (or entry). While statements involving one variable can be plotted on a number line, ordered pairs of real numbers (which is the case when dealing with two variables) must be plotted in a coordinate plane. When plotting order pairs in a coordinate plane, the first component is always x, and the second component is always y. To draw the coordinate plane, we draw two number lines, one horizontal and one vertical. The point where they intersect is the origin. The origin represents zero on each line. As usual, the positive direction on the horizontal line is to the right. On the vertical line, the positive direction is up. Theses two number lines are called axes. The horizontal line represents the first variable and is therefore called the x-axis. The vertical line represents the second variable and is therefore called the y-axis. We can therefore think of the replacement set for the equation y = 3x + 2 as the solution set of all ordered pairs (x, y) that make the equation true.

Note that in the figure below, the number one is marked on each axis to show what length represents one unit. This length may or may not be the same for the two axes.

To plot the point (x, y) in the plane, we first locate the x-coordinate on the x-axis and mark it. Similarly, we mark off the y-coordinate on the y-axis. Finally, the point (x, y) lies directly above (or below) the x-coordinate on the x-axis, but at a height corresponding to the y-coordinate that was marked on the y-axis.

Computing x- and y- Intercepts (pages 86, 91, 311312) Any line which is not vertical

Functions (page 95)

Chapter 7 DERIVING LINEAR EQUATIONS

Chapter 7 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the pointslope formula. What is a point that lies on the line defined by 3x + 6y = 2 What is the equation of the line that has a slope of 4 and passes through the point (3, â&#x20AC;&#x201C;10)? (number 29) The data in the table show the cost of renting a bicycle by the hour, including a deposit. Renting a Bicycle Hours (h) Cost in dollars (c) 2 15 5 30 8 45 If hours, h, were graphed on the horizontal axis and cost, c, were graphed on the vertical axis, what would be the equation of a line that fits the data? Some ordered pairs for a linear function of x are given in the table below.

x 1 3 5 7

y 1 7 13 19

Which of the following equations was used to generate the table above?

Lines (page 81)

Point-slope Formula (page 90)

Chapter 8 PARALLEL AND PERPENDICULAR LINES

Chapter 8 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. The equation of line l is 6x + 5y = 3, and the equation of line q is 5x â&#x20AC;&#x201C; 6y = 0. Which statement about the two lines is true?

Which equation represents a line that is parallel to y = â&#x2C6;&#x2019;

5 x +2? 4

Chapter 9 SOLVING SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES (PAGE 209) A problem that has several conditions on its variables cannot be expressed in just one equation. To satisfy all the relationships simultaneously requires a system of equations or inequalities. Purple book page 231 The general form of a linear equation in two variables isâ&#x20AC;Ś

ax + by = c Often the conditions of a problem cannot be expressed in just one equation. There may be several conditions on the variables, in all these relationships must be satisfied simultaneously. We are that led to the concept of a system of equations or inequalities. A solution to this system must satisfy each of these equations or inequalities. Once again it is helpful to study the situation graphically. We find that that set theoretic concept of intersection lies at the heart of the solution process for each system.

Graphical Approach To solve linear equations with graphs, simply draw the graph each equation and see if the lines intersect at one point, are parallel, or are the same. The graph of an equation gives a picture of the solutions that. More in equation into variables, the point (x, y) lies undergrad if and only if he ordered pair (x, y) is in the solution set to the equation. Often we have to consider two or more equations at once. Usually we are looking for those elements which are solutions to not just one of the giving equations, but to all of them. This type of problem is called a system of equations. Thus we could make the following in the formal definition: A system of equations is defined by any set of equations. And element is in the solution set to the system if and only if it is in the solution set at each of the equations in the system. Page 210 Page 211

This example illustrates one on the most important types of systems of equations. As we noted in Chapter 4, any equation of the form

Ax + B y = C with A and B not both 0 is called a linear equation in two variables. Its graph is always a line.

Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. A problem that has several conditions on its variables cannot be expressed in just one equation. To satisfy all the relationships simultaneously requires a system of equations or inequalities.

Solving systems of linear equations graphically To solve linear equations with graphs, simply draw the graph of each equation and see if the lines intersect at one point, our parallel or are theysame.

Solving systems of linear equations and inequalities by substitution (Transforming systems of equations) To solve a system of linear equations by suffocation you proceed in a series of steps in which he replaced the system in a simpler system that has the same solution set. This is done by replacing one of the equations with one of the following: 1. An equivalent equation 2. The sum (or difference) of the corresponding members of both sides of equations 3. An equation formed by substituting an expression and one of the original equations with a value a sign to it by the other equation. 1 To put it another way, with this technique, a value for a variable

is obtained from one equation. For This value is then substituted for that variable in the other equation.

Which graph best represents the solution to this system of inequalities? 2 x ≥ y − 1  2 x − 5 y ≥ 10

What is the solution to this system of equations?  y = −3 x − 2  6 x + 2 y = −4

Which ordered pair is the solution to the system of equations below? x + 3 y = 7   x + 2 y = 10

Marcy has a total of 100 dimes and quarters. If the total value of the coins is $14.05, how many quarters does she have? Which of the following best describes the graph of this system of equations?  y = −2 + 3  5 y = −10 + 15

Chapter 10 SOLVING PROBLEMS All about polynomials? Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. A polynomial is an expression in which literal numbers (variables) appear only in sums, differences or products. It is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power. Polynomials are written in descending powers of one of the variables. Terms are usually written so that the variables are in alphabetical order.

EXAMPLES: 30abx5y5 (–2x4y4)

DIRECTIONS: Solve. Simplify. 5 x3 10 x 7

(4 x

) (

)

− 2 x + 8 − x 2 + 3x − 2 =

The sum of two binomials is 5x2 – 6x. If one of the binomials is 3x2 – 2x, what is the other binomial? Which of the following expressions is equal to (x + 2) + (x – 2)(2x + 1)? A volleyball court is shaped like a rectangle. It has a width of x meters and a length of 2x meters. Which expression gives the area of the court in square meters?

A monomial is a polynomial that consists of a single term, which could be a number (I thought a polynomial had to be a constant multiplied times a variable), a variable, or the product of a number with one or more variables.

EXAMPLES 3 –3.85 –7

x 4abc − 2 yz

If a variable in a monomial appears with an exponent, the exponent must be a positive integer. The degree of a monomial is the sum of the exponents of all the variable which appear in the monomial/ The degree of a polynomial is the greatest of the degrees of he monomial which appear.

Certain types of polynomial products occur very frequently. Because they are used so often, the formulas should be memorized. (x + y)2 = (x – y)2 =

Solving Word Problems (page 53) One of the most important reasons for studying algebra is that it is helpful for solving practical problems. The aesthetic merit of the subject may be a matter of opinion, but no one can doubt that algebra is good for something. To use algebra in problem solving, we must be able to translate the problem into mathematical terms; that is, we make a mathematical model of the problem. We then saw the algebra problem presented by the mathematical model of the word problem. If the model is correct, we will have the solution to the original word problem. Skill in setting up the mathematical model comes mostly through practice, but the basic steps are always the same. 1. Read the problem very carefully. If possible, draw a picture. 2. Be sure you understand the information you are given and what you are looking for. 3. That a variable represents some unknown quantity. 4. Express other quantities in terms of this variable. Referred to the picture whenever possible. 5. Defined in the problem and equality relationship, either between two quantities or else as two different ways of expressing the same quantity. 6. Express this equality relationship as an equation. 7. Solve the equation. 8. Check that the solution is really correct. The solution must be what is asked for and it must satisfy the conditions of the problem. This last step is very important. Do not blithely accept an answer!

