Chapter 1 Review of Real Numbers.

Chapter 1 Review of Real Numbers

Tips for Success in Mathematics
1.1 Tips for Success in Mathematics

Getting Ready for This Course
Positive Attitude Believe you can succeed. Scheduling Make sure you have time for your classes. Be Prepared Have all the materials you need, like a lab manual, calculator, or other supplies. Objective A

General Tips for Success
Details Get a contact person. Exchange names, phone numbers or addresses with at least one other person in class. Attend all class periods. Sit near the front of the classroom to make hearing the presentation, and participating easier. Do you homework. The more time you spend solving mathematics, the easier the process becomes. Check your work. Review your steps, fix errors, and compare answers with the selected answers in the back of the book. Learn from your mistakes. Find and understand your errors. Use them to become a better math student. Objective B Continued

General Tips for Success
Details Get help if you need it. Ask for help when you don’t understand something. Know when your instructors office hours are, and whether tutoring services are available. Organize class materials. Organize your assignments, quizzes, tests, and notes for use as reference material throughout your course. Read your textbook. Review your section before class to help you understand its ideas more clearly. Ask questions. Speak up when you have a question. Other students may have the same one. Hand in assignments on time. Don’t lose points for being late. Show every step of a problem on your assignment. Objective B Continued

Using This Text Resource Details Continued Practice Problems.
Try each Practice Problem after you’ve finished its corresponding example. Chapter Test Prep Video CD. Chapter Test exercises are worked out by the author, these are available off of the CD this book contains. Lecture Video CDs. Exercises marked with a CD symbol are worked out by the author on a video CD. Check with your instructor to see if these are available. Symbols before an exercise set. Symbols listed at the beginning of each exercise set will remind you of the available supplements. Objectives. The main section of exercises in an exercise set is referenced by an objective. Use these if you are having trouble with an assigned problem. Objective C Continued

Using This Text Resource Details Icons (Symbols).
A CD symbol tells you the corresponding exercise may be viewed on a video segment. A pencil symbol means you should answer using complete sentences. Integrated Reviews. Reviews found in the middle of each chapter can be used to practice the previously learned concepts. End of Chapter Opportunities. Use Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews to help you understand chapter concepts. Study Skills Builder. Read and answer questions in the Study Skills Builder to increase your chance of success in this course. The Bigger Picture. This can help you make the transition from thinking “section by section” to thinking about how everything corresponds in the bigger picture. Objective C Continued

Get help as soon as you need it.
Getting Help Tip Details Get help as soon as you need it. Material presented in one section builds on your understanding of the previous section. If you don’t understand a concept covered during a class period, there is a good chance you won’t understand the concepts covered in the next period. For help try your instructor, a tutoring center, or a math lab. A study group can also help increase your understanding of covered materials. Objective D

Preparing for and Taking an Exam Steps for Preparing for a Test
Review previous homework assignments. Review notes from class and section-level quizzes you have taken. Read the Highlights at the end of each chapter to review concepts and definitions. Complete the Chapter Review at the end of each chapter to practice the exercises. Take a sample test in conditions similar to your test conditions. Set aside plenty of time to arrive where you will be taking the exam. Objective E Continued

Preparing for and Taking an Exam Steps for Taking Your Test
Read the directions on the test carefully. Read each problem carefully to make sure that you answer the question asked. Pace yourself so that you have enough time to attempt each problem on the test. Use extra time checking your work and answers. Don’t turn in your test early. Use extra time to double check your work. Objective E Continued

Tips for Making a Schedule
Managing Your Time Tips for Making a Schedule Make a list of all of your weekly commitments for the term. Estimate the time needed and how often it will be performed, for each item. Block out a typical week on a schedule grid, start with items with fixed time slots. Next, fill in items with flexible time slots. Remember to leave time for eating, sleeping, and relaxing. Make changes to your workload, classload, or other areas to fit your needs. Objective F

Place Value, Names for Numbers, and Reading Tables
§ 1.2 Place Value, Names for Numbers, and Reading Tables

Place Value The position of each digit in a number determines its place value. Ones Hundred-thousands Hundred-billions Ten-billions Billions Hundred-millions Ten-millions Millions Ten-thousands Thousands Hundreds Tens

Writing a Number in Words
A whole number such as 35,689,402 is written in standard form. The columns separate the digits into groups of threes. Each group of three digits is a period. Millions Thousands Billions Ones Hundred-thousands Hundred-billions Ten-billions Hundred-millions Ten-millions Ten-thousands Hundreds Tens

Writing a Number in Words
To write a whole number in words, write the number in each period followed by the name of the period. Ones Hundred-thousands Hundred-billions Ten-billions Billions Hundred-millions Ten-millions Millions Ten-thousands Thousands Hundreds Tens thirty-five million, six hundred eighty-nine thousand, four hundred two

Helpful Hint The name of the ones period is not used when reading and writing whole numbers. Also, the word “and” is not used when reading and writing whole numbers. It is used when reading and writing mixed numbers and some decimal values as shown later.

Chapter 1 / Whole Numbers and Introduction to Algebra
Expanded Form Standard Form Expanded Form 4, = The place value of a digit can be used to write a number in expanded form. The expanded form of a number shows each digit of the number with its place value.

Comparing Whole Numbers
Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Whole Numbers We can picture whole numbers as equally spaced points on a line called the number line. 1 2 3 4 5 A whole number is graphed by placing a dot on the number line. The graph of 4 is shown.

Comparing Numbers For any two numbers graphed on a number line, the number to the right is the greater number, and the number to the left is the smaller number. 5 4 1 2 3 2 is to the left of 5, so 2 is less than 5 5 is to the right of 2, so 5 is greater than 2

Comparing Numbers . . . 2 is less than 5 can be written in symbols as 2 < 5 5 is greater than 2 is written as 5 > 2

are both true statements.
Helpful Hint One way to remember the meaning of the inequality symbols < and > is to think of them as arrowheads “pointing” toward the smaller number. For example, 2 < 5 and 5 > 2 are both true statements.

Reading Tables Most Medals Olympic Winter (1924 – 2002) Games Gold
Silver Bronze Total 107 104 86 297 113 83 78 274 94 92 74 260 69 71 51 191 41 57 64 162 Germany Russia Norway USA Austria Source: The Sydney Morning Herald, Flags courtesy of used with permission

Adding and Subtracting Whole Numbers, and Perimeter
1.3 Adding and Subtracting Whole Numbers, and Perimeter

The sum of 0 and any number is that number.
Addition Property of 0 The sum of 0 and any number is that number. 8 + 0 = 8 and = 8

Commutative Property of Addition
Changing the order of two addends does not change their sum. 4 + 2 = 6 and = 6

Associative Property of Addition
Changing the grouping of addends does not change their sum. 3 + (4 + 2) = = 9 and (3 + 4) + 2 = = 9

Subtraction Properties of 0
Chapter 1 / Whole Numbers and Introduction to Algebra Subtraction Properties of 0 The difference of any number and that same number is 0. 9 – 9 = 0 The difference of any number and 0 is the same number. 7 – 0 = 7

Polygons A polygon is a flat figure formed by line segments connected at their ends. Geometric figures such as triangles, squares, and rectangles are called polygons. triangle square rectangle

Perimeter The perimeter of a polygon is the distance around the polygon.

