IndexChapter 1: The Real Number System

Section Pages 1.1 Fractions 2-211.2 Exponents, Order of Operations, and Inequaity 22-261.3 Variables, Expressions, and Equations 27-331.4 Real Numbers and the Number Line 34-391.5 Adding and Subtracting Real Numbers 40-461.6 Multiplying and Dividing Real Numbers 47-521.7 Properties of the Real Numbers 53-561.8 Simplifying Expression 57-59Practice Test 60-66

1

§1.1 Fractions

OutlineDefinitions Natural Numbers Numerator Denominator Proper Fractions Improper Fractions Mixed numbers Factors Factor Tree Multiple Prime Composite Least Common Multiple (Denominator) Prime Factorization Reciprocal Lowest Terms (Reducing) Product Pie Chart (Circle Graph) Multiplicative Identity ElementWriting fractions in lowest terms Prime Factorization Greatest Common Factor Finding all factors Divisibility of a number Method 1: Prime Factorization Method 2: GCF MethodMultiplying & Dividing Fractions Mixed NumbersImproper Fractions Improper Fractions Mixed Numbers Method for Multiplying Reciprocals & Method for DividingAdding & Subtracting Fractions and Mixed Numbers Adding/Subtracting with Like Denominators Method 1: Adding/Subtracting Mixed #'s using Improper Fractions Method 2: Adding/Subtracting Mixed #'s Vertically Fundamental Principle of Fractions Building Higher Terms Finding Least Common Multiples(Denominator) Adding/Subtracting with Unlike DenominatorsApplication Problems Word Problems – Handy Pointers Pie Charts Approximation

Homework p. 10-14 All except #64, 78, 79 & 80

Review of fractions is the most important arithmetic review that we need for algebra. We will use the core concepts of fractions many times to come, especially in our study of rational expressions. You should focus on mastering the concepts of prime factorization, reducing using prime factorization, finding a least common denominator and building a higher term in order to add fractions with unlike denominators.

First, we need to review some vocabulary for fractions. Recall that

2 Numerator 3 Denominator

Remember also that a fraction can represent a division problem!

Example: What is the numerator of 5/8 ?

2

Example: What is the denominator of 19/97 ?

Fractions represent a part of something. The numerator represents how many pieces of the whole are represented. The denominator tells us how many pieces that the whole has been divided into.

We like to represent fractions in what we refer to as lowest terms, which means that the numerator and denominator have no factors in common except one. There are two technical ways of putting a fraction into lowest terms. The first way uses greatest common factors and the other uses prime numbers. Prime factorization is not shown extensively in your book but I will be giving examples using both. First, we must digress and discuss some definitions and some methods of factoring.

There are 2 classifications of all natural numbers {1,2,3,4…} greater than 1. They are either prime or composite. (Note that one is not considered either prime or composite!)

A prime number is a number that has only one and itself as factors. A factor is a number used in a product. A product is the answer to a multiplication problem.

Example: 7 – 1719 – 11929 – 129 Only one times the number itself yields a

prime!

It is helpful to have some of the prime numbers memorized, I believe that it is most useful to know that 2,3,5,7,11,13,17,19,23, and 29 are prime and the most important of those are 2,3,5,7 and 11.

A composite number is a number that has more factors than one and itself. The definition of composite in the English language is “something that is made up of many things”. This holds true for math as well, it is a number made up of many factors.

To find all the factors of a number (not the prime factors), simply 1) start at one times the number itself and write them down with a good amount of space between, 2) then go on to the two and ask is the number at hand divisible by two (is two a factor?), 2a) if it is write two next to one and the other factor that yields your product next to the number itself (there should still be space between 2 and the other factor), 3) continue on with each successive natural number until you've "met in the middle."

3

Example: 14 – 1, ,1414 2 = 7, so 2 & 7 are factors of 14

1, 2, 7, 1414 isn't divisible by 314 isn't divisible by 414 isn't divisible by 514 isn't divisible by 6Now I've "met in the middle" since the next # is 7 which is in my list already, so I've found all the factors.

Example: Find the factors of 9 and 18

All the numbers in the examples above are composite numbers because they have factors other than one and themselves. Said another way, each composite number contains 1 and itself as factors as well as at least one other number.

In order to find all the prime factors of a composite number, we will use a method called prime factorization. The method goes like this: 1) What is the smallest prime number that our number is divisible by? 2) What times that prime gives our number? 3) Once we have these two factors we circle the prime number and focus on the one that isn’t prime. 4) If there is one that isn’t prime, we ask the same two questions again, until we have found all the prime numbers that our number is divisible by. 5) Then we rewrite our composite number as a product of all the circled primes. 6) Finally, we can use exponential notation to write them in a simplified manner. When multiplied together all the primes must yield the composite number or there is an error.

12/ \

2 6 / \

2 3 12 = 223 = 22 3

Whether you use a factor tree as I have here, or use one of the other methods is up to you, but I find that the very visual factor tree works nicely.

4

Example: Find the prime factorization of 15 and 24

Here are some important hints about whether a number is divisible by 2, 3, 5, and 10. You should take some time to go over these, they certainly help with division problems as well as factoring.

Hints for 2: Is the number even? Does it end in 0, 2, 4, 6, or 8?Example: 248

Hints for 3: Add the digits. Is the sum evenly divisible by 3?Example: 123

Hints for 5: Does the number end in 0 or 5?Example: 205

Hints for 10: Does the number end in 0?Example: 350

5

NOW, back to the task we began to discuss on page 2, putting fractions in lowest terms. First, I will discuss the method that your book does not – Prime Factorization.

Prime Factorization MethodStep 1: Factor numerator and denominator into prime factorsStep 2: Cancel all factors in common in both numerator and denominator.Step 3: Rewrite the fraction.

Example: Reduce 12/24 to its lowest terms.

Step 1: 12 = 2 2 3 . 24 2 2 2 3

Step 2: Cancel the 2 of the 2's and the 3'sStep 3: Rewrite 1/2 .

Note: Canceling is division and when you divide any number by itself the result is always one! This can cause problems if we do not realize that we are dividing out the common factors leaving a one behind when all the factors in the numerator cancel! The tendency then becomes to write a 0, but it is always 1!

Example: Reduce 27/81 to its lowest terms.

Now the method discussed in your book, the greatest common factor (GCF) method. First you must know that the GCF of two numbers is the largest number that is a factor of both. So if you find all the factors of the numerator and denominator (see above for finding all factors of a number) the GCF is the largest factor that is in both lists. We will now put the same fractions as above (Prime Factorization Method) in lowest terms using the GCF method.

6

G reatest Common Factor Method Step 1: Find the GCF of numerator and denominatorStep 2: Factor the numerator and denominator using GCFStep 3: Cancel the GCF from the denominator and numeratorStep 4: Rewrite the fraction

Example: Reduce 12/24 to its lowest terms.

Step 1: 12 – 1, 2, 3, 4, 6, 12 24 – 1, 2, 3, 4, 6, 8, 12, 24

GCF = 12Step 2: 12 = 12 1

24 12 2Step 3: Cancel the 12'sStep 4: Rewrite 1 .