Chapter 11 FACTORING POLYNOMIALS In the preceding chapters we made a detailed study of linear expressions and went into variables. Now it is time to examine a more general kind of expression, called a polynomial. Every linear expression is a polynomial, but the class of polynomials also includes many nonlinear expressions. As we proceed through this chapter, we will be increasing our technical skill in dealing with more complicated mathematical expressions. At the same time, the class of practical problems which we can solve will be enlarged. Mathematics is much easier when the notation is concise and convenient. An excellent example of good mathematical notation is provided by exponents. In this chapter, we will offer aid in writing polynomials in a simple and concise fashion.

Polynomials and Factoring Students apply basic factoring techniques to second-and simple thirddegree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

FACTORING INTEGERS (Purple book, p. 133) When you break down an expression into its component factors (the numbers that were multiplied to obtain the original product) you are “factoring” the expression. DIRECTIONS: Factor. 3a2 – 24ab + 48b2

x2 – 11x + 24 9t2 + 12t + 4 32 – 8z2

Chapter 12 SIMPLIFYING FRACTIONS WITH POLYNOMIALS Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

DIRECTIONS: Reduce to lowest terms. x 2 − 4 xy + 4 y 2 3 xy − 6 y 2

DIRECTIONS: Reduce to lowest terms. 6 x 2 + 21x + 9 4x2 − 1

x2 − 4x + 4 x 2 − 3x + 2 12a 3 − 20a 2 16a 2 + 8a

Chapter 13 SOLVING RATIONAL EXPRESSIONS AND FUNCTIONS Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. 7z2 + 7z z2 − 4 ⋅ 2 = 4z + 8 z + 2z 2 + z

DIRECTIONS: Multiply.  x + 5   2x − 3    =  3x + 2   x − 5 

DIRECTIONS: Solve. x 2 + 8 x + 16 2 x + 8 ÷ 2 = x+3 x −9

3x 5 x x + 4 2

Chapter 14 QUADRATIC EQUATIONS (PAGE 304) The next step beyond a linear equation is an equation containing quadratic (second-degree) terms, such as x2. When we derive the quadratic formula, we will become binding polynomials and factoring, fractions, radicals, and complex numbers to completely solve the quadratic equation in one variable.

Graphing y = ax2 + bx + c Assume that a, b, and c are real numbers with a ≠ 0. State whether each of the following is, or is not, a quadratic polynomial in x, and provide a justification for your answer. a. 2x 2 – 7x + 4 b. π x 2 –

c. x 2 d. 3x e. x 3 + x 2 A.

2 = a, – 7 = b, and 4 = c, so this is a quadratic polynomial.

π = a, 0 = b, and – 2 = c, so this is a quadratic polynomial.

1 = a, 0 = b, and 0 = c, so this is a quadratic polynomial.

0 = a, so this is not a quadratic polynomial.

E. This is not a quadratic polynomial because it is in the third degree. An equation of the form y = ax2 + bx + c is an example of a quadratic equation involving two variables that defines y as a function of x. We can learn much about the function defined by y = ax2 + bx + c from its graph.

Perhaps the easiest specific example is the equation y = x2, the graph of which appears below. The graph of this equation forms a parabola. In this example, the lowest point on the graph is called the vertex. Cutting the parabola and half is called the axis of symmetry. Were we to continue graphing quadratic equations, you would find that… the graph of a quadratic equation y = ax2 + bx + c is always a parabola.

x-intercepts Whenever a function y = f (x) is graphed, the x-intercepts of the graph are the x-coordinates of the points for which y = 0. Therefore, the xintercepts of the graph are the solutions to the equation f (x) = 0.

Solving Quadratic Equations (page 316) Factoring (page 316) Solving by Granite equations by graphing is not always the best way to find the solution set. Solution techniques that do not rely on a graph are called analytic methods. One important method of this type is factoring. As we know, solution by factoring is based on the following principle: If ab = 0, then either a = 0 or b = 0.

EXAMPLE: x2 – 4 = 0 (x – 2)(x + 2) = 0 Then x – 2 = 0 or x + 2 = 0 and x = 2 or x = – 2

Extraction of Roots A contraction of roots is used only when the equation is in the form of perfect square = number.

Mathematically, this concept is represented by… If x2 = d, then either x =

d or x = – d

Completing the Square (page 318) When you cannot solve a quadratic region by factoring, you always have the option of trying to complete the square. Sometimes, a little ingenuity is needed to create a perfect square. For example, x2 + 6x = 7 is not a perfect square. However, we can solve the equation by adding nine to each side. x2 + 6x = 7 x2 + 6x + 9 = 16 (x + 3) 2 = 16 Then x + 3 = 4

x+3=–4

Therefore x = 1

x=–7

Completing the square is based on the following formula: 2

b b  x + bx +   =  x +  THIS IS WRONG! 2 2  2

In other words, to convert x2 + bx into a perfect square, we should add (b/2)2 to both sides of the equation.

Solving Quadratic Equations In this chapter we will look at different ways of solving quadratic equations: by factoring or by completing the square. If x2 is added to x, the sum is 42. Which of the following could be the value of x? What quantity should be added to both sides of this equation to complete the square? x2 â&#x20AC;&#x201C; 8x = 5 What are the solutions for the quadratic equation x2 + 6x = 16? Carter is solving this equation by factoring. 10x2 â&#x20AC;&#x201C; 25x + 15 = 0

Chapter 15 PERCENT MIXTURE, RATE AND WORK PROBLEMS Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. A pharmacist mixed some 10% saline solution with some 15% saline solution to obtain 100 mL of a 12% saline solution How much of the 10% saline solution to the pharmacist used in the mixture? Andyâ&#x20AC;&#x2122;s average driving speed for a four-hour trip was 45 mph. During the first three hours he drove 40 mph. What was his average speed for the last hour of his trip? One pipe can fill a tank in 20 minutes, while another takes 30 minutes to fill the same tank. How long would it take the two pipes together to fill the tank? Two airplanes left the same airport traveling in opposite directions. If one airplane average is 400 mph and the other airplane averages 250 mph, and how many hours will the distance between the two planes be 1625 miles? Lisa will make punch that is 25% fruit juice by adding pure fruit juice to a 2-liter mixture that is 10% pure fruit juice. How many liters of pure fruit juice does she need to add?

Chapter 16 FUNCTIONS AND RELATIONS Functions (page 95) Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Which relation is a function? What is a function? Why should I care and what does it matter? A function is a way of getting answers. It's sort of like a vending machine. If you put in the right amount, and push the right buttons, you’ll get what you’re looking for. Let’s say I want an orange soda. To get it, I'm going to have to put some money in the machine. In fact, I'm going to have to put the correct amount of money into the machine. In the language of functions, we call this our input. But that's not enough. I can put money into that machine all day and all night. I can put money into the machine until I’m blue in the face. I can put money into that machine until I’m broke, and I still won't get a cotton picking thing (which, in the language of functions, is called output) until I start pushing the right buttons. So, in addition to entering some kind of input, I’m going to have to tell the machine what to do with it. When it comes to functions, you might say we are told what to do with the input by some kind of rule, or you might say, by some kind of formula. So then, there are four key concepts involved when it comes to functions. You have an input.

You have directions that tell you what to do with the input And you have an output that is generated when the input is acted upon—when you do something to the input, which means that the input is related to the output by some kind of procedure. So then, functions are all about relationships—the relationships become inputs and outputs, as defined by particular rules of correspondence. We’ll get the money from my wallet, but we won’t call it money…we’ll call it x. And we won’t call my wallet a wallet…we call it Set A. As for the pushing of buttons…we’ll call that our formula. To finish off the analogy, we’ll call all the sodas in the machine Set B, and the particular flavor of soda I want we will call y. A function is a correspondence from set A to set B in which each element x in set A is assigned exactly one element y from set B. I say exactly one because that is a condition of being a function. In any element x in Set A is assigned more than one element y from set B, the correspondence is, by definition, no longer a function. So, to define a function might say something like

A function is a correspondence from a set of x’s to a set of y’s that assigns exactly one member from y to each member x.