Addition Problems Descriptions of problems solved through addition may include any of these key words or phrases: Key Words Examples Symbols added to 3 added to 9 3 + 9 plus 5 plus 22 5 + 22 more than 7 more than 8 7 + 8 total total of 6 and 5 6 + 5 increased by 16 increased by 7 16 + 7 sum sum of 50 and 11

Subtraction Problems Descriptions of problems solved by subtraction may include any of these key words or phrases: Key Words Examples Symbols subtract subtract 3 from 9 9 – 3 difference difference of 8 and 2 8 – 2 less 12 less 8 12 – 8 take away 14 take away 9 14 – 9 decreased by 16 decreased by 7 16 – 7 subtracted from 5 subtracted from 9 9 – 5

Helpful Hint Be careful when solving applications that suggest subtraction. Although order does not matter when adding, order does matter when subtracting. For example, 10 – 3 and 3 – 10 do not simplify to the same number.

Helpful Hint Since subtraction and addition are reverse operations, don’t forget that a subtraction problem can be checked by adding.

Number of Endangered Species
Reading a Bar Graph The graph shows the number of endangered species in each country. Number of Endangered Species 146 89 83 73 Country 72 64 Source: The Top 10 of Everything, Russell Ash.

Rounding and Estimating
1.4 Rounding and Estimating

Rounding 20 30 23 23 rounded to the nearest ten is 20. 40 50 48
10 20 15 15 rounded to the nearest ten is 20.

Rounding Whole Numbers
Chapter 1 / Whole Numbers and Introduction to Algebra Rounding Whole Numbers Step 1: Locate the digit to the right of the given place value. Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and replace each digit to its right by 0. Step 3: If this digit is less than 5, replace it and each digit to its right by 0.

Estimates Making estimates is often the quickest way to solve real-life problems when their solutions do not need to be exact.

Helpful Hint Estimation is useful to check for incorrect answers when using a calculator. For example, pressing a key too hard may result in a double digit, while pressing a key too softly may result in the number not appearing in the display.

Multiplying Whole Numbers and Area
1.5 Multiplying Whole Numbers and Area

Multiplication Multiplication is repeated addition with a different notation. = 5 ∙ 4 = 20 5 fours factor product

Multiplication Property of 0
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplication Property of 0 The product of 0 and any number is 0. 9  0 = 0 0  6 = 0

Multiplication Property of 1
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplication Property of 1 The product of 1 and any number is that same number. 9  1 = 9 1  6 = 6

Commutative Property of Multiplication
Chapter 1 / Whole Numbers and Introduction to Algebra Commutative Property of Multiplication Changing the order of two factors does not change their product. 6  3 = 18 and 3  6 = 18

Associative Property of Multiplication
Chapter 1 / Whole Numbers and Introduction to Algebra Associative Property of Multiplication Changing the grouping of factors does not change their product. 5  ( 2  3) = 5  6 = 30 and (5  2)  3 = 10  3 = 30

Distributive Property
Chapter 1 / Whole Numbers and Introduction to Algebra Distributive Property Multiplication distributes over addition. 5(3 + 4) = 5   4

Area 1 square inch 1 5 inches 3 inches
Area of a rectangle = length  width = (5 inches)(3 inches) = 15 square inches

Helpful Hint Remember that perimeter (distance around a plane figure) is measured in units. Area (space enclosed by a plane figure) is measured in square units. 5 inches 4 inches Rectangle 5 inches + 4 inches + 5 inches + 4 inches = 18 inches Perimeter = Area = (5 inches)(4 inches) = 20 square inches

Multiplication Words There are several words or phrases that indicate the operation of multiplication. Some of these are as follows: Key Words Examples Symbols multiply multiply 4 by 3 4  3 product product of 2 and 5 2  5 times 7 times 6 7  6

Dividing Whole Numbers
1.6 Dividing Whole Numbers

Division quotient divisor dividend
The process of separating a quantity into equal parts is called division. quotient dividend divisor

Division Properties of 1
The quotient of any number, except 0, and that same number is 1. 6 1 5 7 =

The quotient of any number and 1 is that same number. 6 1 5 7 =

The quotient of 0 and any number (except 0) is 0. 6 7 = 5

The quotient of any number and 0 is not a number. We say that 6 5 7 are undefined.

Helpful Hint Since division and multiplication are reverse operations, don’t forget that a division problem can be checked by multiplying.

Division Words Here are some key words and phrases that indicate the operation of division. Key Words Examples Symbols divide divide 15 by 3 15  3 quotient quotient of 12 and 6 divided by 8 divided by 4 divided or shared equally $20 divided equally among five people 20  5

Average How do you find an average? A student’s prealgebra grades at the end of the semester are: 90, 85, 95, 70, 80, 100, 98, 82, 90, 90. How do you find his average? Find the sum of the scores and then divide the sum by the number of scores. Sum = 880 Average = 880 ÷ 10 = 88

Exponents and Order of Operations
1.7 Exponents and Order of Operations

An exponent is a shorthand notation for repeated multiplication.
Exponents An exponent is a shorthand notation for repeated multiplication. 3 • 3 • 3 • 3 • 3 3 is a factor 5 times Using an exponent, this product can be written as exponent base

Exponential Notation exponent base Read as “three to the fifth power” or “the fifth power of three.” This is called exponential notation. The exponent, 5, indicates how many times the base, 3, is a factor. 3 • 3 • 3 • 3 • 3 3 is a factor 5 times

Reading Exponential Notation 4 = 41 is read as “four to the first power.” 4  4 = 42 is read as “four to the second power” or “four squared.”

Reading Exponential Notation 4  4  4 = 43 is read as “four to the third power” or “four cubed.” 4  4  4  4 = 44 is read as “four to the fourth power.”

Helpful Hint Usually, an exponent of 1 is not written, so when no exponent appears, we assume that the exponent is 1. For example, 2 = 21 and 7 = 71.

Evaluating Exponential Expressions To evaluate an exponential expression, we write the expression as a product and then find the value of the product. 35 = 3 • 3 • 3 • 3 • 3 = 243

Helpful Hint An exponent applies only to its base. For example,
4 • 23 means 4 • 2 • 2 • 2. Don’t forget that 24 is not 2 • 4. 24 means repeated multiplication of the same factor. 24 = 2 • 2 • 2 • 2 = 16, whereas 2 • 4 = 8

Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

Introduction to Variables, Algebraic Expressions, and Equations
1.8 Introduction to Variables, Algebraic Expressions, and Equations

Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra Algebraic Expressions A combination of operations on letters (variables) and numbers is called an algebraic expression. Algebraic Expressions 5 + x  y  y – 4 + x 4x means 4  x and xy means x  y

Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra Algebraic Expressions Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable.

Evaluating Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra Evaluate x + y for x = 5 and y = 2. Replace x with 5 and y with 2 in x + y. x + y = ( ) + ( ) 5 2 = 7

Equation Statements like = 7 are called equations. An equation is of the form expression = expression An equation can be labeled as Equal sign x = 9 left side right side

Solutions When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable. A solution of an equation is a value for the variable that makes an equation a true statement.

Solutions Determine whether a number is a solution: Is –2 a solution of the equation 2y + 1 = –3? Replace y with –2 in the equation. 2y + 1 = –3 ? 2(–2) + 1 = –3 ? –4 + 1 = –3 –3 = –3 True Since –3 = –3 is a true statement, –2 is a solution of the equation.

Solutions Determine whether a number is a solution: Is 6 a solution of the equation 5x – 1 = 30? Replace x with 6 in the equation. 5x – 1 = 30 ? 5(6) – 1 = 30 ? 30 – 1 = 30 29 = 30 False Since 29 = 30 is a false statement, 6 is not a solution of the equation.