2

Example: Reduce 27/81 to its lowest terms.

Of all the operations with fractions, multiplication is the easiest! It is extremely straight forward.

7

Multiplying Fractions or Mixed NumbersStep 1: Convert any mixed numbers to improper fractions.Step 2: Multiply the numerators (tops)Step 3: Multiply the denominators (bottoms)Step 4: Simplify by writing as a mixed number and/or reducing

First, recall that a mixed number is a whole number added to a fraction, denoted simply by a whole number with a fraction written next to it. These can be changed to an improper fraction – a fraction in which the numerator is larger than the denominator. (We should also note that a proper fraction also exists and it is a fraction in which the numerator is smaller than the denominator.) Let's quickly review how to change a mixed number to an improper fraction.

Mixed Number Improper Fraction Step 1: Multiply the whole number by the denominator of the fractionStep 2: Add the numerator of the fraction to the product in step1Step 3: Put sum over the original denominator

Example: Convert 1½ to an improper fraction.

Let's quickly review writing an improper fraction as a mixed number. We just finished our discussion of reducing (also called lowest term) and I believe that it is always easiest to convert to a mixed number when the improper fraction is in its lowest terms, but it is not necessary, therefore step one can become step three. Improper Fraction Mixed Number Step 1: Reduce the improper fraction to its lowest termsStep 2: Divide the denominator into the numeratorStep 3: Write the whole number and put the remainder over the

denominator.

Let's try one example before we resume our discussion of multiplication.

Example: Change to a mixed number 27/4

8

Now let's return to multiplication.

Example: Multiply 21/25 5/7

Note: You may also use principles of canceling and therefore eliminate the need for reducing. Because the original numerators are the factors of the resulting numerator and the same for the denominators the principles of converting to lowest terms by method of GCF or prime factorization apply!

Example: Multiply 3/4 1 1/4

Note: The denominators multiply even though they are the same! This does not seem difficult until after we cover addition, but try to keep it in mind!!!

Before we discuss dividing fractions we must define a reciprocal. A reciprocal can be defined as flipping the fraction over, which means making the denominator the numerator and the numerator the denominator. Another way that I frequently speak of taking a reciprocal is saying to invert it. The true definition of a reciprocal, however, is the number that when multiplied by the number at hand will yield the multiplicative identity element, one.

9

Dividing FractionsStep 1: Convert all mixed numbers to improper fractionsStep 2: Take the reciprocal the divisor (that is the second number that

you're dividing by)Step 3: Multiply the inverted divisor by the dividend (the number that you

are dividing into, the first number)Step 4: Simplify the answer if necessary by reducing and/or changing

to a mixed number.

Example: Divide: 5/8 2/3

Example: Divide: 5/8 3/4

Example: Divide: 5/8 3/5

Note: If you tend to do your canceling before multiplying then do not jump the gun! Canceling can only be done when there is a multiplication problem!!! In this problem, you can not cancel the 5's because it is division at this point, not multiplication!

10

Example: Divide: 2 1/8 3/5

Note: Since division is multiplication by the reciprocal, dividing mixed numbers is no different from multiplying them! We must always convert them to improper fractions before performing any operation on them.

In addition (subtraction) we have two cases to consider. The first case is the easiest, when the denominators are alike, and the second requires using the LCD to add fractions with unlike denominators. We will discuss the LCD shortly.

Adding (Subtracting) Fractions with Common DenominatorsStep 1: Add the numeratorsStep 2: Bring along the common denominatorStep 3: Simplify if necessary by reducing and/or changing to a mixed

number

Example: 4/5 + 3/5

11

Example: 3/8 + 1/8

Note: The truly unfortunate thing about addition of fraction is that you can only simplify after the problem is complete, so you always have to keep your eyes open for an answer that is not in simplest form!

Along with adding/subtracting fractions we need to discuss adding/subtracting mixed numbers. There are two approaches for doing this. The first is to change both to improper fractions and follow the steps for adding fractions. This method is fine for adding/subtracting fractions that have small whole numbers and/or small denominators, but it can get messy when those numbers get large. The second method handles large numbers well, but can get tricky when subtracting a larger fraction from a smaller one. (If you want to know which fraction is the larger of two find their cross products and the larger indicates the larger fraction. Cross products are the product of the denominator of one fraction and the numerator of the other, and are written next to the numerator factor's fraction.)

Adding/Subtracting Mixed Numbers using Improper FractionsStep 1: Convert to improper fractionsStep 2: Find a least common denominator if necessaryStep 3: Build the higher term if necessary (step 2 was taken)Step 4: Add/subtract numerators and place over LCDStep 5: Simplify by reducing and/or changing to a mixed number

Example: Subtract 5 2/3 3 1/3

12

Adding/Subtracting Mixed Numbers using Mixed NumbersStep 1: Put in a vertical fashion. First on & top second on bottom.Step 2: Find a least common denominator if necessaryStep 3: Build the higher term if necessary (step 2 was taken)Step 4: Add/subtract numerators of fraction portion and place over LCDNote: If the numerator of the top fraction is less than that of the bottom, you can't reverse the order of subtraction, you must borrow.

Step 4a: To borrow subtract one from the whole number (like when subtracting large numbers) and add that whole to your fraction by making a fraction of the LCD over the LCD.

Step 5: Add whole numbersStep 6: Simplify by reducing and/or changing the fractional portion to

a mixed number.Note: If you change the fractional portion to a mixed number you must further simplify the answer by adding the whole number and this mixed number.

Step 6a: Add the fractional part's mixed number and the whole number by adding the wholes and rewrite as a mixed number.

Example: Add 1 ¾ + 2 ¾

Example: Subtract 306 15/100 125 58/100

Note: I think that you could see how much more difficult this could become if we used the first approach with these large whole numbers and large denominators!

13

Now let's proceed to the discussion of necessary elements required to add/subtract two fractions with unlike denominators. First we need to know how to build a higher term and then we need to know how to find a least common multiple (known as a least common denominator when applied to fractions).

To build a higher term you must know the Fundamental Principle of Fractions. Essentially this principle says that as long as you do the same thing (multiply or divide by the same number) to both the numerator and denominator you will get an equivalent fraction. Here it is in symbols:

Fundamental Principle of Fractions a c = a or a c = a b c b b c b

This is used to build an equivalent fraction – a fraction that represents the same quantity.

Example: 1/4 and 2/8

are equivalent fractions.

Building the Higher Term (Creating Equivalent Fractions)Step 1: Decide or know what the new denominator is to be.Step 2: Use division to decide what the "c" will be (as in the fundamental principle of

fractions.)Step 3: Multiply both the numerator and denominator by the "c"Step 4: Rewrite the fraction.

Example: Write 1/8 as an equivalent fraction with a denominator of 24

Next we must discuss how to find a least common multiple (LCM). The least common multiple is the smallest number for which 2 or more numbers are factors. Another way of saying the same thing is to use the word multiple and say that the LCM is the smallest number which is a multiple of 2 or more numbers. A multiple is a number created when you multiply a certain number by the natural numbers. For instance, the multiples of 4 are 4, 8, 12, 16, … We will use LCM’s to find the least common denominator (LCD) so that we can add fractions with unlike denominators. A least common denominator is the smallest number which is a multiple of all the denominators.