A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. Functions are good for finding the results in an, “If I do this, what will happen?” kind of situations. They are especially useful in a “How much of a change in that over there is caused when I change this over hear?” scenarios. Like how will the volume a box made from this piece of cardboard change as I change the (The distance Charles covers every 5 minutes during a 2-hour stretch of time

So, for any function you have to start with a formula. Like the rate at which a ball drops, etc is given by the formula. How do you write functions? What you want to say is… y is what you get when you do all this stuff to x. Of course, instead of saying “is,” we use and equal sign. That gives us y = what you get when you do all this stuff to x. But remember, we don’t call y, y anymore. We call it, the function of x, so now we have… f(x) = what you get when you do all this stuff to x. Now all that is left is to actually do the stuff to x. What so you want to do to it. Square it. Subtract 1 from it? The possibilities are endless, but you get the idea. There are four ways you can represent a function: With an equation, a table, a graph or with words.

Relations (page 103) A function is a special case of a more general concept, known as a relation.

Chapter 17 DETERMINING DOMAINS AND RANGES A Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.

For which equation graphed below are all of the y-values negative? What is the domain of the functions shown below?

Chapter 18 FUNCTIONS, RELATIONS, ORDER PAIRS AND SYMBOLIC EXPRESSIONS Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. Which of the following graphs represent a relation that is not a function of x?

Unit 1 CHAPTER 19 Solving Quadratic Equations Algebra I students are expected to know the quadratic formula and to be familiar with its proof by completing the square. A quadratic equation is any second degree polynomial equation—that’s when the highest power of x; or whatever other variable is used, is 2.

EXAMPLE: ax2 + bx + c = 0 (where a ≥ 0) Completing the square involves creating a perfect square trinomial that you can then solve by taking its square root… Solve this equation by completing the square. 3x2 = 24x + 27 STEP 1

The first step is to put the x2 and the x terms on one side and the constant on the other. 3x2 – 24x = 27 STEP 2

Then divide both sides of the coefficient of x2 (unless, of course, it’s 1). x2 – 8x = 9 STEP 3

Next, take half of the coalition of x (– 4), square it, then add that both sides. x2 – 8x + 16 = 25 STEP 4

Factor the left side. Notice that the factor always contains the same number you have found in Step 3. (x – 4)2 = 25 STEP 5

Take the square root of both sides, remembering at a ± sign on the right side. ( x − 4) 2 = 25 x − 4 = ±5 STEP 6

Solve

x=4 ± 5 x = 9 or –1

x2 – 8x + 16 = 25 Factoring …

STEP 1 The first step is to bring all terms to one side of the equation, needing a zero on the other side. Solve 2x2 – 5x = 12 2x2 – 5x – 12 = 0

What are the four steps to derive the quadratic formula?

Chapter 19 QUADRATIC FORMULA (PAGE 324) In addition to solving quadratic equations by factoring or completing the square, the is solving quadratic equations by means of the quadratic formula. If a, b, and c are numbers with a ≠ 0, then we see that the solutions of the equation ax2 + bx + c = 0 are given by the formula…

Proof by Completing the Square (page 322) The method of completing the square can be used to solve any quadratic equation of the form ax2 + bx + c = 0,

a≠0

The steps in the solution are always the same to matter what numbers are represented by a, b, and c. It seems very inefficient to have to repeat the steps each time we need a particular quadratic equation. Instead, let us apply the same steps to the general equation ax2 + bx + c = 0. We should be in obtaining a formula for x in terms of a, b, and c. We can use this formula to solve any particular equation by substituting the appropriate values for a, b, and c in formula. Let us begin with the equation ax2 + bx + c = 0 The next step is to write the equation as ax2 + bx = – c We may safely assume that a ≠ 0 (otherwise our equation is linear, not quadratic). Therefore, we may divide each side of the equation by a, so that the coefficient of x2 will be 1. The resulting equation is…

c b x2 +  x = − a a

Now we must complete the square. To do so, we square half of

b and add the result. a 2

 b  In other words, we add   each side of the equation…  2a 

c b  b   b  x +  x +   =   – a  2a   2a  a 2

When

CHAPTER 20 THE QUADRATIC FORMULA

..20.0 Students use the quadratic formula to find the roots of a seconddegree polynomial and to solve quadratic equations. Which is one of the solutions to the equation 2x2 – x – 4 = 0? Which statement best explains why there is no real solution to the quadratic equation 2x2 + x + 7 = 0 What is the solution set of the quadratic equation 8x2 + 2x + 1 = 0?

Chapter 20 USING THE QUADRATIC FORMULA At

CHAPTER 21 G

..21.0 Students graph quadratic functions and know that their roots are the x- intercepts. The graph of the equation y = x2 – 3x – 4 is shown below. For what value or values of x is y = 0 Which best represent the graph of y = – x2 + 3? Which quadratic function, when graphed, has x-intercepts of 4 and – 3?

Chapter 21 GRAPHING QUADRATIC FORMULAS ..21.0 Students graph quadratic functions and know that their roots are the x- intercepts. The graph of the equation y = x2 – 3x – 4 is shown below. For what value or values of x is y = 0 Which best represent the graph of y = – x2 + 3? Which quadratic function, when graphed, has x-intercepts of 4 and – 3?

CHAPTER 22 G

..22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. How many times does the graph of y = 2x2 â&#x20AC;&#x201C; 2x + 3

Chapter 22 FACTORING WITH QUADRATIC FORMULAS ..22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. How many times does the graph of y = 2x2 â&#x20AC;&#x201C; 2x + 3

Unit 1 Chapter 23 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

An object that is projected straight downward with initial velocity v feet per second traveled a distance s = vt + 16t2, where t = time in seconds. If Raymond is standing on a balcony 84 feet above the ground and throws a penny straight down with an initial velocity of 10 fps, and how many seconds will it reach the ground? The height of a triangle is 4 inches greater than twice its base. That area of the triangle is 168 square inches. What is the base of the triangle?

Chapter 23 APPLYING QUADRATIC EQUATIONS At

Chapter 24 LOGICAL ARGUMENTS ..24.0 Students use and know simple aspects of a logical argument: 24.1

24.2

24.3

CHAPTER 21.1 G

24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

CHAPTER 24.2 MUL

24.2 Students identify the hypothesis and conclusion in logical deduction What part of this statement is the conclusion? If x2 = 4, then x = â&#x20AC;&#x201C;2 or x = 2

Which of the following is a valid conclusion to this statement â&#x20AC;&#x153;If a student is a high school band member, then the student is a good musicianâ&#x20AC;?? A

All good musicians are high school band members.

A student is a high school band member.

All students are good musicians.

All high school band members are good musicians.

CHAPTER 24.3 . 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

Chapter 25 USING PROPERTIES OF THE NUMBER SYSTEM CHAPTER 25.1 G

..25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: 25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. 25.1

25.2 CHAPTER 25.2 G

25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

25.3 CHAPTER 25.3 G

25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

√

APPENDIX VERBAL PHRASES To take word statements and translate them into equations that you can solve using algebra, you will also need to be able to represent typical verbal phrases and questions with their algebraic equivalents. A list is provided below. Make sure you are familiar with everything on it because for you next assignment you will be given the same list except that YOU will need to provide the algebraic equivalents.