Solutions To solve an equation, we will use properties of equality to write simpler equations, all equivalent to the original equation, until the final equation has the form x = number or number = x Equivalent equations have the same solution. The word “number” above represents the solution of the original equation.

Keywords and Phrases Keywords and phrases suggesting addition, subtraction, multiplication, division or equals. Addition Subtraction Multiplication Division Equal Sign sum difference product quotient equals plus minus times into gives added to less than of per is/was/ will be more than less twice divide yields total decreased by multiply divided by amounts to increased by subtracted from double is equal to

Translating Word Phrases
Chapter 1 / Whole Numbers and Introduction to Algebra Translating Word Phrases the product of 5 and a number 5x twice a number 2x a number decreased by 3 n – 3 a number increased by 2 z + 2 four times a number 4w

Additional Word Phrases
Chapter 1 / Whole Numbers and Introduction to Algebra Additional Word Phrases x + 7 three times the sum of a number and 7 3(x + 7) the quotient of 5 and a number the sum of a number and 7

Helpful Hint Remember that order is important when subtracting. Study the order of numbers and variables below. Phrase Translation a number decreased by 5 x – 5 subtracted from 5 5 – x

Integers and Introduction to Integers
Chapter 2 Integers and Introduction to Integers

Introduction to Integers
2.1 Introduction to Integers

Positive and Negative Numbers
Numbers greater than 0 are called positive numbers. Numbers less than 0 are called negative numbers. negative numbers zero positive numbers 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6

Integers Some signed numbers are integers. negative numbers zero
positive numbers 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 The integers are { …, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, …}

Negative and Positive Numbers
–3 indicates “negative three.” 3 and + 3 both indicate “positive three.” The number 0 is neither positive nor negative. zero negative numbers positive numbers 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6

Comparing Integers We compare integers just as we compare whole numbers. For any two numbers graphed on a number line, the number to the right is the greater number and the number to the left is the smaller number. < means “is less than” > means “is greater than”

Graphs of Integers 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6
The graph of –5 is to the left of –3, so –5 is less than –3, written as 5 < –3 . We can also write –3 > –5. Since –3 is to the right of –5, –3 is greater than –5. 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 –1 –2 –3 –4 –5 –6

Absolute Value The absolute value of a number is the number’s distance from 0 on the number line. The symbol for absolute value is | |. is 2 because 2 is 2 units from 0. 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 is 2 because –2 is 2 units from 0.

Helpful Hint Since the absolute value of a number is that number’s distance from 0, the absolute value of a number is always 0 or positive. It is never negative. zero a positive number

Opposite Numbers Two numbers that are the same distance from 0 on the number line but are on the opposite sides of 0 are called opposites. 5 units 5 and –5 are opposites. 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6

Opposite Numbers 5 is the opposite of –5 and –5 is the opposite of 5. The opposite of is – 4 is written as –(4) = –4 The opposite of – 4 is 4 is written as –(– 4) = 4 –(–4) = 4 If a is a number, then –(– a) = a.

Helpful Hint Remember that 0 is neither positive nor negative. Therefore, the opposite of 0 is 0.

2.2 Adding Integers

Adding Two Numbers with the Same Sign
Start End 2 + 3 = 5 2 3 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 End Start – 2 + (– 3) = – 5 –3 –2 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

Adding Two Numbers with the Same Sign
Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: Add their absolute values. Step 2: Use their common sign as the sign of the sum. Examples: – 3 + (–5) = – 8 5 + 2 = 7

Adding Two Numbers with Different Signs
2 + (–3) = –1 – 3 End Start 2 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 3 End – = 1 – 2 Start 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6

Adding Two Numbers with Different Signs
Chapter 1 / Whole Numbers and Introduction to Algebra Adding Two Numbers with Different Signs Step 1: Find the larger absolute value minus the smaller absolute value. Step 2: Use the sign of the number with the larger absolute value as the sign of the sum. Examples: –4 + 5 = (–8) = –2

Helpful Hint If a is a number, then –a is its opposite. a + (–a) = 0 –a + a = 0 The sum of a number and its opposite is 0.

Helpful Hint Don’t forget that addition is commutative and associative. In other words, numbers may be added in any order.

Chapter 1 / Whole Numbers and Introduction to Algebra Evaluate x + y for x = 5 and y = –9. Replace x with 5 and y with –9 in x + y. x + y = ( ) + ( ) 5 –9 = –4

2.3 Subtracting Integers

Subtracting Integers Chapter 1 / Whole Numbers and Introduction to Algebra To subtract integers, rewrite the subtraction problem as an addition problem. Study the examples below. = 4 9 + (–5) = 4 Since both expressions equal 4, we can say = 9 + (–5) = 4

Subtracting Two Numbers
Chapter 1 / Whole Numbers and Introduction to Algebra Subtracting Two Numbers If a and b are numbers, then a b = a + (–b). To subtract two numbers, add the first number to the opposite (called additive inverse) of the second number.

Subtracting Two Numbers
Chapter 1 / Whole Numbers and Introduction to Algebra Subtracting Two Numbers subtraction = first number + opposite of second number 7 – 4 7 (– 4) 3 – 5 – 3 – 5 (– 3) – 8 3 – (–6) 6 9 – 8 – (– 2) 2 – 6

Adding and Subtracting Integers
Chapter 1 / Whole Numbers and Introduction to Algebra Adding and Subtracting Integers If a problem involves adding or subtracting more than two integers, rewrite differences as sums and add. By applying the associative and commutative properties, add the numbers in any order. 9 – 3 + (–5) – (–7) = 9 + (–3) + (–5) + 7 6 + (–5) + 7 1 + 7 8

Evaluate x – y for x = –6 and y = 8. Replace x with –6 and y with 8 in x – y. x – y = ( ) – ( ) –6 8 = ( ) + ( ) –6 –8 = –14

Multiplying and Dividing Integers
2.4 Multiplying and Dividing Integers

Consider the following pattern of products.
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Integers Consider the following pattern of products. First factor decreases by 1 each time. 3  5 = 15 Product decreases by 5 each time. 2  5 = 10 1  5 = 5 0  5 = 0 This pattern continues as follows. – 1  5 = - 5 – 2  5 = - 10 – 3  5 = - 15 This suggests that the product of a negative number and a positive number is a negative number.

Multiplying Integers Observe the following pattern. 2  (– 5) = –10 Product increases by 5 each time. 1  (– 5) = –5 0  (– 5) = 0 This pattern continues as follows. –1  (–5) = 5 –2  (–5) = 10 – 3  (–5) = 15 This suggests that the product of two negative numbers is a positive number.

Multiplying Integers The product of two numbers having the same sign is a positive number. 2  4 = 8 –2  (– 4) = 8 The product of two numbers having different signs is a negative number. 2  (– 4) = –8 – 2  4 = –8

Multiplying Integers Product of Like Signs ( + )( + ) = + (–)(–) = + Product of Different Signs (–)( + ) = – ( + )(–) = –

Helpful Hint If we let ( – ) represent a negative number and ( + ) represent a positive number, then ( – ) ( – ) = ( + ) The product of an even number of negative numbers is a positive result. ( – ) ( – ) ( – ) = ( – ) The product of an odd number of negative numbers is a negative result. ( – ) ( – ) ( – ) ( – ) = ( + ) ( – ) ( – ) ( – ) ( – ) ( – ) = ( – )

Division of integers is related to multiplication of integers.
Chapter 1 / Whole Numbers and Introduction to Algebra Dividing Integers Division of integers is related to multiplication of integers. 3 2 6 = because = – 3 2 – 6 because – 3 (– 2) 6 = because – 2 = 3 – 6 because (– 2) – 6 – 2

Dividing Integers Chapter 1 / Whole Numbers and Introduction to Algebra The quotient of two numbers having the same sign is a positive number. 12 ÷ 4 = 3 –12 ÷ (–4 ) = 3 The quotient of two numbers having different signs is a negative number. – 12 ÷ 4 = –3 12 ÷ (– 4) = – 3

Dividing Numbers Chapter 1 / Whole Numbers and Introduction to Algebra Quotient of Like Signs Quotient of Different Signs

2.5 Order of Operations

Order of Operations Chapter 1 / Whole Numbers and Introduction to Algebra 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

Using the Order of Operations
Simplify 4(5 – 2) + 42. 4(5 – 2) + 42 = 4(3) + 42 Simplify inside parentheses. = 4(3) + 16 Write 42 as 16. = Multiply. Objective D Continued = 28 Add.