14

Prime Factorization Method to Find LCM (LCD)Step 1: Factor the numbers (denominators for LCD) using prime factorizationStep 2: Take note of the unique factors in the factorizationsStep 3: Create a LCM (LCD) by using each unique prime factor the

number of times that it appears the most (largest exponent, variables included), for any one number (not the total number of times that it appears!)

Example: Find the LCD of 1/12 & 2/15

Note: A negative is not considered to be a part of an LCD, they are only positive.

Example: Find the LCD of 2/3, 3/5 and 5/7

Note: If they are all primes then the LCM is their product!

15

Example: Find the LCD of 2/5, 7/25, 13/50

Note: If the largest is a multiple of the smaller one(s) then the largest is the LCM

Now that we know how to build a higher term and find an LCD we can put these two steps together in order to find an LCD, build the higher term using the LCD and finally add or subtract the fractions at hand. Let's go over the steps for putting it all together.

Adding/Subtract Fractions with Unlike DenominatorsStep 1: Find the LCDStep 2: Build the higher term using LCDStep 3: Add/Subtract the new fractions with common denominators.Step 4: Simplify the fraction if possible by reducing and/or converting to a

mixed number.

Example: 2/5 1/3

16

Example: 1 1/4 + 3 2/3

Example: 9 2/15 2 13/25

We can also apply our working knowledge of fractions to the real world and to interpretation of a special kind of graph called a Pie Chart. First, we should discuss some common sense approaches to application problems. What I mean is that we should not read an application problem (Yes, I am talking about the dreaded word problem) and automatically say I can't. You don't know until you've tried with an open mind! The first key is translation from English to math. Just with anything in life you must ask yourself what is being expected of you; mastering a few simple phrasings and using common sense in relation to wording can accomplish this. Here is a simple process for working word problems:

17

Solving Word ProblemsStep 1: Read the problem carefully before doing anything. Understand

what the question is that needs to be answered.*Step 2: Write down in shorthand all information given and needed.

(*This is a step that I must see on paper or points are deducted.)

*Step 3: Decide how the information given, can be used to obtain the information needed (this is actually part of what you need to do in step 1 to understand the problem, but it comes in written form here). Write down an equation using shorthand notation and symbols that will be used to find the information needed.

*Step 4: Plug given information (usually numbers at this time) into the equation in step 3.

*Step 5: Solve the equation (at this time it will be a simple computation).*Step 6: Find the answer to the problem (most times this will be the same as step 5).

Give the answer using correct labels.Step 7: Check your answer, making sure that your answer makes sense.

Some helpful hints in deciding the operation being done to arrive at the answer (the operation in the formula) are as follows:

Addition – More than, altogether, somehow you are combining things to see a total, perimeter (the distance around an object)

Subtraction – Less or smaller than, taking away from, how many are left, a total prior to (before) something happening that increased the amount, decreasing, difference

Multiplication – Repeated addition, number of times some amount, number of a single serving or single type (key word being of)

Division – Parts of, Subdividing into sections, pieces or parts, Equal parts

Example: If I have a recipe that calls for 2 ½ cups of flour and I want to double the recipe (make 2 times the number of servings), how many cups of flour should I use?

18

Example: The three sides of a triangle measure as follows. Side 1is 2 1/8 feet, side 2 is 3 5/8 feet, and side 3 is 7 5/12 feet. Find the perimeter of the triangle.

Example: If I have a string that is 25 ¾ inches long and I need it to be divided into 4 equal pieces, how long will each piece be?

19

Example: At the end of the day I have more garbage than I started with. My can is marked off in fractional increments to one. It is at ¾ at the end of the day and it has increased by 3/8. At what fractional increment did it begin the day?

Example: At the end of the day my garbage can is at the ¾ mark and at the beginning of the day it was at 1/8, what was the diference between these two measurements?

20

Pie charts are used to represent parts of a whole. This is what fractions do for us so a pie chart is a nice visual representation of fractional parts of things. We use pie charts to show many statistics, and most commonly you will see them as percentages, but since a percentage is a fraction, it is the same. The whole can be one or one hundred thousand or any other whole number. This opens pie charts up for applications of fractions using addition, subtraction and multiplication.

Example: Look at the chart on page 13 of your book. It shows the fractional portion of the 8 million immigrants in the US between 1990 and 1997 broken down by the major "cultural" groups.

Asian – 3/10

European – 13/100

Latin – 13/25

Other – Remaininga) How much larger was the fractional portion of Latin

immigration than European?

b) Together what fraction did the Asian, European and Latin immigrants make up?

c) How many of the 8 million people were Asian?

21

§1.2 Exponents, Order of Operations, and Inequality

OutlineDefinitions Exponent Base Operators Order of Operations Braces Brackets Parentheses Grouping Symbols Translation Mathematical Statements Mathematical Expression Inequality SymbolsExponents Base Repeated MultiplicationOrder of Operations PEMDASInequality Symbols <,>, , , Translation Mathematical Statement Mathematical ExpressionBar Graph Practical Application of Inequality Symbols

Homework p. 20-22 #1-4,6,10,18-22,24-33mult.of3,38,40,41,42-54even,55-61odd,71,72,75,76,78,81

What is an exponent? An exponent represents repeated multiplication. The exponent, the little number that is written above and to the right of the base, tells us how many times to use the base as a factor. The base, the main number, can be any real number. Using a number as a factor means multiplying it by itself.

Base 5 2 Exponent

Example: What is the base of 23 ?

Example: What is the exponent of 23 ?

Example: What is 23 ?

22

There are also some special names for exponents 2 and 3. The exponent 2 is called the square and 3 is called the cube. There are also several ways to refer to an exponent – we can say "to the (exponent's) power" or "to the power of (exponent)"

Not only can the base be a whole number but it can also be a fraction, an expression, etc. Let's try an example using fractions.

Example: (3/5)2

I would highly recommend that you memorize the perfect squares for quickly squaring simple numbers and for future reference.12=1, 22=4, 32=9, 42=16, 52=25, 62=36, 72=49, 82=64, 92=81, 102=100, 112=121, 122=144, 132=169, 142=196, 152=225, 252=625.

The following perfect cubes may also be helpful: 13=1, 23=8, 33=27, 43=64, & 53=125

When many operators (addition, subtraction, multiplication & division) are used in a single mathematical expression it can become confusing as to which one to do first. We solve this by using order of operations and grouping symbols. Grouping symbols such as brackets [ ], braces {}, and parentheses ( ) help us to tell others what we wish them to do first, which leads directly into the order of operations – the order in which operations are done when in an expression together.

Order of OperationsParentheses, Brackets or BracesExponentsMultiplication and Division in order from left to rightAddition and Subtraction in order from left to right

A trick for reminding yourself which operations in what order is the following: Please Elect My Dear Aunt Sally or simply PEMDAS.