1. A number increased by 6 x+6 2. A number decreased by 3 x–3 3. A number is twice as much 2x 4. A number is half as much ½x 5. A number is four more than twice as much 2x + 4 6. A number is two less than four times as many 4x – 2

TYPICAL SITUATIONS Make sure you can also represent these common situations…

7. You are told that two numbers add up to a given a sum. number = x

Second number = The sum – x

8. Three consecutive integers. x x+1

x+2

First

9. Three consecutive even (or odd) integers. x x+2

x+4

EXAMPLE The sum of two numbers is 65. First number

Second number

65 â&#x20AC;&#x201C; x

Translating Word Problems to Number Problems Step One When translating a word problem into a number problem, your first step will be to determine the unknown quantity and represent it by a variable. The unknown quantity will be some piece of information that requires a value, but has absolutely no number associated with it whatsoever. For example, say you were told twice as many people visited a museum on Tuesday than did on Monday. People visited the museum on both days, so both days require some value to be associated with them. The numeral 2 is associated with Tuesday, due to the phrase “twice as many,” but so far, absolutely no number is associated with Monday. Now, let’s say you are also told that five more people visited on Wednesday then did on Tuesday. We now have the numeral 5 associated with Wednesday, but there is still absolutely no number associated with Monday. We can therefore conclude that the number of people who visited the museum on Monday is the piece of information that must be represented by our variable because we have absolutely no other values to attach to it.

x = the number of people that visited the museum on Monday.

Step Two Now, our next step is to represent (or express) all of the other unknown quantities in terms of that same variable. Since twice as many people visited on Tuesday than did on Monday, we have...

2x = the number of people that visited the museum on Tuesday And since five more people visited on Wednesday then did on Tuesday, we have

2x + 5 = the number of people that visited the museum on Wednesday

In summaryX Monday = x Tuesday = 2x Wednesday = 2x + 5 DIRECTIONS: In each of the situations below, circle the piece of information that has absolutely no number of associated with it whatsoever. However, do NOT solve the problems. Merely identify the piece of information in each problem that needs to be represented by a variable. 1. The Pacific Youth Club charges in initial fee of $22.50 along with a monthly fee of $12.50. The Brighton Youth Club charges only a monthly fee of $20.00. At how many months will the two clubs cost the same? 2. Last month, Mayfield Paper Company sold 58 more than three times as many reams of paper as Superior Paper Products. 3. Madeleine makes three times as much money per month as her brother Jorge. 4. A total of 19 seventh and eighth graders arrived for a walkathon on the school track at 9:00 A.M. By the end of the event, there were three times as many seventh graders and five times as many eighth graders, for a total of 77 students, as there had been at 9:00 A.M. How many seventh graders were at the event at 9:00 A.M.? 5. Weaver Excavating charges $80.00 for a full load of 2 tons of gravel and $40.00 for delivery of each node. How many full loads of gravel can be delivered for $3,000. 6. Based on the present balance in their account, the sophomore class cannot exceed a budget of $2500 for a field trip. The expenses for the trip include $750 for transportation and $10.50 for each student who goes on the trip. The four students who are class officers will go on the trip but will not be charged for their tickets. What is the maximum number of students who could go on the trip? 7. Muslin fabric cost $3.80 a yard. Bert already has 5 1/2 of the night in three fourths yards that he needs. What will be the cost of the muslin that he needs to purchase?

8. The San Joaquin Valley Cab Company charges $5.00 for the first mile and an additional $0.25 per mile for the rest of any ride. What is the maximum number of whole miles that someone can travel if he or she has $11? 9. Alison bought groceries for her family with half of her paycheck from her part-time job. She saved three fourths of the remaining money and had $60 left. What is the amount of Allison's paycheck? 10. Selena and James collect baseball cards. Together, they had 160 cards. After Selena had tripled her collection and James had doubled his, they then had a total of 396 cards. How many cards did Celina have originally? 11. The length of Andrewâ&#x20AC;&#x2122;s rectangular laundry room is 2 feet less than twice the width. 12. Astrid won a prize of $6,720 in a baking contest. She used all the money I spending three times as much on home renovations as on furniture and four times as much on a saving certificate as on furniture. How much did she spend on home renovations? 13. Jason has four more dimes than nickels and two fewer than 5 times as many quarters as nickels. 14. Jake is five years younger than twice the age of Cal, and Lila is four years older than Jake. 15. A fund-raising committee has $10,000 to spend on a banquet. The band is charging $3000; the convention center is charging $2400 for the banquet room and $20.50 per person. What is the maximum number of people who can attend the evening? 16. Emily has a total of 58 nickels and quarters. The total value of the coins is $10.50. How many quarters does Emily have? 17. For a wedding, Shereda bought several dozen roses and several dozen carnations. The roses cost $15 per dozen, and the carnations cost $8 per dozen. Shereda bought a total of 17 dozen flowers and paid a total of $192. How many roses did she buy? 18. A restaurant manager bought 20 packages of bagels. Some packages contained 6 bagels each, and the rest contained 12 bagels each. There were 168 bagels in all. How many packages of 12 bagels did the manager buy?

If this publication was like most other Algebra I textbooks we would, at this point, begin preparing you to solve a bunch of abstract equations. However, this publication is NOT like most other Algebra I textbooks in that we want you to understand why you are doing what you are doing. Consequently, we will begin preparing you to solve real-life problems and show you how to apply abstract equations later on when they become truly relevant. The first step is to learn how to represent real-life situations using the language of mathematics. Again, unlike most textbooks, we will explain how this is done explicitly and systematically to make the process unbelievably easy—or at least as easy as is humanly possible.

When you are given a sumX Given any two addends, the value of either one will equal their sum minus the value of the other. Therefore, if you know what their sum is, you can always express the value of one addend as x and the other addend as their sum minus x.

HERE IS AN EXAMPLE: The sum of two numbers is 65. Eight times the smaller number is 12 less than six times the larger number. Find it to numbers. We can let either number be x. In this case, let’s make it the smaller number. Therefore... Smaller number Larger number

= =

x 65 – x

When we see the word “times” we know that we are probably multiplying, and whenever we see the word “less” we know that we are probably subtracting, so translating the above information into mathematical language yields... 8 x = 6 (65 – x)

See if you can represent these relationships algebraically:

1. One number is seven less than another number. 2. The sum of two numbers is a 48. 3. Three consecutive odd integers.

Arithmetic Properties & True and False Assertions The state of California expects you to be able to use properties of numbers to demonstrate whether assertions are true or false. Directions: are the following mathematical statements true or false?

3+1=1+3 The equation below is an example of what property? a+b=b+a The commutative property of addition

FREE WORLD U True and False Assertions Directions: are the following mathematical statements true or false?

3+1=1+3 The equation below is an example of what property? a+b=b+a The commutative property of addition

By the community of property of addition, the order in which two or more numbers are added does not affect the answer.

The equation below is an example of what property? (a + b) + c = a + (b + c) The associative property of addition (2 + 4) + 1 = 2 + (4 + 1) 9 + (3 + 1) = (9 + 3) + 1

By the associative property of addition, how many numbers are grouped with parentheses will not change the answer when the numbers are added. By the community property of multiplication, numbers can be multiplied in any order. 2Ă&#x2014;6=6Ă&#x2014;2 Property of multiplication, for any number a and b, ab = ba.

By the associative property of multiplication, for any numbers a, b, c, (ab)c = a(bc) (4 × 2) × 3 = 4 × (2 × 3) By the associative property of multiplication, how numbers are grouped with parentheses will not change the answer when the numbers are multiplied. By which property are the left and right sides of the equation equivalent?