Helpful Hint Chapter 1 / Whole Numbers and Introduction to Algebra When simplifying expressions with exponents, parentheses make an important difference. (–5)2 and –52 do not mean the same thing. (–5)2 means (–5)(–5) = 25. –52 means the opposite of 5 ∙ 5, or –25. Only with parentheses is the –5 squared.

Solving Equations and Problem Solving
Chapter 3 Solving Equations and Problem Solving

Simplifying Algebraic Expressions
3.1 Simplifying Algebraic Expressions

Constant and Variable Terms A term that is only a number is called a constant term, or simply a constant. A term that contains a variable is called a variable term. 3y2 + (–4y) + 2 x + 3 Constant terms Variable terms

Coefficients The number factor of a variable term is called the numerical coefficient. A numerical coefficient of 1 is usually not written. 5x x or 1x –7y y 2 Numerical coefficient is 5. Numerical coefficient is –7. Understood numerical coefficient is 1. Numerical coefficient is 3.

Like Terms Terms that are exactly the same, except that they may have different numerical coefficients are called like terms. Like Terms Unlike Terms 3x, 2x –6y, 2y, y –3, 4 5x, x 2 7x, 7y 5y, 5 6a, ab 2ab2, –5b 2a The order of the variables does not have to be the same.

Distributive Property A sum or difference of like terms can be simplified using the distributive property. Distributive Property If a, b, and c are numbers, then ac + bc = (a + b)c Also, ac – bc = (a – b)c

Distributive Property
Chapter 1 / Whole Numbers and Introduction to Algebra Distributive Property By the distributive property, 7x + 5x = (7 + 5)x = 12x This is an example of combining like terms. An algebraic expression is simplified when all like terms have been combined.

Addition and Multiplication Properties The commutative and associative properties of addition and multiplication help simplify expressions. Properties of Addition and Multiplication If a, b, and c are numbers, then Commutative Property of Addition a + b = b + a Commutative Property of Multiplication a ∙ b = b ∙ a The order of adding or multiplying two numbers can be changed without changing their sum or product.

Associative Properties The grouping of numbers in addition or multiplication can be changed without changing their sum or product. Associative Property of Addition (a + b) + c = a + (b + c) Associative Property of Multiplication (a ∙ b) ∙ c = a ∙ (b ∙ c)

Helpful Hint Examples of Commutative and Associative Properties of Addition and Multiplication 4 + 3 = 3 + 4 6 ∙ 9 = 9 ∙ 6 (3 + 5) + 2 = 3 + (5 + 2) (7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8) Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication

We can also use the distributive property to multiply expressions.
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Expressions We can also use the distributive property to multiply expressions. The distributive property says that multiplication distributes over addition and subtraction. 2(5 + x) = 2 ∙ ∙ x = x or 2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x

Simplifying Expressions
To simply expressions, use the distributive property first to multiply and then combine any like terms. Simplify: 3(5 + x) – 17 Apply the Distributive Property 3(5 + x) – 17 = 3 ∙ ∙ x + (–17) = x + (–17) Multiply = 3x + (–2) or 3x – 2 Combine like terms Note: 3 is not distributed to the –17 since –17 is not within the parentheses.

Finding Perimeter 7z feet 3z feet 9z feet Perimeter is the distance around the figure. Perimeter = 3z + 7z + 9z = 19z feet Don’t forget to insert proper units.

Finding Area (2x – 5) meters 3 meters A = length ∙ width = 3(2x – 5)
= 6x – 15 square meters Don’t forget to insert proper units.

Helpful Hint Don’t forget . . . Area: surface enclosed
measured in square units Perimeter: distance around measured in units

3.2 Solving Equations: Review of the Addition and Multiplication Properties

Equation vs. Expression
Chapter 1 / Whole Numbers and Introduction to Algebra Equation vs. Expression Statements like = 7 are called equations. An equation is of the form expression = expression. An equation can be labeled as Equal sign x = 9 left side right side

Addition Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra Addition Property of Equality Let a, b, and c represent numbers. If a = b, then a + c = b + c and a – c = b - c In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.

Multiplication Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplication Property of Equality Let a, b, and c represent numbers and let c  0. If a = b, then a ∙ c = b ∙ c and In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

Solve for x. x - 4 = 3 To solve the equation for x, we need to rewrite the equation in the form x = number. To do so, we add 4 to both sides of the equation. x = Add 4 to both sides. x = Simplify.

Check To check, replace x with 7 in the original equation. x - 4 = Original equation 7 - 4 = Replace x with 7. 3 = True. Since 3 = 3 is a true statement, 7 is the solution of the equation. ?

Solve for x 4x = 8 To solve the equation for x, notice that 4 is multiplied by x. To get x alone, we divide both sides of the equation by 4 and then simplify. 1∙x = 2 or x = 2

Check To check, replace x with 2 in the original equation. 4x = 8 Original equation 4 ∙ 2 = 8 Let x = 2. 8 = 8 True. The solution is 2. ?

Using Both Properties to Solve Equations
Chapter 1 / Whole Numbers and Introduction to Algebra 2(2x – 3) = 10 Use the distributive property to simplify the left side. 4x – 6 = 10 Add 6 to both sides of the equation 4x – = 4x = 16 Divide both sides by 4. x = 4

Check To check, replace x with 4 in the original equation. 2(2x – 3) = 10 Original equation 2(2 · 4 – 3) = 10 Let x = 4. 2(8 – 3) = 10 (2)5 = True. The solution is 4. ? ?

Fractions and Mixed Numbers
Chapter 4 Fractions and Mixed Numbers

Introduction to Fractions and Mixed Numbers
4.1 Introduction to Fractions and Mixed Numbers

Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used. A fraction is a number of the form , where a and b are integers and b is not 0. The parts of a fraction are numerator fraction bar denominator

Helpful Hint Remember that the bar in a fraction means
division. Since division by 0 is undefined, a fraction with a denominator of 0 is undefined.

Visualizing Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra Visualizing Fractions One way to visualize fractions is to picture them as shaded parts of a whole figure.

Visualizing Fractions
Picture Fraction Read as part shaded equal parts one-fourth parts shaded equal parts five-sixths parts shaded equal parts seven-thirds

Types of Fractions A proper fraction is a fraction whose numerator is less than its denominator. Proper fractions have values that are less than 1. An improper fraction is a fraction whose numerator is greater than or equal to its denominator. Improper fractions have values that are greater than or equal to 1. A mixed number is a sum of a whole number and a proper fraction.