Example: 7 + 22 2

23

Example: 5 (1 + 9) 21

Example: 32[3 + 3(22)] 5

Example: 4 7 / 3 + 1 2/3 1/2 + (1/3)2

24

Next on our agenda is a small discussion of the inequality symbols, which will lead us directly into a discussion of translation problems.Inequality Symbols Does Not Equal< Strictly Less Than> Strictly Greater Than Less Than or Equal To Greater Than or Equal To

Some helpful hints in reading the greater than and less than symbols:The arrow points to the smaller number.Put teeth in the mouth and the smaller eats the larger (you may have learned this as a child, and therefore I pose this here to jog a memory!)

We have both strictly less than and less than or equal to, and both can be used to refer to the same case, but the less than or equal to can also indicate equality. The less than or equal to refers to a case in which either the strict inequality or the equality holds true.

Example: State whether each statement is true or false.a) 7 < 9

b) 9 > 11

c) 15 7 + 8

Example: Write the statement using symbols (translate).a) 12 is less than 21

b) 15 is greater than or equal to 13

Now we need to discuss the most important concept in this section. Without total control of translation word problems will become unmanageable. Translation means taking a sentence written in English and putting it into mathematical symbols. We will be translating mathematical sentences like those above that contain the words and phrases that refer to the inequality symbols. These should not be confused with translation of mathematical expressions that contain operators. Two phrasing for the operators addition and subtraction can appear similar to those for mathematical sentences [10 greater than 4 (4 +

25

10) versus 10 is greater than 4 (10 < 4)]. Mathematical expressions always contain operators, whereas mathematical sentences contain inequality symbols and not operators.

The authors now choose to discuss translation of bar graph in their relation to inequality symbols and translation of mathematical sentences. I will not present a formal discussion here. I will remind you that more is the same as greater, at most means less than or equal to, and not less than means greater than or equal to.

26

§1.3 Variables, Expressions and Equations

OutlineDefinitions Equation expression evaluation solution Variable elements set ellipsis notation

root sequence set-builder notation define the variable all phrases in chartsAlgebraic Expression Evaluation Translation Differentiate from EquationAlgebraic Equation Is a number or a member of a set a solution? Translation Differentiate from ExpressionTranslation Phrases for Operators Key for order in subtraction Phrases for Equality Phrases for Variables Phrasing for ParenthesesSets

Homework p. 27-28 #1-9all, 12-42mult.of3, 43-58all, 60, 63, 66, #68-76all

In this section we will be discussing algebraic expressions and algebraic equations. The difference between an algebraic expression and an algebraic equation is an equal sign. An algebraic expression does not contain an equal sign and an algebraic equation does! An algebraic expression is a collection of numbers, variables (a letter of the alphabet used to denote an unknown number), operators and grouping symbols.

Example: All the following are algebraic EXPRESSIONSa) 3x

b) 3x2 5

c) 6 ( y 4 ) + 2

d) 5z 16 (Recall that fraction bars are grouping symbols)

5

27

An algebraic expression can be evaluated. This means that given specific values for the variables, we can give a numeric answer to the expression.

Example: Evaluate 3x2 + 5y z given x = 2 y = 1 and z = 3

Example: Evaluate 2c + 15 given c = 3 and k = 7k

Example: Evaluate 2 ( z + 5 ) 4/z + k2 given z = 2 and k = 5

An algebraic equation sets up equivalence between an algebraic expression and a numeric expression (a number or numbers with operators) or between two algebraic expressions. Since an equation sets up an equivalence relationship it can be solved. This means that a solution or root can be found which will make the equation true. A solution or root is a value that can be put in place of the variable to make the equation true. We are not developing a method for solving equations yet, but we do need to practice

28

deciding if the solution is correct. This is done in the same manner as evaluating an expression, except we then must decide if the resulting statement is true, if it is then we have found a solution and if it is false then we don't have a solution.

Example: The following are examples of algebraic EQUATIONSa) 5 + x = 9

b) 9 z = 5 2z

Example: Is 2 a solution to the following equation? (Your book will pose this question in this manner: x2 + 2 = 8; 2)

x2 + 2 = 8

This section also helps you to practice your skills for writing algebraic expressions or equations by translating the following words and phrases into operators. Following is a helpful chart, after which we will practice translating some expression. Remember that translation is the key to all word problems.

AdditionWord Phrasing Symbols

sum The sum of 7 and 2 7 + 2more than 5 more than 10 10 + 5added to 6 added to 10 10 + 6greater than 7 greater than 9 9 + 7increased by 4 increased by 20 4 + 20years older than

15 years older than John. John is 20. 20 + 15

total of The total of 6 and 28 6 + 28plus 8 plus 281 8 + 281

SubtractionWord Phrasing Symbols

difference of The difference of 5 and 2The difference of 2 and 1

5 22 1

*years younger than

Sam's age if he is 3 years younger than John. John is 7.

7 3

diminished by

15 diminished by 921 diminished by 15

15 921 15

29

*less than 17 less than 497 less than 17

49 1717 7

decreased by 29 decreased by 1515 decreased by 7

29 1515 7

*subtract(ed) from

Subtract 13 from 51Subtract 51 from 103

51 13103 51

take away 79 take away 61 79 61subtract 54 subtract 2 54 2less 16 less 4 16 4* - Means that the numbers come in opposite order than they appear in the sentence.

MultiplicationWord Phrasing Symbols

product The product of 6 and 5 65times 24 times 7 24(7)twice Twice 24 2(24)multiplied by 8 multiplied by 15 8*15at 9 items at $5 a piece ($5)9"fractional part" of

A quarter of 8 (¼)(8) or 8/4 .

"Amount" of "$" or "¢"

Amount of money in 25 dimes (nickels, quarters, pennies, etc.)

($0.1)(25) or (10)(25) ¢

percent of 3 percent of 15 0.03(15)

DivisionWord Phrasing Symbols

divide Divide 81 by 9 81 9quotient The quotient of 6 and 3

The quotient of 24 and 66 324 6

divided by 100 divided by 2020 divided by 5

100 2020 5

ratio of The ratio of 16 to 8The ratio of 8 to 2

16 88 2

shared equally among

65 apples shared equally among 5 people

65 5

Note: Division can also be written in the following equivalent ways, i.e. x 6 = x/6 = 6x = x

6Exponents

Words Phrasing Algebraic Expressionsquared Some number squared x2

square of The square of some number x2

cubed Some number cubed x3

cube of The cube of some number x3

(raised) to Some number (raised) to the power of x6

30

the power of 6Equality

Words Phrasing Algebraic Equationyields A number and 7 yields 17. Let x = #. x + 7 = 17equals 7 and 9 equals 16 7 + 9 = 16is The sum of 5 and 4 is 9. 5 + 4 = 9will be 12 decreased by 4 will be 8. 12 4 = 8was The quotient of 12 and 6 was 2. 12 6 = 2Note: Any form of the word is can be used to mean equal.