By which property are the left and right sides of the equation equivalent? 5xy = 5yx The community property of multiplication By which property or the left and right sides of the equation equivalent? X + 11 = 11 + x

By which property or the left and right sides of the equation equivalent? The associative property of addition

By which property are the left and right sides of the equally to equivalent? The community property of addition By which property are the left and right sides of the equation equivalent? The associative property of addition

By the distributive property of multiplication over addition, adding to numbers and then multiplying a third is the same as multiplying each of the two numbers by the third number and then adding. By the distributive property of multiplication over addition, for any numbers a, b, and c, a(b + c) = (a)(b) + (a)(c) 3(1 + 4) = (3)(1) + (3)(4) 3(1 + 4) = (3·1) + (3·4)

By the distributive property of multiplication oversubtraction, for any numbers a, b, and c, a(b – c) = (a)(b) – (a)(c) a(b – c) = ab – ac By which property are the two sides of the equation equivalent? 2(3 – 1) = 4 The distributive property of multiplication over subtraction By the distributive of multiplication over addition, what number is missing in the equation?

By which property are the two equations equipment?

The commutative property of addition By which property are the two equations equipment?

The associative property of multiplication By which property are the two equations equivalent? The two equations are equivalent by which property?

The distributive property of multiplication over addition

Which one property this shows that the two equations are equivalent?

The associative property of addition

By which property are the two equations equivalent?

The associative property of multiplication

OPPOSITES AND RECIPROCALS

To find the reciprocal of a number, reverse its sign from positive to negative, or from negative to positive. To find the opposite of a variable, reverse its sign from positive to negative, or from negative to positive. To find the opposite of a number or variable, either change the sign to its opposite or multiply both sides by negative one. A number or variable and its opposite added together equals zero. The opposite of a number or veritable is called its additive inverse. To change in expression to its un-simplified opposite, multiply or divide the entire expression by negative one or change the sign of the entire expression.

A negative in front of parentheses is applied into each term just like negative one.

A negative in front of the parentheses changes the sign of every term inside the parentheses when it is distributed.

To simplify in expression in parentheses, change every term inside the parentheses by the number and/or sign outside.

What is the opposite of (x + 3)? â&#x20AC;&#x201C;x â&#x20AC;&#x201C; 3

What is the opposite of (x – 5)? –x + 5

Another way to change in expression to its opposite is to change the sign of every term in the expression to its opposite.

What is the opposite of 2x – 3? –2x + 3 What is the reciprocal of ⅓ ? The reciprocal of any whole number is one over the original number. The product of a number end its reciprocal is one.

What is the reciprocal of 1/x? What is the reciprocal of x2?

What is the reciprocal of a/b? What is the reciprocal of zero? There is not a reciprocal for zero.

ROOTS AND FRACTIONAL POWERS

Since there is a range of possible numbers that each boy could have, we need to variables to represent them. Here's how it works. The first step is to figure out how many numbers you need

Steps to solving word problems

Directions: Use the following guidelines to help you translate real-life situations into language. The first thing you do is ask yourself, "How many numbers am I representing?" I Identify what it is that needs a value but is missing one.

Number Problems One of the primary goals of this textbook is to help you learn how to use algebra to solve real-life problems, which will require you to take word statements and translate them into equations that you can solve using algebra. The key to accomplishing this is figuring out what piece of information has to be represented by a variable, and to do that, you have to find the piece of information that has absolutely no number attached to it (absolutely no value or quantity associated with it).

FOR EXAMPLEX Let’s say you have two pieces of rope, and the first piece of rope is twice as long as the second piece of rope. Well, you at least know something about the first piece of rope. (You know that it is twice as long as the other piece of rope, which means that the first piece of rope has the numeral 2 associated with it.) But, since you know absolutely nothing about the second piece of rope, it is the second piece of rope that will have to be represented by a variable.

The second piece of rope = x You always begin by figuring out what needs to be represented by a single variable—what it is that has absolutely no number attached to it. All of the other quantities will have to be expressed in terms of this single variable. In our example here, since the first piece of rope is twice as long as the second piece of rope, and since the second piece of rope = x, the first piece of rope is twice as long as x, or…

The first piece of rope = 2x However, before we start asking you to represent all of the values in a problem, let’s make sure you understand how to identify what it is that must be represented by a single

Name: _____________________________________________________ variable—all by itself.

Date: ______/______/______

ALGEBRA I CHAPTER 1

Linear Equations in One Variable ♦ Word Problem

DIRECTIONS: Here are some common problem situations. In each circumstance, id piece of information that must be represented by a single variable—all by itself. Do N solve any problems. Only identify the piece of information that needs to be represente more than a variable. In other words, identify what it is that equals x. And good luck.

SAMPLE: You have two pieces of rope, and the first piece of rope is twice as long a piece of rope. ANSWER: The second piece of rope = x

1. The height of a lamp is 4 feet longer than the base. __________________________________________________ 2. Gilbert sold 40 more chocolate bars this week than last week. __________________________________________________ 3. Charlie is six years older than Cynthia. __________________________________________________ 4. There are fewer students in grade 6 then in grade 5. __________________________________________________ 5. Thomas has half as many baseball cards as Spencer. __________________________________________________ 6. Shirley has three times as many earrings as Keisha. __________________________________________________ 7. There are 10 more pigs on Old Man Zuckerman's farm than on Eugene's farm.

VERBAL PHRASES To take word statements and translate them into equations that you can solve using algebra, you will also need to be able to represent typical verbal phrases and questions with their algebraic equivalents. Copyright © 2008 Jewels Educational Services for Up-and-coming Scholars A list is provided below. Make sure you are familiar with everything on it because for you next assignment you will be given the same list except that YOU will need to provide the algebraic equivalents.

10. A number increased by 6 x+6 11. A number decreased by 3 x–3 12. A number is twice as much 2x 13. A number is half as much ½x 14. A number is four more than twice as much 2x + 4 15. A number is two less than four times as many 4x – 2

TYPICAL SITUATIONS Make sure you can also represent these common situations…

16. You are told that two numbers add up to a given a sum. number = x

Second number = The sum – x

17. Three consecutive integers. x

First

x+1

x+2

18. Three consecutive even (or odd) integers. x x+2

x+4

EXAMPLE The sum of two numbers is 65. First number

Second number

65 â&#x20AC;&#x201C; x

x = the number of people that visited the museum on Monday.

2x = the number of people that visited the museum on Tuesday And since five more people visited on Wednesday then did on Tuesday, we have

2x + 5 = the number of people that visited the museum on Wednesday

In summaryX Monday = x Tuesday = 2x Wednesday = 2x + 5 DIRECTIONS: In each of the situations below, circle the piece of information that has absolutely no number of associated with it whatsoever. However, do NOT solve the problems. Merely identify the piece of information in each problem that needs to be represented by a variable. 19. The Pacific Youth Club charges in initial fee of $22.50 along with a monthly fee of $12.50. The Brighton Youth Club charges only a monthly fee of $20.00. At how many months will the two clubs cost the same? 20. Last month, Mayfield Paper Company sold 58 more than three times as many reams of paper as Superior Paper Products. 21. Madeleine makes three times as much money per month as her brother Jorge. 22. A total of 19 seventh and eighth graders arrived for a walkathon on the school track at 9:00 A.M. By the end of the event, there were three times as many seventh graders and five times as many eighth graders, for a total of 77 students, as there had been at 9:00 A.M. How many seventh graders were at the event at 9:00 A.M.? 23. Weaver Excavating charges $80.00 for a full load of 2 tons of gravel and $40.00 for delivery of each node. How many full loads of gravel can be delivered for $3,000. 24. Based on the present balance in their account, the sophomore class cannot exceed a budget of $2500 for a field trip. The expenses for the trip include $750 for transportation and $10.50 for each student who goes on the trip. The four students who are class officers will go on the trip but will not be charged for their tickets. What is the maximum number of students who could go on the trip? 25. Muslin fabric cost $3.80 a yard. Bert already has 5 1/2 of the night in three fourths yards that he needs. What will be the cost of the muslin that he needs to purchase? 26. The San Joaquin Valley Cab Company charges $5.00 for the first mile and an additional $0.25 per mile for the rest of any