Fractions on Number Lines
Chapter 1 / Whole Numbers and Introduction to Algebra Fractions on Number Lines Another way to visualize fractions is to graph them on a number line. 3 5 equal parts 1

Fraction Properties of 1
Chapter 1 / Whole Numbers and Introduction to Algebra Fraction Properties of 1 If n is any integer other than 0, then If n is any integer, then

Fraction Properties of 0
Chapter 1 / Whole Numbers and Introduction to Algebra Fraction Properties of 0 If n is any integer other than 0, then If n is any integer, then

Writing a Mixed Number as an Improper Fraction
Step 1: Multiply the denominator of the fraction by the whole number. Step 2: Add the numerator of the fraction to the product from Step 1. Step 3: Write the sum from Step 2 as the numerator of the improper fraction over the original denominator. 2 3 4 8 11 = ∙ +

Writing an Improper Fraction as a Mixed Number or a Whole Number
Step 1: Divide the denominator into the numerator. Step 2: The whole number part of the mixed number is the quotient. The fraction part of the mixed number is the remainder over the original denominator. remainder original denominator quotient =

Factors and Simplest Form
4.2 Factors and Simplest Form

Prime and Composite Numbers
Chapter 1 / Whole Numbers and Introduction to Algebra Prime and Composite Numbers A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . A composite number is a natural number greater than 1 that is not prime.

Helpful Hint The natural number 1 is neither prime nor composite.

Prime Factorization A prime factorization of a number expresses the number as a product of its factors and the factors must be prime numbers.

Helpful Hints Remember a factor is any number that divides a number evenly (with a remainder of 0).

Prime Factorization Every whole number greater than 1 has exactly one prime factorization. 12 = 2 • 2 • 3 2 and 3 are prime factors of 12 because they are prime numbers and they divide evenly into 12.

Divisibility Tests A whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. 3 if the sum of its digits is divisible by 3. 196 is divisible by 2 117 is divisible by 3 since = 9 is divisible by 3.

Divisibility Tests A whole number is divisible by 5 if the ones digit is 0 or 5. 10 if its last digit is 0. 2,345 is divisible by 5. 8,470 is divisible by 10.

Equivalent Fractions Graph on the number line. 1 Graph on the number line.

Equivalent Fractions Fractions that represent the same portion of a whole or the same point on the number line are called equivalent fractions.

Fundamental Property of Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra Fundamental Property of Fractions If a, b, and c are numbers, then and also as long as b and c are not 0. If the numerator and denominator are multiplied or divided by the same nonzero number, the result is an equivalent fraction.

Simplest Form A fraction is in simplest form, or lowest terms, when the numerator and denominator have no common factors other than 1. Using the fundamental principle of fractions, divide the numerator and denominator by the common factor of 7. Using the prime factorization of the numerator and denominator, divide out common factors.

Writing a Fraction in Simplest Form
Chapter 1 / Whole Numbers and Introduction to Algebra To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors. The process of writing a fraction in simplest form is called simplifying the fraction.

Helpful Hints When all factors of the numerator or denominator are divided out, don’t forget that 1 still remains in that numerator or denominator.

Multiplying and Dividing Fractions
4.3 Multiplying and Dividing Fractions

Multiplying Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Fractions of is 1 The word “of” means multiplication and “is” means equal to.

Multiplying Fractions
of is means

Multiplying Two Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Two Fractions If a, b, c, and d are numbers and b and d are not 0, then In other words, to multiply two fractions, multiply the numerators and multiply the denominators.

Examples If the numerators have common factors with the denominators, divide out common factors before multiplying. 1 or 2

Examples or 2 1

Helpful Hint Recall that when the denominator of a fraction contains a variable, such as we assume that the variable is not 0.

Expressions with Fractional Bases
Chapter 1 / Whole Numbers and Introduction to Algebra The base of an exponential expression can also be a fraction.

Reciprocal of a Fraction
Chapter 1 / Whole Numbers and Introduction to Algebra Reciprocal of a Fraction Two numbers are reciprocals of each other if their product is 1. The reciprocal of the fraction is because

Dividing Two Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra If b, c, and d are not 0, then In other words, to divide fractions, multiply the first fraction by the reciprocal of the second fraction.

Helpful Hint Every number has a reciprocal except 0. The number 0 has no reciprocal. Why? There is no number that when multiplied by 0 gives the result 1.

Helpful Hint When dividing by a fraction, do not look for common factors to divide out until you rewrite the division as multiplication. Do not try to divide out these two 2s.

Fractional Replacement Values
Chapter 1 / Whole Numbers and Introduction to Algebra Fractional Replacement Values If x = and y = , evaluate Replace x with and y with .

4.4 Adding and Subtracting Like Fractions, Least Common Denominator, and Equivalent Fractions

Like and Unlike Fractions
Fractions that have the same or common denominator are called like fractions. Fractions that have different denominators are called unlike fractions. Like Fractions Unlike Fractions

Adding or Subtracting Like Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra If a, b, and c, are numbers and b is not 0, then To add or subtract fractions with the same denominator, add or subtract their numerators and write the sum or difference over the common denominator.

Adding or Subtracting Like Fractions Chapter 1 / Whole Numbers and Introduction to Algebra = Start End 1 To add like fractions, add the numerators and write the sum over the common denominator.

Helpful Hint Chapter 1 / Whole Numbers and Introduction to Algebra Do not forget to write the answer in simplest form. If it is not in simplest form, divide out all common factors larger than 1.

Equivalent Negative Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra

Least Common Denominator Chapter 1 / Whole Numbers and Introduction to Algebra To add or subtract fractions that have unlike, or different, denominators, we write the fractions as equivalent fractions with a common denominator. The smallest common denominator is called the least common denominator (LCD) or the least common multiple (LCM).

Least Common Multiple Chapter 1 / Whole Numbers and Introduction to Algebra The least common denominator (LCD) of a list of fractions is the smallest positive number divisible by all the denominators in the list. (The least common denominator is also the least common multiple (LCM) of the denominators.)

Least Common Denominator Chapter 1 / Whole Numbers and Introduction to Algebra To find the LCD of First, write each denominator as a product of primes. 12 = 2 • 2 • 3 18 = 2 • 3 • 3 Then write each factor the greatest number of times it appears in any one prime factorization. The greatest number of times that 2 appears is 2 times. The greatest number of times that 3 appears is 2 times. LCD = 2 • 2 • 3 • 3 = 36

Adding and Subtracting Unlike Fractions
4.5 Adding and Subtracting Unlike Fractions

Adding or Subtracting Unlike Fractions
Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: Find the LCD of the denominators of the fractions. Step 2: Write each fraction as an equivalent fraction whose denominator is the LCD. Step 3: Add or subtract the like fractions. Step 4: Write the sum or difference in simplest form.

Adding or Subtracting Unlike Fractions Chapter 1 / Whole Numbers and Introduction to Algebra Add: Step 1: Find the LCD of 9 and 12. 9 = 3 ∙ 3 and 12 = 2 ∙ 2 ∙ 3 LCD = 2 ∙ 2 ∙ 3 ∙ 3 = 36 Step 2: Rewrite equivalent fractions with the LCD. Continued.

Adding or Subtracting Unlike Fractions Chapter 1 / Whole Numbers and Introduction to Algebra Continued: Step 3: Add like fractions. Step 4: Write the sum in simplest form.

Writing Fractions in Order One important application of the least common denominator is to use the LCD to help order or compare fractions. Insert < or > to form a true sentence. The LCD for these fractions is 35. Write each fraction as an equivalent fraction with a denominator of 35. Continued.