ParenthesesParentheses are indicated in four ways. The first is the use of a comma, such as: The product of 5, and 16 less than a number. The second is the use of two operators' phrases next to one another. An example of this would be 17 decreased by the sum of 9 and 2. Notice how decreased by is followed by the sum of and not a number, this indicates that we will be doing the sum first; hence a set of parentheses will be needed.Next, you may notice that the expected 'and' between the two numbers being operated upon is after a prepositional phrase [A phrase that consists of a preposition (usually of in our case) and the noun it governs (usually number in our case) and acts like an adjective or adverb]. For example: The sum of 9 times a number and the number. Usually we would see the 'and' just after the number 9, but it does not appear until after the prepositional phrase 'of 9 times a number'. If you think of this in a logical manner, what you should see is that you have to have two numbers to operate upon before you can complete the operation, which would require the use of parentheses to tell you to find a number first! Finally, you may notice a phrase containing another operator after the 'and' where you would expect a number. An example here might be: The difference of 51 and the product of 9 and a number. The note about thinking in a logical manner applies here too! You must have two numbers to operate on!

VariablesVariables are an undisclosed number. If you are not told what variable to use by the author, you should always define the variable as part of the problem. We define the variable by saying "Let x = #" or whatever letter you choose to be your unknown, the most commonly used variable is x, but you will find that sometimes it makes more sense to use another letter.

Example: Write an algebraic expression for the following. Let x = #a) The difference of five times a number and two

b) The sum of five and a number increased by two

31

c) The product of a number and two, decreased by seven

d) The quotient of seven and two times a number

Example: Write an algebraic equation for the following:a) Two times a number yields ten

b) The ratio of a number and two is the same as eight

c) The total of eighteen and five times a number equals eleven times the number

d) The product of five and the sum of two and a number is equivalent to the quotient of forty and the number

A set of numbers is a group of numbers that meet some predefined criteria. A set is denoted by braces and can list all elements (in math usually numbers), describe the elements using an algebraic expression or words (called set-builder notation), or list a sequence (a pattern by which numbers are related using addition or multiplication) using ellipsis notation [the use of three periods (…) to denote continuation of a pattern]. Your authors will ask you to determine which numbers from a is a solution to an equation that your have translated.

32

Example: Which of the number from {2, 5, 9, 12} is a solution to the following algebraic equations?a) The product of 2 and the sum of x and 3 is 30.

b) The square of x less 9 yields 16.

33

§1.4 Real Numbers and the Number Line

OutlineDefinitions Whole Numbers Natural Numbers Integers Positive Numbers Negative Numbers Signed Numbers Rational Numbers Irrational Numbers Number Line Real Numbers Negative Opposite Additive Inverse Absolute Value Order Property of Real Numbers Coordinate Double Negative Identity Element of AdditonReal Numbers Whole Numbers {0, 1, 2, 3, …} Natural Numbers {1, 2, 3, …} Integers {…, -3, -2, -1,0, 1, 2, 3, …} Positive Numbers (see natural numbers) Negative Numbers {x | x integers < 0} Signed Numbers {x | x integers, x0} Usefulness in everyday life Rational Numbers {x/y | x integers, y integers, y0} Irrational Numbers {x | x can't be represented by a rational number}Number Line Visual representation of Real Numbers Graphing on Whole numbers Rational Numbers Opposites Definition Negative (shorthand notation) Additive Inverse Double Negative is opposite of a negative number Absolute Values Definition Notation Opposite of Comparing Opposites and Absolute Values Simplify 1st then compareDecimals Appendix B, p. 649

Homework p. 36-38 #9-18all, 21, 22, 29, 30, #35-41all, 44, 47, 50, 53, 56, 59, 62, #65-68all p. 652 #2-18even, #22-38even Real Numbers are all numbers that can be represented on a number line.

A number line is a continuing line that represents the real numbers. At the center is zero, moving to the left from zero are the negative numbers and moving to the right from zero are the positive numbers. The fact that moving left on the number line encounters increasingly smaller numbers, while moving right encounters increasingly larger numbers is known as the order property of real numbers.

34

Graphing on a Number LineIn order to graph a number on a number line all we must do is locate the number and then put a dot on the point and then label it with the coordinate. A coordinate is the number associated with the point being labeled.

Example: Label 0, -1 and 2 on the number line below.

We have now discussed 3 set of numbers thus far – the natural numbers {1, 2, 3, …}, the whole numbers {0, 1, 2, 3, …}, and the real numbers {x | x is any number on the number line}. The number line is going to help us with our next set of number, the integers. The integers are the natural numbers, their opposites (negatives) and 0 {…, -3, -2, -1, 0, 1, 2, 3, …}. Opposites are numbers that are the same distance from zero but on the opposite side of zero when consulting a number line. Said another way, place a negative sign in front of a number and you have created its opposite (although it may need to be simplified when considered in this manner, but we'll discuss that a little later).

The integers are made up of positive numbers, negative numbers and zero. The positive and negative numbers (the integers to the right of zero are called the positive numbers and those to the left are negative) make up a set of numbers called the signed numbers {x | x integers, x 0}. The signed numbers are useful to us in describing everyday circumstances. Temperatures above zero degrees are positive and those below zero are negative. If we have money in our checking account we have positive number balances and if we are overdrawn we have negative number balances. If we are above sea level we have heights measured in positive numbers, but if we go below sea level we have heights (depths) measured in negative numbers.

On the number line there is always one unit between each whole number that is marked. Between these whole numbers are the fractions. Fractions are part of the irrational numbers – numbers that can be represented as a quotient of two real numbers, where the bottom (divisor) can’t be zero {x/y | x , y , y 0}. The other part of the irrational numbers is the integers, because every integer can be written as a fraction by placing it over 1.

35

Graphing FractionsGraphing fractions is the same as graphing any other number; it just requires that we have marks between whole numbers. There will be one less mark than the number in the denominator since one whole is represented by the denominator over itself. For instance, if we wish to represent thirds, you will have two marks between 0 and 1, 1 and 2, etc. because the marks break the line into 3 parts between 0 and 1, 1 and 2, etc. Let’s graph some fractions.

Example: Graph the following fractions.

a) ½

b) ¾

Note: Both ¾ and ½ can be graphed on the same line if you realize that 2/4 is equivalent to ½.

Graphing a mixed number is the same as graphing a fraction. You must recall that it is the whole number plus the fraction, meaning that it is the fractional portion greater than the whole portion of the mixed number. Therefore, in order to graph a mixed number you would label your number line to at least 1 number larger than the whole portion and then divide the line between the whole portion and the next largest number into the fractional parts represented by the fraction.

Example: Draw your own number line and graph 12/5.

36

Note: If you were asked to graph an improper fraction it is easiest to divide the numerator by the denominator to change it to a mixed number and graphing as described above. The last set of numbers that we must discuss is the irrational numbers. The irrational numbers are numbers that can't be written as a quotient of integers. Any number that is a non-repeating, non-terminating decimal is an irrational number. Some examples of such numbers are as follows: , - 7 , 0.325079325774…, etc.

The real numbers can now be described as the set of rational and irrational numbers {x | xrational numbers or x irrational numbers}. The only note left is that although our discussions in this class will be restricted to the real numbers, there are more sets outside the reals. One such set we have a chance to encounter in Chapter 9. They are the imaginary numbers and they represent square roots of negative numbers such as -3.