ride. What is the maximum number of whole miles that someone can travel if he or she has $11? 27. Alison bought groceries for her family with half of her paycheck from her part-time job. She saved three fourths of the remaining money and had $60 left. What is the amount of Allison's paycheck? 28. Selena and James collect baseball cards. Together, they had 160 cards. After Selena had tripled her collection and James had doubled his, they then had a total of 396 cards. How many cards did Celina have originally? 29. The length of Andrewâ&#x20AC;&#x2122;s rectangular laundry room is 2 feet less than twice the width. 30. Astrid won a prize of $6,720 in a baking contest. She used all the money I spending three times as much on home renovations as on furniture and four times as much on a saving certificate as on furniture. How much did she spend on home renovations? 31. Jason has four more dimes than nickels and two fewer than 5 times as many quarters as nickels. 32. Jake is five years younger than twice the age of Cal, and Lila is four years older than Jake. 33. A fund-raising committee has $10,000 to spend on a banquet. The band is charging $3000; the convention center is charging $2400 for the banquet room and $20.50 per person. What is the maximum number of people who can attend the evening? 34. Emily has a total of 58 nickels and quarters. The total value of the coins is $10.50. How many quarters does Emily have? 35. For a wedding, Shereda bought several dozen roses and several dozen carnations. The roses cost $15 per dozen, and the carnations cost $8 per dozen. Shereda bought a total of 17 dozen flowers and paid a total of $192. How many roses did she buy? 36. A restaurant manager bought 20 packages of bagels. Some packages contained 6 bagels each, and the rest contained 12 bagels each. There were 168 bagels in all. How many packages of 12 bagels did the manager buy?

= =

x 65 – x

See if you can represent these relationships algebraically: 4. One number is seven less than another number.

5. The sum of two numbers is a 48. 6. Three consecutive odd integers.

Name: ___________________________________________________

Date: ______/______/______

Score: _______

Questions and Phrases DIRECTIONS: Translate these verbal phrases into their algebraic equivalents.

1. A number increased by six

_____________

2. A number decreased by three

_____________

3. One number is eight more then a second number

_____________ _____________

4. One number is three less than a second number

_____________ _____________

5. The sum of two numbers is 71

_____________ _____________

6. One number is one-half a second number

_____________ _____________

7. One number is three times a second number

_____________ _____________

8. A number is three less than twice a second number

_____________ _____________

9. One number is five more than three times a second number

_____________ _____________

DO NOT DO THE REMAINING PROBLEMS! 10. The number a is 10 less than the number b

____________________________

11. The number whose units (ones) digit is 3x and tens digit is x

____________________________

12. The number whose tens digit is twice its units digit

____________________________

13. Three consecutive integers

____________________________

14. Three consecutive odd integers

____________________________

15. Three consecutive even integers

____________________________

SOLUTIONS TO PAGE 78

1. x + 6 2. x â&#x20AC;&#x201C; 3 3. First number: x + 8

Second number: x

4. First number: x â&#x20AC;&#x201C; 3

Second number: x

Lesson 5 Word Problems TRANSLATING REAL-LIFE PROBLEMS INTO ALGEBRAIC EQUATIONS The state of California expects you to be able to solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable, and provide justification for each step. That is because this skill is used to solve word problems. Word problems are nothing more than statements that express relationships among numbers, and our ultimate goal is to translate those words into algebraic equations so that we can find a solution using certain standard procedures. The “standard procedures” to which we refer are as follows: 1. First, determine the unknown quantity and represent it as a variable. 2. All other unknown quantities must then be expressed in terms of the same variable. 3. Next, translate the statements from the problem which relate to the variable into an algebraic equation. 4. Solve the equation for the unknown variable and then find the other required quantities. 5. Check your answer in the word problem—not in the equation. You already had some practice with translating verbal phrases and questions into their algebraic equivalents. Now let’s add an additional piece of information so that we can translate those mathematical phrases into equations. For example, we can take “A number increased by six” and specify that “sixty is a number increased by six.”

That enables us to come up with the equation: 60 = x + 6

Name: ___________________________________________________

Date: ______/______/______

Score: _______

Forming Equations

Now, all we need to do is apply certain arithmetic properties to solve for x.how But to before we do, let’sphrases make sure you too are equivalents, able to DIRECTIONS: You already know translate verbal intothat their algebraic so you translate real-life situations into the appropriate algebraic equations. are now going to be presented with the exact same situations you translated before except that this time you will also be provided with a quantity, making it possible for you to translate the phrases into matching equations. (Remember, whenever you see the word “is,” you should translate it as an equal (=) sign, and whenever you see phrases such as “a number, another number” or “a second number” you should translate them as: x.)

1. Sixty is a number increased by six

____________________________

2. Forty-nine is a number decreased by three

____________________________

3. Thirteen is eight more then a second number

____________________________

4. Fifty-five is three less than a second number

____________________________

5. The sum of twenty-eight and another number is 71

____________________________

6. Seventeen is one-half a second number

____________________________

7. Seventy-two is three times a second number

____________________________

8. Twenty is three less than twice a second number

____________________________

9. One hundred four is five more than three times a number

____________________________

10. Nineteen is 10 less than a number

____________________________

Using the properties of the real number system THE EQUATION ADDITION PROPERTY We are allowed to use additive inverse because of a theorem (a statement generally accepted without proof) that some algebra books call the “equation addition property” and others call the “addition property of equality.” Essentially, the equation addition property says that if two sides of an equation are equal (if the two sides weren’t equal, they wouldn’t be equations), then if you add the same amount to both sides, the two sides will still be equal. You see, an equation is sort of like a balance scale because it shows that two quantities are equal. The scales remain balanced when the same weight is added to each side.

THE ADDITION PROPERTY OF EQUALITY SAYS; For every real number a, b, and c, if a = b, then a + c = b + c.

EXAMPLE 4 + 4 = 6 + 2, so (4 + 4) + 9 = (6 + 2) + 9.

We use the addition property to reach one of the interim goals when solving word problems, which is to express the problem algebraically, so we can manipulate the information provided until we end up with a single variable on one side of an equation, all by itself. We can then replace the variables in problems with actual values, all of which is called “solving for x.” Let’s see the addition property of equality in action! Using the Addition Property of Equality Solve x – 3 = – 8

PRACTICE DIRECTIONS: The real-life situation below has been translated into an algebraic equation. Solve the equation using the addition property of equality. Make sure you show each step of your solution .

1. After losing 49 pounds this year, Katrina now weighs 136 pounds. What did she weigh at the beginning of the year?

x – 49 = 136

Similar to the equation addition property of equality is “the equation subtraction property” or “subtraction property of equality.” Essentially, the equation subtraction property says that if two sides of an equation are equal, if you subtract the same amount to both sides, the two sides will still be equal. When you solve an equation involving addition, subtract the same number from each side of the equation. Using the Subtraction Property of Equality Geometry The triangle below is isosceles with sides congruent. Find the value of a.

and

PRACTICE DIRECTIONS: The real-life situation below has been translated into an algebraic equation. Solve the equation using the subtraction property of equality. Make sure you show each step of your solution 2. Victor and Mark used up all the copy paper in the supply closet. There were 20 reams of paper in the closet at the beginning of the week. Mark used up 8 reams of paper. How many reams of paper did Victor use?

x + 8 = 20 3. Charlotte and Veronica want to move into an apartment that rents for $1250 a month. If Veronica can afford to pay $815 each month, for how much of the rent will Charlotte need to be responsible?