Writing Fractions in Order Continued: Compare the numerators of the equivalent fractions. Since 21 > 20, then Thus,

Evaluating Expressions Evaluate x – y if x = and y = . Replacing x with and y with , then, x – y Martin-Gay, Prealgebra, 5ed

Solving Equations Containing Fractions Solve: To get x by itself, add to both sides. Continued.

Solving Equations Containing Fractions Continued: Write fraction in simplest form.

Complex Fractions and Review of Order of Operations
4.6 Complex Fractions and Review of Order of Operations

Complex Fraction Chapter 1 / Whole Numbers and Introduction to Algebra A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction.

Method 1: Simplifying Complex Fractions
This method makes use of the fact that a fraction bar means division. 1 3 1 4 When dividing fractions, multiply by the reciprocal of the divisor.

Recall the order of operations. Since the fraction bar is a grouping symbol, simplify the numerator and denominator separately. Then divide. 2 8 When dividing fractions, multiply by the reciprocal of the divisor. 1

Chapter 1 / Whole Numbers and Introduction to Algebra This method is to multiply the numerator and the denominator of the complex fraction by the LCD of all the fractions in its numerator and its denominator. Since this LCD is divisible by all denominators, this has the effect of leaving sums and differences of terms in the numerator and the denominator and thus a simple fraction. Let’s use this method to simplify the complex fraction of the previous example.

Chapter 1 / Whole Numbers and Introduction to Algebra Method 2: Simplifying Complex Fractions Step 1: The complex fraction contains fractions with denominators of 2, 6, 4, and 3. The LCD is 12. By the fundamental property of fractions, multiply the numerator and denominator of the complex fraction by 12. Step 2: Apply the distributive property Continued.

Chapter 1 / Whole Numbers and Introduction to Algebra Method 2: Simplifying Complex Fractions Continued: The result is the same no matter which method is used. Step 3: Multiply. Step 4: Simplify.

Reviewing the Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

Operations on Mixed Numbers
4.7 Operations on Mixed Numbers

Mixed Numbers Recall that a mixed number is the sum of a whole number and a proper fraction. 3 4 5 = + 1 2 3 4 5 19 5 3 4 =

Multiplying or Dividing with Mixed Numbers
To multiply or divide with mixed numbers or whole numbers, first write each mixed number as an improper fraction. Multiply: Write the solution as a mixed number if possible. Remove common factors and multiply. Change mixed numbers to improper fractions.

Adding or Subtracting Mixed Numbers We can add or subtract mixed numbers by first writing each mixed number as an improper fraction. But it is often easier to add or subtract the whole number parts and add or subtract the proper fraction parts vertically.

Adding or Subtracting Mixed Numbers
The LCD of 14 and 7 is 14. Add the fractions, then add the whole numbers. Write equivalent fractions with the LCD of 14. Notice that the fractional part is improper. Since is , write the sum as Make sure the fractional part is always proper.

When subtracting mixed numbers, borrowing may be needed. 3 1 1 2 5 4 3 3 1 = + 2 1 3 = + 2 1 3 = + 2 3 1 = 2 4 3 Borrow 1 from 3.

The LCD of 14 and 7 is 14. To subtract the fractions, we have to borrow. Write equivalent fractions with the LCD of 14. Subtract the fractions, then subtract the whole numbers. Notice that the fractional part is proper.

Solving Equations Containing Fractions
4.8 Solving Equations Containing Fractions

Addition Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra Let a, b, and c represent numbers. If a = b, then a + c = b + c and a – c = b - c In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.

Multiplication Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra Let a, b, and c represent numbers and let c  0. If a = b, then a ∙ c = b ∙ c and In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

Solving an Equation in x
Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: If fractions are present, multiply both sides of the equation by the LCD of the fractions. Step 2: If parentheses are present, use the distributive property. Step 3: Combine any like terms on each side of the equation.

Solving an Equation in x
Chapter 1 / Whole Numbers and Introduction to Algebra Step 4: Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side. Step 5: Divide both sides of the equation by the numerical coefficient of x to solve. Step 6: Check the answer in the original equation.

Solve for x Chapter 1 / Whole Numbers and Introduction to Algebra Multiply both sides by 7. Simplify both sides.

Solve for x Multiply both sides by 5. Simplify both sides. Add – 3y to both sides. Add – 30 to both sides. Divide both sides by 7.

Chapter 5 Decimals

Introduction to Decimals
5.1 Introduction to Decimals

Decimal Notation Chapter 1 / Whole Numbers and Introduction to Algebra Like fractional notation, decimal notation is used to denote a part of a whole. Numbers written in decimal notation are called decimal numbers, or simply decimals. The decimal has three parts. 16.743 Whole number part Decimal part Decimal point

Place Value The position of each digit in a number determines its place value. hundred-thousandths hundreds tens tenths hundredths thousandths ten-thousandths ones 1 10 1 1000 1 100,000 Place Value 100 10 1 1 100 1 10,000 decimal point

Place Value 1 Notice that the value of each place is 10
of the value of the place to its left.

Place Value 16.734 The digit 3 is in the hundredths place, so
its value is 3 hundredths or 3 100

Writing a Decimal in Words
Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: Write the whole number part in words. Step 2: Write “and” for the decimal point. Step 3: Write the decimal part in words as though it were a whole number, followed by the place value of the last digit.

Writing a Decimal in Words
Write the decimal in words. whole number part decimal part one hundred forty-three and fifty-six thousandths

Writing Decimals in Standard Form
A decimal written in words can be written in standard form by reversing the procedure. Write one hundred six and five hundredths in standard form. one hundred six and five hundredths whole-number part decimal decimal part 5 must be in the hundredths place 106 . 05

Helpful Hint When writing a decimal from words to decimal notation, make sure the last digit is in the correct place by inserting 0s after the decimal point if necessary. For example, three and fifty-four thousandths is 3.054 thousandths place

Writing Decimals as Fractions Once you master writing and reading decimals correctly, then you write a decimal as a fraction using the fractions associated with the words you use when you read it. 0.9 is read “nine tenths” and written as a fraction as

Writing Decimals as Fractions
0.21 is read as twenty-one hundredths and written as a fraction as 21 100 0.011 is read as eleven thousandths and written as a fraction as 11 1000

Comparing Two Positive Decimals
37 100 . = 029 29 1000 . = 2 decimal places 2 zeros 3 decimal places 3 zeros Notice that the number of decimal places in a decimal number is the same as the number of zeros in the denominator of the equivalent fraction. We can use this fact to write decimals as fractions.

Comparing Decimals One way to compare decimals is to compare their graphs on a number line. Recall that for any two numbers on a number line, the number to the left is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs. 0.3 0.7 1 0.3 < 0.7 or > 0.3 3 10 7 10

Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Two Positive Decimals Comparing decimals by comparing their graphs on a number line can be time consuming, so we compare the size of decimals by comparing digits in corresponding places.

Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Two Positive Decimals Compare digits in the same places from left to right. When two digits are not equal, the number with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the right of the decimal point to continue comparing. Compare hundredths place digits. 35.638 35.657 < 5 3 35.638 < 35.657

Helpful Hint For any decimal, writing 0s after the last digit to the right of the decimal point does not change the value of the number. 8.5 = 8.50 = 8.500, and so on When a whole number is written as a decimal, the decimal point is placed to the right of the ones digit. 15 = = 15.00, and so on

Rounding Decimals We round the decimal part of a decimal number in nearly the same way as we round whole numbers. The only difference is that we drop digits to the right of the rounding place, instead of replacing these digits by 0s. For example, rounded to the nearest hundredth is 63.78

Rounding Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: Locate the digit to the right of the given place value. Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and drop all digits to the right. If this digit is less than 5, drop all digits to the right of the given place.