You will have exercises to do that ask you to choose numbers from a set that meet the criteria of the various sets. Realize that the sets overlap and a number may be a member of more than one set of numbers! I never take valuable class time to go over examples like this, but feel free to ask questions after attempting homework, if you feel it necessary.-----------------------

When talking about the number line, there are two properties that come about, the absolute value and the opposite. These two properties are very similar and can become confused, but if you learn their definitions you should never have a problem!!

The absolute value of a number is defined as the number of units a coordinate is located away from zero on the number line. Absolute value does not take into account whether a number is positive or negative – it is strictly the number without its sign! We indicate absolute value in the following way

| number | The absolute value of number

Example: Simplify

a) | 7 |

b) | -3 |

37

The opposite is defined as the coordinate located the same distance on the opposite side of zero. If a number is positive, then its opposite is negative, and likewise if a number is negative then its opposite is positive. A shorthand way of writing the opposite of a is –a and hence your authors' reference to the opposite as the negative. There is one more name for the opposite and that is the additive inverse. The additive inverse is defined as a number that when added to the number at hand will yield the identity element of addition (zero). The number that does this is a number's opposite; hence the additive inverse and the opposite (negative) are the same!

a + -a = 0 a's Additive Inverse/Opposite is -a

To possibly head off some complications, let's define something called a double negative. A double negative it the opposite of a negative number. And when considered in this respect while taking into account the definition of an opposite, it should be quite clear that the opposite of a negative number is a positive number, but when it comes at you in shorthand it can be confusing (some of you may also have some additional amo to use in this regards in that you already know that a negative times a negative yields a positive).

- ( -a) = a For all a integers

Example: Simplify

a) - (-5)

b) - (2)

c) - (- (-7))

d) - | - 3 |

Note: This is the most confusing of all opposite problems because it is read the opposite of the absolute value of negative 3. Since the absolute value is defined as the number without its sign, we can’t determine the opposite of the absolute value until that portion is sorted through. Thus, unlike its counterpart –(-3), the answer is –3 for this problem because the absolute value of –3 is 3 and the opposite of 3 is –3. You

38

must always remember that absolute value signs are like parentheses, but they have a special meaning that must be taken into account before anything to their left.In order to compare absolute values and/or opposites you must first simplify their meaning. I must see any simplification for full credit on problems! Then you must use the order property of the real numbers (recall that this says that any number to the left of a number is less than the number and any number to the right of a number is greater than the number).

Example: Use < , > or = to compare each of the following (show any

simplification of the numbers that you are comparing).

a) -4 1

b) -(-6) 13/2

c) - | 5 | - | -5|

d) ¼ ¾

e) - ¼ - ¾

f) -32 (-3)2

Note: This is a very important distinction! These are not equal because in the first case the base is 3, not –3, because –3 = -1 3, and the exponent only applies to the number to which it is written directly to the right of. It is the same as saying 3x2, in which case it is only the x that is being squared not the 3! On the other hand in (-3)2 the parentheses indicate that the base is indeed –3, which means that –3 is to be used as a factor twice.

As a final note in this section – you should review working with decimals in Appendix B on page 649 of your book. You will be responsible for operations with decimals without the use of a calculator and therefore you may need to review the basics. If you experience difficulty, please come to me for help during office hours or ask questions in class. Some homework has been assigned from this appendix and it is expected that you have done it without the aid of a calculator.

39

§1.5 Adding and Subtracting Real Numbers

OutlineDefinitions Addend Subtrahend MinuendAdding Real Numbers Using the Number Line Using Rules Same signs Opposite signs Order of Operations Still PEMDASSubtracting Real Numbers Redefine subtraction as addition of the opposite Use rules of addition Order of Operations Still PEMDASTranslation Problems w/ Integers Review §1.3 words and phrasesApplications w/ Integers Decreases can be negative integers Any number below zero is negative

Homework p. 46-50 #1-6all, #9-33mult.of3, #32, #42-96mult.of3, #98-108even

One way that you can add real numbers is using a number line. This is a very helpful visual tool for learning how positive and negative numbers are added. Recall the order property of the number line – recall that if we have a negative number we are traveling in the negative direction on the number line (left) and we have a positive number we are traveling in a positive direction on the number line.

Adding on the Number LineStep 1: Start at the first addendStep 2: Move the appropriate number of units in the positive or

negative direction as indicated by the 2nd addend

Example: Add each of the following using a number line

a) 7 + (-3)

40

0-5 5

b) -3 + (-4)

c) -10 + 4

The other alternative has some rules to remember:

Adding IntegersIf the signs are the same

Step 1: Add the absolute valuesStep 2: Keep the common sign

If the signs are differentStep 1: Subtract the absolute value of the smaller from the largerStep 2: Keep the sign of the larger

Example: Use the rules for adding integers to add the following

a) 15 + 88

41

b) 9 + (-5)

Note: If you need to know which fraction is the larger of the two you may put them side by side find the cross products and the one with the larger cross product is the larger fraction. Here 1/5 is larger as 91= 9 is larger than 15=5. Since it is the absolute values that we are dealing with, just disregard the sign until the end.

c) -15 + (-42 )

d) -80 + 15

Now, let’s do some that are a little more complex. Let’s throw in real numbers and add more than 2 integers. Remember when adding more than 2 integers that you can use the addition properties of numbers, they also apply to integers. Recall that 2 important

42

properties say that we can regroup numbers and move them around. I will use these properties to add all numbers with the same signs and then add numbers with opposite signs last when necessary.

Example: Add the followinga) 1/9 + (-1/5)

Note: If you need to know which fraction is the larger of the two you may put them side by side find the cross products and the one with the larger cross product is the larger fraction. Here 1/5 is larger as 91= 9 is larger than 15=5. Since it is the absolute values that we are dealing with, just disregard the sign until the end.

b) -15 ½ + (-42 ¼ )

c) -80.5 + 15

d) -9 + 10 + (-7)

43

e) [-7 + (-8)] + [7 + (-3)]

Note: Given this problem without the brackets I would choose to add all the negative numbers first, because that is a simple addition problem that yields a negative number and then I would add the positive number to that result. Try it and see if you get the same answer as we just got!

The first key to subtracting two real numbers is seeing subtraction problems as addition problems. This is crucial in subtracting integers and does not conflict with anything that you have already learned. Please do not fight this process as it is crucial in understanding. If you are beyond this, please do not voice your objections, just bear with me. In order to subtract two integers we need to rewrite all subtraction problems as adding the opposite of the subtrahend (that’s the second number)!!

Example: Rewrite each subtraction problem as an addition problem and solve.

a) 3 9

b) 7 (-4)

c) -9 6

d) -12 (-6)

44

Once we see a subtraction problem as an addition problem there are no more rules to learn! We already know how to add integers in two different ways – using the number line and using the rules. Now all we must do is practice our new skill.

Let's practice some more with some problems that include real numbers, and absolute values.

Example: Simplify each of the following.a) | -5 7| 8

b) | 19 7/8 | | -11 + - 2/3 |

c) | -19.07 (-2.125) | + (-205 + -3.1)

45

Your authors also include some translation problems in this section. They are the same as those included in section 3, and although I think that they are very important I will skip more examples in the interest of time. Some other problems that I would rather skip in interest of time are the application problems involving signed numbers. I would like to remind you that negative numbers represent decreases or going below zero, but I am going to leave the actual problems up to you. I will give you an example of a word problem and remind you that word problems are all about translation (watch order).