815 + x = 1250 4. $628 of Alfredo's monthly mortgage goes towards paying the principal. If his monthly mortgage payment totals $1100, how much goes towards paying the interest?

x + 628 = 1100 5. To buy the book she wants Ruby needs $25.99. So far she has saved $16.23. How much more money will she need before she will be able to buy the book?

12.23 + x = 25.99

Lesson 2e Arithmetic Properties of Numbers USING THE PROPERTIES OF THE REAL NUMBER SYSTEM We said that algebra was invented as a tool for solving real-life problems in real-world situations and pointed out the importance of variables in accomplishing this task. We also noted that, even so, our ultimate goal is to replace the variables in problems with actual values. This requires intimate familiarity with even more arithmetic properties of the real number system than just the addition and subtraction properties of equality. One of the most useful “tricks” for accomplishing this task is the use of additive inverse. Additive inverse is sort of like subtracting. It says that… For each real number a, there is a unique real number – a such that a + – a = 0.

Let’s see how we can apply the concept of additive inverse to solve equations. Although any letter of the alphabet can serve as a variable, we will most often use the letter x. And rather than use the phrase “replace the variable in the problem with an actual value” we will instead say something like “solve for x.” To solve for x we have to come up with an equation that expresses a given word problem algebraically, and then get the variable on one side of the equation all by itself.

USING ADDITIVE INVERSE The phrase “Sixty is a number increased by six” translates into… 60 = x + 6 On the previous page, we would have used the subtraction property to solve this problem. However, generally speaking, rather than use the subtraction property of equality, we add the additive inverse in combination with the addition property of equality. Perhaps you know that there are a series of theorems which state that (in simple terms) if two amounts are identical, as long as you add to, subtract from, multiply times or divide both quantities using the exact same value, their amounts, though changed, will nonetheless continue to be identical.

So, by adding the additive inverse of 6 to both sides, we are able to get the variable on one side all by itself, while still maintaining the integrity of our equation. 60 + (–6) = x + 6 + (–6) 54 = x So, you can use the addition property of equality, the subtraction property of equality, and additive inverse to write and solve equations describing real-world situations, using estimation to check whether your solution is reasonable. Here is another example using the subtraction property of equality. Real-World Connection Weighing a Baby A mother holds her baby and steps on a scale. The scale reading is 147 lb. Alone, the mother weighs 129 lb. How much does the baby weigh? Our first step is to decide what piece of information in the problem has absolutely no quantity associated with it whatsoever. The problem indicates that the mother AND the baby are 147 lb. It also tells us that the mother alone is 129 lb. So then, he only piece of information that has absolute no value associated with it is the baby all by itself, so the weight of the baby is the piece of information that will need to be represented by our variable. We have a 129 lb mother (129) holding her baby (+ w) weighing a total of 147 lb (= 147). Again…

The baby weights 18 lb. But remember, instead of subtracting, we add the additive inverse. So, we have…

w + 129 = 147 w + 129 + –129 = 147 + –129

w = 147

CHECK Is the solution reasonable? The baby weighs about 20 lb, and the mother weighs about 130 lb. The baby’s weight plus the mother’s weight is about 150 lb, which is close to 147 lb. The answer is reasonable. REVIEW:

DIRECTIONS: Solve the following review problems. 6.

7 2 − = 8 3 2 1 5 −2 = 9 4

2 1 − = 3 4

2 3 − = 3 5

Inverse Property of Addition For every real number n, there is an additive inverse -n such that n + (-n) = 0.

Multiplication Property of Zero For every real number n, n · 0 = 0.

Here is another situation…

n · 60 = 60 Joe was traveling at 60 miles per hours. At the end of the trip he traveled a distance of 60 miles. How long was the trip? Comes in hand wihen you want to multipliy a quantity time a number without changing the number’s value.

Identity Property of Multiplication For every real number n, 1 · n = n.

hang the information that will immediately follow, and make sense of all the words you went through the trouble of memorizing. In

14. A 120-foot-long rope is cut into 3 pieces. The first piece of rope is

twice as long as the second piece of rope. The third piece of rope is three

times as long as the second piece of rope. What is the length of the longest piece of rope?

The Symmetric Property of Equality So, we said that in this first lesson we would begin looking at the way real numbers behave, starting with the properties of equality, or more specifically, the symmetric property of equality. Now that we know about variables, let’s look at the symmetric property of equality. The symmetric property of equality says that: For every two real numbers a and b, if a = b, then b = a But, that’s just a bunch of mathematical mumbo-jumbo. So, let’s apply the concept to something less abstract, like the concepts of brother and sister.

EXAMPLE: Does the symmetric property of equality exist between two brothers? Well, let’s see... If Carlton is the brother of Samuel, is Samuel also the brother of Carlton? The answer is yes, and so, the symmetric property of equality does exist between two brothers. What about between a brother and a sister? If Victor is the brother of Charlotte, is Charlotte also the brother of Victor? Hah! I think not! Well then, the symmetric property of equality does not exist between a brother and a sister. Again, the symmetric property of equality says that: For every two real numbers a and b, if a = b, then b = a Now, the second rule we will look at will be the transitivity property of equality. The transitivity property of equality says that… For every three real numbers a, b, and c, if a = b and b = c, then a = c.

So, in plain English, the transitivity property of equality says that if two things are both equal to a third thing, they are also equal to each other. Now, let’s apply that concept to a concrete example…

EXAMPLE: Let’s say we have a friend named Darnell who is nine years old. We also have a friend named Keisha, whose age we don't know, except we are told she is the same age as someone else named Emanuel. However, as it turns out, Emanuel and Darnell are both the same age. Well, since Darnell is the same age as Emanuel… And Keisha is also the same age as Emanuel… We can conclude that Darnell and Keisha are the same age too, so Keisha must also be nine years old. So, the transitivity property of equality says that if two things are both equal to something else, they are also equal to each other. Transit means to move, similar to the word transportation. So the transitivity property of equality says that, if a relationship of equality exists between a and b, and a relationship of equality exists between b and c, we can move that relationship of equality so that it is between a and c as well. Now to review… We've learned that a real number is any number that does not have the lowercase letter-i in front of it. Also, we learned that a variable is… That enable us to begin looking at properties, and when we talk about the properties of real numbers, we are talking about the rules that govern the way real numbers behave, such as the symmetric property of equality. The symmetric property of equality says that: For every two real numbers a and b, if a = b, then b = a And finally, the second property we learned about was the transitivity property of equality, which, in plain English, says: if two numbers are both equal to another number, then they are also equal to each other.

Now let’s practice applying what we learned.

Tell me what property or properties of equality justify the following statements? 1. If x + 1 = 3, and 3 = y + 4, then x + 1 = y + 4 If you said the transitivity property of equality…you were right! 2. If 2 ½ = x, then x = 2 ½ If you said symmetric property of equality…you were right. 3. If x = y, and y = 4, then x = 4 If you said the transitivity property of equality…you were right! 4. If x – 7 = 11, then 11 = x – 7 If you said symmetric property of equality…you were right. 5. If x + 2 = 11, and y – 5 = 11, then x + 2 = y – 5 If you said symmetric property of equality…you were right.

So, what do we know? We know a real number variable properties symmetric property of equality transitivity property of equality Excellent job!

So, I’ll see you next time. Bye bye.