Rounding Decimals to a Place Value
Round to the nearest tenth. Locate the digit to the right of the tenths place. tenths place digit to the right Since the digit to the right is less than 5, drop it and all digits to its right. rounded to the nearest tenths is 326.4

Adding and Subtracting Decimals
5.2 Adding and Subtracting Decimals

Adding or Subtracting Decimals
Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: Write the decimals so that the decimal points line up vertically. Step 2: Add or subtract as with whole numbers. Step 3: Place the decimal point in the sum or difference so that it lines up vertically with the decimal points in the problem.

Helpful Hint Recall that 0s may be inserted to the right of the decimal point after the last digit without changing the value of the decimal. This may be used to help line up place values when adding or subtracting decimals. becomes two 0s inserted 71.74

Helpful Hint Don’t forget that the decimal point in a whole number is after the last digit.

Estimating Operations on Decimals Estimating sums, differences, products, and quotients of decimal numbers is an important skill whether you use a calculator or perform decimal operations by hand.

Estimating When Adding Decimals Add Exact Estimate rounds to rounds to This is a reasonable answer.

Helpful Hint When rounding to check a calculation, you may want to round the numbers to a place value of your choosing so that your estimates are easy to compute mentally.

Evaluating with Decimals Evaluate x + y for x = 5.5 and y = 2.8. Replace x with 5.5 and y with 2.8 in x + y. x + y = ( ) + ( ) 5.5 2.8 = 8.3

Multiplying Decimals and Circumference of a Circle
5.3 Multiplying Decimals and Circumference of a Circle

Multiplying Decimals Multiplying decimals is similar to multiplying whole numbers. The difference is that we place a decimal point in the product. 7 10 3 100  21 1000 0.7  = = 1 decimal place 2 decimal places = 3 decimal places

Multiplying Decimals Step 1: Multiply the decimals as though they were whole numbers. Step 2: The decimal point in the product is placed so the number of decimal places in the product is equal to the sum of the number of decimal places in the factors.

Estimating when Multiplying Decimals Multiply 32.3  1.9. Exact Estimate rounds to rounds to This is a reasonable answer.

Multiplying Decimals by Powers of 10
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Decimals by Powers of 10 There are some patterns that occur when we multiply a number by a power of ten, such as 10, 100, 1000, 10,000, and so on.

Multiplying Decimals by Powers of 10
Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Decimals by Powers of 10  10 =  100 =  100,000 = 7,654,300 Decimal point moved 1 place to the right. 1 zero Decimal point moved 2 places to the right. 2 zeros Decimal point moved 5 places to the right. 5 zeros The decimal point is moved the same number of places as there are zeros in the power of 10.

Multiplying Decimals by Powers of 10 Move the decimal point to the right the same number of places as there are zeros in the power of 10. Multiply:  100 Since there are two zeros in 100, move the decimal place two places to the right.  100 = = 343.05

Multiplying Decimals by Powers of 10 Move the decimal point to the left the same number of places as there are decimal places in the power of 10. Multiply: 8.57  0.01 Since there are two decimal places in 0.01, move the decimal place two places to the left. 8.57  0.01 = = 0.0857 Notice that zeros had to be inserted.

The Circumference of a Circle The distance around a polygon is called its perimeter. The distance around a circle is called the circumference. This distance depends on the radius or the diameter of the circle.

The Circumference of a Circle r d Circumference = 2·p ·radius or Circumference = p ·diameter C = 2 p r or C = p d

p The symbol p is the Greek letter pi, pronounced “pie.” It is a constant between 3 and 4. A decimal approximation for p is 3.14. A fraction approximation for p is 22 7

The Circumference of a Circle Find the circumference of a circle whose radius is 4 inches. 4 inches C = 2pr = 2p ·4 = 8p inches 8p inches is the exact circumference of this circle. If we replace  with the approximation 3.14, C = 8  8(3.14) = inches. 25.12 inches is the approximate circumference of the circle.

5.4 Dividing Decimals

Dividing by a Decimal Division of decimal numbers is similar to division of whole numbers. The only difference is the placement of a decimal point in the quotient. If the divisor is a whole number, divide as for whole numbers; then place the decimal point in the quotient directly above the decimal point in the dividend. 8 4 quotient divisor 63 52.92 dividend 2 5 2 -2 52

Dividing by a Decimal If the divisor is not a whole number, we need to move the decimal point to the right until the divisor is a whole number before we divide. divisor 6 3 52 92 . dividend 63 529 2 . 8 4 63 52 9.2 - 504 25 2 -252

Dividing by a Decimal Step 1: Move the decimal point in the divisor to the right until the divisor is a whole number. Step 2: Move the decimal point in the dividend to the right the same number of places as the decimal point was moved in Step 1. Step 3: Divide. Place the decimal point in the quotient directly over the moved decimal point in the dividend.

Estimating When Dividing Decimals Divide ÷ 2.8 Exact Estimate rounds to This is a reasonable answer.

Dividing Decimals by Powers of 10
There are patterns that occur when dividing by powers of 10, such as 10, 100, 1000, and so on. . 45 62 10 = 456.2 The decimal point moved 1 place to the left. 1 zero 1 000 4562 , . = 456.2 The decimal point moved 3 places to the left. 3 zeros The pattern suggests the following rule.

Dividing Decimals by Powers of 10
Move the decimal point of the dividend to the left the same number of places as there are zeros in the power of 10. Notice that this is the same pattern as multiplying by powers of 10 such as 0.1, 0.01, or Because dividing by a power of 10 such as 100 is the same as multiplying by its reciprocal , or 0.01. To divide by a number is the same as multiplying by its reciprocal.

Fractions, Decimals, and Order of Operations
5.5 Fractions, Decimals, and Order of Operations

Writing Fractions as Decimals To write a fraction as a decimal, divide the numerator by the denominator.

Comparing Fractions and Decimals To compare decimals and fractions, write the fraction as an equivalent decimal. Compare and Therefore, < 0.25

Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

Using the Order of Operations
Simplify ( –2.3) ( ) ( –2.3) ( ) = ( –2.3) (5.3) Simplify inside parentheses. = (5.3) Write ( –2.3)2 as 5.29. Objective D Continued = Multiply. = 27.02 Add.

Finding the Area of a Triangle
height A base • height = 1 2 base A bh = 1 2

Equations Containing Decimals
5.6 Equations Containing Decimals

Steps for Solving an Equation in x
Chapter 1 / Whole Numbers and Introduction to Algebra Step 1: If fractions are present, multiply both sides of the equation by the LCD of the fractions. Step 2: If parentheses are present, use the distributive property. Step 3: Combine any like terms on each side of the equation.

Steps for Solving an Equation in x
Chapter 1 / Whole Numbers and Introduction to Algebra Step 4: Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side. Step 5: Divide both sides by the numerical coefficient of x to solve. Step 6: Check the answer in the original equation.

Solving Equations with Decimals
–0.01(5a + 4) = 0.04 – 0.01(a + 4) –1(5a + 4) = 4 – 1(a + 4) Multiply both sides by 100. –5a – 4 = 4 – a – 4 Apply the distributive property. –4a – 4 = 4 – 4 Add a to both sides. –4a = 4 Add 4 to both sides and simplify. a = –1 Divide both sides by 4.

Decimal Applications: Mean, Median, and Mode
5.7 Decimal Applications: Mean, Median, and Mode

Measures of Central Tendency The mean, the median, and the mode are called measures of central tendency. They describe a set of data, or a set of numbers, by a single “middle” number.