Example: In Death Valley the average winter temperature is 82 and several hundred miles away at the top of Mt. Whitney, the average winter temperature is -23. What is the difference in average temperatures between Death Valley and Mt. Whitney in the winter?

46

§1.6 Multiplying and Dividing Real Numbers

OutlineDefinitons Quotient Dividend Divisor Multiplicative Inverse Reciprocal Division Average Sigma NotationMultiplication & Divison Redefine Division – Multiplication by the reciprocal (see fractions) Basic Truths Multiplication by Zero – Zero Division by Zero – Undifined Zero divided by Anything – ZeroMultiplication & Division w/ Reals + + = + - - = + + - = Combining All Concepts Thus Far Order of Op ProblemsAveragesDivisibility Tests Only Covered in Book on p. 62

Homework p.60-62 #1-6all, #43, #46-70even, #83, #88-102even, #110, #116-119all, #121&122

Let's first talk about some of the ways that multiplication and division can be written.

All of the following mean multiplication:

3 x 2 3 2 3(2) (3)(2) (3)2

In each of these the 3 and the 2 are called factors and the answer would be called the product

Note: We will try to phase out the 1st in this list as it will become confusing when variables are involved.

All of the following mean division:

10 2 10 10/2 210 2

In each of these the 10 is called the dividend, the 2 is called the divisor and the answer would be called the quotient

47

The multiplicative inverse, also called the reciprocal, is any number which when multiplied by the number itself yields 1. We are going to be redefining division just as we did subtraction. Division is multiplication by the reciprocal (multiplicative inverse) of the divisor (second number, outside number or denominator). When we write a/b we are actually saying “a 1/b”, so we see that multiplying by the inverse of a number is the same as division by that number. Recall that division by zero is undefined, therefore there is no reciprocal of zero!

Example: Rewrite each as a multiplication problema) 56 8

b) 7/8

c) 8 24

NOW SOME IMPORTANT TRUTHS :

At this time I would like to remind you: the product of zero and anything is zero,

Multiplication by Zero – ZERO a(0) = 0

division by zero is undefined and

Division by Zero – UNDEFINED

a = undefined since no number, when multiplied by zero yields a 0

zero divided by anything is zero.

Zero Divided by Anything – ZERO

0 = 0 since a(0) = 0 [mult. prop. of zero] a

48

Multiplying & Dividing RealsRule 1: Multiplication or Division of two positive numbers yields a positive number

(+ + = +)Rule 2: Multiplication or Division of two negative numbers yields a positive number

(- - = +)Rule 3: Multiplication or Division of a negative and positive number yields a negative

number (- + = - or + - = -)

Example: -5 x 3

Example: -15 -3

Example: -3 -5

Example: -21/3

Note: A negative fraction is negative whether the negative is associated with the numerator, the denominator or just the entire fraction! -21/3 = 21/-3 = -(21/3)

Example: 3(5)

Example: 7-56

Example: 28 -7

49

Really this section is quite simple in the new rules that it brings up. There are only the four and this business about division being multiplication really should be old hat since we have already seen this in the discussion on fractions. Now all there is left to do is practice all the more difficult things with the use of integer multiplication and division. We will do some order of operation problems that involve integers, exponents, parentheses, absolute values, and both multiplication/division and addition/subtraction. I will leave all translation problems up to you without review, but I will stress that you should ask for review if you are having difficulty. Try to review section 3 before trying any translations.

Example: Simplify each of the following.a) -2/5 1 7/8

b) - 5.1/0.3

c) (5 9) | -8 2/3 |

50

d) 9 5 (3)(-7) 2 ½ 7 9

e) 7 -8 + (-7) 2 + 7 | -8 9 |

f) | (9)(-1) + -11 | -5 + -22 + | -9 |

51

The final thing that we will do before closing out the section is to cover averages. An average is computed by finding the sum of elements and dividing it by the number of elements. An average is the point that would "balance" all the elements if they were weighted. I will define an average using sigma notation below (capital sigma is a Greek letter that mathematicians use to denote addition of elements).

Average = x 1 + x2 + x3 + x4 + … + xn x , n = # of elements n

Note: The little numbers below and to the right of the variable are called subscripts and are used to denote different elements without using a different variable name for each. By naming the last xn it indicates that all the numbers in the set are added together, no matter how many elements there are, since n is used to denote the number of elements.

Example: Find the average of the following numbers.7, 8, -9, -1, 12, -53

52

If I feel that we have time I may assign board problems from p.63 for each person to do in class, to count as a homework assignment.§1.7 Properties of Real Numbers

OutlineProperties of Real Numbers Commutative Property of Addition a + b = b + a Allowed us to learn our addition facts without twice the work Commutative Property of Multiplication a b = b a Allowed us to learn our multiplication facts without twice the work Associative Property of Addition (a + b) + c = a + (b + c) Useful in adding columns or numbers Associative Property of Multiplication ( a b) c = a ( b c ) Useful in multiplying multiple factors Distributive Property a(b + c) = ab + ac Multiplication distributes over addition Allows simplification of algebraic expressions(equations) Identity Property of Addition a + 0 = a Allows us to solve algebraic equations (addition property of equality) Identity Property of Multiplication a 1 = a Allows us to solve algebraic equations (multiplication property of equality) Helps in building higher terms and reducing Inverse Property of Addition a + -a = 0 Further allows us to solve algebraic equations (addition property of equality) Inverse Property of Multiplication a 1/a = 1 Further allows us to solve algebraic equations (multiplication property of equality)

Homework p. 70-73 #11-30all, #36-50even, #51-54all, #64-78even, #87 & 88

Commutative PropertiesThis special property states that no matter in which order you add or multiply two numbers the sum or product is still the same.

a + b = b + a Commutative Property of Additiona b = b a Commutative Property of Multiplication

Example: 0 + 8 = 8 + 0Example: 15 + 7 = 7 + 15Example: x + 7 = 7 + x

53

Example: 7(8) = 8(7)Example: 8x = x8

Associative Properties This property tells us that we can group numbers together in any way and add or multiply them and still get the same answer. You learned this property when you learned to add columns of numbers and found that it was easier to group numbers together and then add the groups' sums. Or when you learned that the multiplication table was symmetric.

( a b) c = a ( b c ) Associative Prop. of Mult.(a + b) + c = a + (b + c) Associative Prop. of Add.

Example: 5 + 4 + 7 + 3 = ( 5 + 4 ) + ( 7 + 3 )

Example: 6 + ( 1 + 8 ) = ( 6 + 1 ) + 8

Example: (x + 7) + 3 = x + (7 + 3)

One very nice thing about the associative property of addition is that we can use it to add terms that are alike, like terms. A term is any number, variable(s) or a variable(s) multiplied by a number (referred to as a numeric coefficient). Like terms are terms that have the exact same variable or variables. We can also use the associative property of multiplication to group numeric coefficients together to create a single product. Note: You might recall that when a number is written next to a variable it indicates multiplication.