If x + 1 = 3, and 3 = y + 4, then x + 1 = y + 4 If 2½ = x, then x = 2½ If x = y, and y = 4, then x = 4 If x – 7 = 11, then 11 = x – 7 If x + 2 = 11, and y – 5 = 11, then x + 2 = y – 5

So, we said that in this first lesson we would begin looking at the way real numbers behave, starting with the properties of equality, or more specifically, the symmetric property of equality. Now that we know about variables, let’s look at the symmetric property of equality. The symmetric property of equality says that: For every two real numbers a and b, if a = b, then b = a But, that’s just a bunch of mathematical mumbo-jumbo. So, let’s apply the concept to something less abstract, like the concepts of brother and sister.

Now let’s practice applying what we learned.

Tell me what property or properties of equality justify the following statements? 6. If x + 1 = 3, and 3 = y + 4, then x + 1 = y + 4 If you said the transitivity property of equality…you were right! 7. If 2 ½ = x, then x = 2 ½ 8. Blah

Any number that does not have a lowercase-i in front of it, you can assume it is a real number. But, what about the term properties? Well, that’s just the fancy way of saying rules. The arithmetic properties are simply the rules that control how positive real numbers behave with one another. In other words, when we say “arithmetic properties” we are simply referring to the rules that govern how positive real numbers may be added, subtracted, multiplied or divided. You might say they are the rules that govern how positive real numbers work together, how they operate, which is why addition, subtraction, multiplication and division are called the four operations. However, the problem with basic arithmetic is that it can only look at what specific numbers do in specific situations.

That is one of the main reasons we need algebra. You see, algebra has the ability to look at what entire categories of numbers do in general. Similarly, it can take into consideration a whole range of numbers all at once. This is because, in addition to numbers, the four operations and the rules that control how the numbers and operations interact—algebra also makes use of variables. A variable is a symbol (usually a letter from the alphabet printed in italics) that can assume any value in a problem. Variables make it possible for algebra to consider the general properties of numbers without having to consider or worry about their specific attributes. (Set notation?) For example, the variable R is use to represent the entire category of real numbers. EXAMPLE The entire range of positive numbers can be represented by an expression using a variable, such as: x > 0. EXAMPLE Algebra uses variables in other ways as well. For example, when it comes to solving problems, there are often times when we don’t know all of the values involved and therefore need something else to stand for the values we don’t have. For example, suppose we have a friend named Roger, but we know nothing about his age, except that he is three years older than Darnell. If we know that Darnell is nine years old, then we can represent Roger’s age with the equation…

x=9+3 Secondly, there are those cases when the solution to a problem has more than one value. For example, if we wanted to refer to every pair of numbers that have a sum greater than 45, he could do so using the following inequality.

a + b > 45 And finally, there are other instances in which the value that satisfies a problem at any given time changes depending on a changing set of

circumstances. Such a situation might be represented by an equation that looks something like this:

y = x2 + 6 Whenever a value is unknown, nonspecific, or unfixed… a variable will always come in quite handy. FIVE WAYS IN WHICH VERIABLES ARE USED: 7. To represent entire categories of numbers. 8. To represent a wide range of numbers. 9. To consider the general properties without respect to their specific attributes. 10. To represent unknown values when solving problems. 11. To represent the different answers to a problem whose requirements are satisfied by more than one solution. 12. To represent the values in a problem whose quantities/solution changes as the situation changes. Whenever a value is unknown, nonspecific, or unfixed… a variable will always be much appreciated. Lets see how variables are used, along with some basic arithmetic properties, to solve some elementary math problems. Kenyan had a little get-together with four of his friends. At 5:00 P.M. he invited some more of his friends by telephone, and at the end of the night a total of five people were at his little get-together. How many more people came after 5:00 P.M.? This situation can be represented by… 5+n=5 It’s obvious that no one else came to Kenyan’s get-together after 5:00. In other words,

n = 0. But on a test you might be asked to justify your answer. You would need to know that this is an example of the…

(x + 7)(x – 4 ) = 0

Inverse Property of Addition For every real number n, there is an additive inverse -n such that n + (-n) = 0.

Multiplication Property of Zero For every real number n, n · 0 = 0.

Here is another situation…

Identity Property of Multiplication For every real number n, 1 · n = n.

We will start with the most general category and move down to the more specific. This means that, initially at least, you'll be memorizing a bunch of meaningless words. Nonetheless this task will be very useful in that the activity will provide you with a hook on which to hang the information that will immediately follow, and make sense of all the words you went through the trouble of memorizing. In

14. A 120-foot-long rope is cut into 3 pieces. The first piece of rope is

twice as long as the second piece of rope. The third piece of rope is three times as long as the second piece of rope. What is the length of the longest piece of rope?

Chapter 2 Solving Equations, Lesson 1 An equation is like a balance scale because it shows that two quantities are equal. The scales remain balanced when the same weight is added to each side.

Similarly, the scales remain balanced when the same weight is taken away from each side. This demonstrates the addition and subtraction properties of equality.

Addition Property of Equality For every real number a, b, and c, if a = b, then a + c = b + c. EXAMPLE 8 = 5 + 3 so… 8 + 4 = 5 + 3 + 4.

Subtraction Property of Equality For every real number a, b, and c, if a = b, then a – c = b – c. Example 8 = 5 + 3, so 8 – 2 = 5 + 3 – 2. To solve an equation containing a variable, you find the value (or values) of the variable that make the equation true. Such a value is a solution of the equation (a value or values that make an equation true). To find a solution, you can use properties of equality to form equivalent equations. Equivalent equations (equations that have the

same solution) are equations that have the same solution (or solutions). One way to solve an equation is to get the variable alone on one side of the equal sign. You can do this using inverse operations, which are operations that undo one another. Addition and subtraction are inverse operations.

Using the Addition Property of Equality

Solve

x–3 = –8

x– 3 +3 = –8 + 3

Add 3 to each side to get the variable

alone

x = –5

Check

x–3 = –8

on one side of the equal sign. Simplify.

Check your solution in the original

equation.

–5–3 = –8

Substitute – 5 for x.

–8 = –8

When you solve an equation involving addition, subtract the same number from each side of the equation. Using the Subtraction Property of Equality Geometry The triangle below is isosceles with sides

congruent. Find the value of a.

and

You can write and solve equations describing real-world situations. Use estimation to check whether your solution is reasonable.

Real-World Connection Weighing a Baby A mother holds her baby and steps on a scale. The scale reading is 147 lb. Alone, the mother weighs 129 lb. How much does the baby weigh?

The baby weights 18 lb. CHECK Is the solution reasonable? The baby weighs about 20 lb, and the mother weighs about 130 lb. The babyâ&#x20AC;&#x2122;s weight plus the motherâ&#x20AC;&#x2122;s

weight is about 150 lb, which is close to 147 lb. The answer is reasonable.

Solving One-Step Equations Multiplication and division are inverse operations. When you multiply or divide to solve equations, you use the following properties. Multiplication Property of Equality For every real number a, b, and c, if a = b, then a c = b c. Example = 3, so 2 = 3 2.

Division Property of Equality For every real number a, b, and c, with c 0, if a = b, then = . Example 3 + 1 = 4, so

= .

Multiplication and division are inverse operations. When you solve an equation involving division, multiply each side of the equation by the same number. Using the Multiplication Property of Equality

In the next example, the coefficient of the variable is a fraction. You can use reciprocals to solve the equation. Using Reciprocals to Solve Equations

When you solve an equation involving multiplication, divide each side of the equation by the same number. Using the Division Property of Equality

Simplify each expression.

1. x – 2 + 2 2. n + 2 – 2 3.

4. For each number, choose its opposite and reciprocal. 5. 2 6. – 2

7. a.

8. – a.