Example: Simplify the following by using the associative property to add like terms

a) 8x + 5 2 + x

b) 8 (7n)

c) 2 3 6n

Distributive PropertyUnlike addition, multiplication has another property called the distributive property. The distributive property only works with multiplication and goes as follows, as it distributes multiplication over addition:

a ( b + c ) = a ( b ) + a ( c )

54

Example: Simplify each of the following using the distributive property

a) 4 ( 3 + 2 )

Note: Do not become confused by this simple example. It is meant to give you a warm fuzzy about using the distributive property. The distributive property should not replace order of operations if you have all numbers!

b) x ( 3 + 5 )

c) 2 ( 2x + 3 )

d) 5 ( x + y + 5z )

Identity Elements (Properties)The identity element is the thing that gives the number itself back. They should not be confused with the Inverse properties, which yield the Identity Elements.

a 1 = a Multiplication's Identity Element is 1a + 0 = a Addition's Identity Element is 0

These properties are used in fractions most often. They are the properties that allow us to reduce and build higher terms. For more information please see your book's explanation on page 66.

55

Inverse PropertiesThe inverse properties are very useful properties that allow us to solve equations. Neither you book nor I will cover this feature directly at this time, but when we cover solving equations if you return to page 67 and review the examples that your book gives you will see the similarities.

a + (-a) = 0 Inverse Property of Additiona 1/a = 1 Inverse Property of Multiplication

The basic properties discussed in this section rarely if ever come up by themselves, although you can expect to see one problem of the following type on any first test in my classes. (Like problems 9 & 11-30 on p. 70-71 in your book)The importance of these properties is how they help us to solve algebra problems.

Example: Match each example with the property that it best represents by writing the corresponding letter next to it.

_____ 5(t + x) = 5t + 5x a. Identity Element of Mult._____ - 6 + 6 = 0 b. Associative Property of Mult._____ 5 + 8 = 8 + 5 c. Commutative Prop. of Mult. _____ (2 3) 5 = 2 (3 5) d. Additive Inverse_____ 1/7 7 = 1 e. Multiplicative Inverse_____ -1/5 3/3 = -3/15 f. Distributive Property_____ (-9 + 5) + 1 = -9 + (5 + 1) g. Commutative Prop. of Addition_____ 3 (4 2) = (4 2) 3 h. Associative Prop. of Addition

56

§1.8 Simplifying Expressions

OutlineDefinition Review Term Like Term Numeric CoefficientNew Definition Simplifying (Combining Like Terms)Combining Like Terms Basics Translation

Homework p. 76-78 #27-66mult.of3, #67-72all, #81-84all

Let’s take a few moments to review some important vocabulary that we will need for this section.

Term – Number, variable, product of a number and a variable or a variable raised to a power.

Example: a) 5b) 5xc) xyd) x2

Numeric Coefficient – The numeric portion of a term with a variable.

Example: What is the numeric coefficient?a) 3x2

b) x/2

c) - 5x/2

d) – z

57

Like Term – Terms that have a variable, or combination of variables, that are raised to the exact same power.

Example: Are the following like terms?a) 7x 10x2

b) - 15z 23z

c) t 15t3

d) 5 5w

e) xy 6xy

f) x2y - 2x2y2

Simplifying an algebraic expression by combining like terms means adding/subtracting terms in an algebraic expression that are alike. Remember that terms in an algebraic expression are separated by an addition sign (recall also that subtraction is addition of the opposite, so once you change all subtraction to addition, you may simplify) and if multiplication is involved the distributive property must first be applied.

Simplifying Algebraic ExpressionsStep 1: Change all subtraction to additionStep 2: Use the distributive property wherever necessaryStep 3: Group like termsStep 4: Add numeric coefficients of like termsStep 5: Don’t forget to separate each term in your simplified expression

by an addition symbol!!

Example: Simplifya) 3x 12x

b) 5y + 2 + 7 2y

58

c) 9 3(2y + 2)

Just for some much needed review and for some more experience in simplifying let's do some translation.

Example: Translate each of the following using x for the variable and then simplify.

a) The product of 5 and a number, added to the difference of 7 and twice the number.

b) The sum of a number and 14, subtracted from nine times the number.

c) Triple the difference of a number and 5, subtracted from twice the number.

59

Chapter 1 Practice Test

Instructions: Put your name on the top before beginning. You will have approximately one hour to complete this exam. You may not use a calculator. Please attach you note card to the back of the test when finished. If you can’t complete a problem in the space allotted, please direct me to the place where you have completed the work and label it clearly. A problem without work will not receive full credit under any circumstances. Word problems must show the appropriate set up, equation, substitution and work to receive full credit. Show all work and box your final answer. Good luck!

1. Simplify each of the following using your knowledge of fractions.a) 1/5 2/5

b) 1 2/3 3/8

c) 2/5 + 4/5

d) 2/27 1/18

e) 12 1/5 + 1 1/3

f) 12 ¼ 5 ¾

60

g) 0.115 + 1.5

h) (2/5)2

2. Simplify the following using your knowledge of decimals.a) 25 0.529

b) 75 0.03

c) 1.25 5

d) (1.2)(0.05)

e) (0.02)3

61

3. Simplify using your knowledge of integers. Change all subtraction to addition before simplifying.a) 9 + -7

b) -8 + -9

c) -9 + 6

d) -9 8

e) -6 (-9)

f) 7 9

g) -9 5

62

h) -54 -9

i) -22

j) (-2)2

Note: There is a difference in the answers between i & j and you should be able to explain this difference. If it is not clear ask in class!

4. Simplify by using order of operations. Show all steps.a) 9 + 15 3 8

b) 8 + -8 8 (8 + -8) 8

c) 2(5 3) + 25 5 + 20 8 4

d) 8 1 / 8 + -1 5 15 + 1/3

63

5. Translate using x as the variable, simplify and then evaluate when x = 2.

a) The sum of seven and twice the number, subtracted from the difference of the number and five.

b) The quotient of a number and 9, less a third of the number.

6. Compare each of the following using <, > or =. Show work in simplifying the numbers that you are comparing.a) -(-5) | -5 |

b) - | -5 | -(-5)

64

c) 0/5 - | 5 |

7. For the equation 2x 9 = 11, is x = 9 a solution. Explain.

8. Using integers and only addition, write down all important information in shorthand, write an expression and solve the following word problem.

John had $51 in his account. He wrote a check for $27, then another one for $28, finally he made a deposit of $30. What was John's final balance.

9. Match each example with the property that it best represents by writing the corresponding letter next to it.

_____ 2(84) = (28)4 a) Inverse Property of Mult.

_____ 0 + 4 = 4 b) Commutative Prop. of Add.

_____ 1/4 4 = 1 c) Associative Prop. of Mult.

65

_____ 2 + (5 + 3) = (5 + 3) + 2 d) Identity Element of Add.

_____ 4z + 4v = 4(z + v) e) Distributive Property

10. Combine like terms to simplify.a) -5c + -5 3c

b) -(a + b) (5a 5b)

c) 5ax + 9 3(5ax + 5a)

d) 7x2 + 9x 5x2 + 9

66