Mean (Average) The most common measure of central tendency is the mean (sometimes called the “arithmetic mean” or the “average”). The mean (average) of a set of number items is the sum of the items divided by the number of items.

Finding the Mean Chapter 1 / Whole Numbers and Introduction to Algebra Find the mean of the following list of numbers. 2.5 5.1 9.5 6.8 Continued.

Finding the Mean The mean is the average of the numbers: 2.5 5.1 9.5 6.8

Median You may have noticed that a very low number or a very high number can affect the mean of a list of numbers. Because of this, you may sometimes want to use another measure of central tendency, called the median. The median of an ordered set of numbers is the middle number. If the number of items is even, the median is the mean (average) of the two middle numbers.

Finding the Median Find the median of the following list of numbers. 2.5 5.1 9.5 6.8 Continued.

Finding the Median List the numbers in numerical order: 2.5 5.1 6.8 9.5 Median

Helpful Hint In order to compute the median, the numbers must first be placed in order.

Mode The mode of a set of numbers is the number that occurs most often. (It is possible for a set of numbers to have more than one mode or to have no mode.)

Finding the Mode Find the mode of the following list of numbers. 2.5 5.1 9.5 6.8 Continued.

Finding the Mode The mode occurs the most often: 2.5 5.1 9.5 6.8 The mode is 2.5.

Helpful Hint Don’t forget that it is possible for a list of numbers to have no mode. For example, the list 2, 4, 5, 6, 8, 9 has no mode. There is no number or numbers that occur more often than the others.

Ratio, Proportion, and Triangle Applications
Chapter 6 Ratio, Proportion, and Triangle Applications

6.1 Ratio and Rates

Writing Ratios as Fractions A ratio is the quotient of two quantities. For example, a percent can be thought of as a ratio, since it is the quotient of a number and 100. 53% = or the ratio of 53 to 100

Ratio The ratio of a number a to a number b is their quotient. Ways of writing ratios are a b a to b, a : b, and

Writing Rates as Fractions A rate is a special kind of ratio. It is used to compare different kinds of quantities.

Finding Unit Rates To write a rate as a unit rate, divide the numerator of the rate by the denominator. 314.5 ÷ 17 = 18.5 The unit rate is

Finding Unit Prices When a unit rate is “money per item,” it is also called a unit price. A store charges $2.76 for a 12-ounce jar of pickles. What is the unit price? ($0.23 per ounce )

6.2 Proportions

A proportion is a statement that two ratios or rates are equal.
Solving Proportions A proportion is a statement that two ratios or rates are equal. If and are two ratios, then is a proportion. a b c d =

Solving Proportions A proportion contains four numbers. If any three numbers are known, the fourth number can be found by solving the proportion. To solve use cross products. bc a b c d = ad a b c d Multiply both sides by the LCD, bd These are called cross products. ad = bc Simplify cross product

Determining Whether Proportions are True
? True proportion

Finding Unknown Numbers in Proportions
Cross multiply. Simplify the left side. Divide both sides by 28. Check: (Rounded)

Proportions and Problem Solving
6.3 Proportions and Problem Solving

Solving Problems by Writing Proportions A 16-oz Cinnamon Mocha Iced Tea at a local coffee shop has 80 calories. How many calories are there in a 28-oz Cinnamon Mocha Iced Tea? Solve the proportion. Cross multiply. Simplify the right side. Divide both side by 140. A 28-oz Cinnamon Mocha Iced Tea has 140 calories.

Helpful Hint When writing proportions to solve problems, write the proportions so that the numerators have the same unit measures and the denominators have the same unit measures. For example,

Square Roots and the Pythagorean Theorem
6.4 Square Roots and the Pythagorean Theorem

The square of a number is the number times itself.
The square of 6 is 36 because 62 = 36. The square of –6 is also 36 because (–6)2 = (–6) (–6) = 36.

The reverse process of squaring is finding a square root.
Square Root of a Number The reverse process of squaring is finding a square root. A square root of 36 is 6 because 62 = 36. A square root of 36 is also –6 because (–6)2 = 36. We use the symbol , called a radical sign, to indicate the positive square root. because 42 = 16 and 4 is positive. because 52 = 25 and 5 is positive.

Square Root of a Number The square root, , of a positive number a is the positive number b whose square is a. In symbols,

Helpful Hint Remember that the radical sign is used to indicate the positive square root of a nonnegative number.

Perfect Squares Numbers like are called perfect squares because their square root is a whole number or a fraction.

Approximating Square Roots
A square root such as cannot be written as a whole number or a fraction since 6 is not a perfect square. It can be approximated by estimating by using a table or by using a calculator.

Right Triangles Chapter 1 / Whole Numbers and Introduction to Algebra One important application of square roots has to do with right triangles. A right triangle is a triangle in which one of the angles is a right angle or measures 90º (degrees). The hypotenuse of a right triangle is the side opposite the right angle. The legs of a right triangle are the other two sides. hypotenuse leg leg

Pythagorean Theorem If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then c a b In other words, (leg)2 + (other leg)2 = (hypotenuse)2.

Congruent and Similar Triangles
6.5 Congruent and Similar Triangles

Congruent Triangles Two triangles are congruent when they have the same shape and the same size. Corresponding angles are equal, and corresponding sides are equal. equal angles e = 11 a = 6 c = 11 d = 6 b = 9 f = 9 equal angles equal angles

Similar Triangles Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size.

Similar Triangles In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion. d = 6 a = 3 b = 5 e = 10 c = 8 f = 16 Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f.

Chapter 7 Percents

Percents, Decimals, and Fractions
7.1 Percents, Decimals, and Fractions

Understanding Percent The word percent comes from the Latin phrase per centum, which means “per 100.” Percent means per one hundred. The “%” symbol is used to denote percent.

Writing a Decimal as a Percent Multiply by 1 in the form of 100%. 0.65 = 0.65(100%) = 65.% or 65%

Writing a Percent as a Decimal Replace the percent symbol with its decimal equivalent, 0.01; then multiply. 43% = 43(0.01) = 0.43 100% = 100(0.01) = 1.00 or 1

Writing a Percent as a Fraction Replace the percent symbol with its fraction equivalent, ; then multiply. Don’t forget to simplify the fraction, if possible.

Writing a Fraction as a Percent Multiply by 1 in the form of 100%.

Helpful Hint We know that 100% = 1 Recall that when we multiply a number by 1, we are not changing the value of that number. Therefore, when we multiply a number by 100%, we are not changing its value but rather writing the number as an equivalent percent.

Summary To write a percent as a decimal, replace the % symbol with its decimal equivalent, 0.01; then multiply. To write a percent as a fraction, replace the % symbol with its fraction equivalent, ; then multiply. To write a decimal or fraction as a percent, multiply by 100%.

Solving Percent Problems with Equations
7.2 Solving Percent Problems with Equations

Key Words of means multiplication (∙) is means equals (=) what (or some equivalent) means the unknown number Let x stand for the unknown number.

Helpful Hint Remember that an equation is simply a mathematical statement that contains an equal sign (=). 6 = 18x equal sign

Solving Percent Problems 20% of = 10 20% • 50 = 10 percent base amount Percent Equation percent ∙ base = amount

Helpful Hint When solving a percent equation, write the percent as a decimal or fraction. If your unknown in the percent equation is a percent, don’t forget to convert your answer to a percent.

Helpful Hint Use the following to see if your answers are reasonable.
100% of a number = the number a percent greater than 100% a number larger than the original number = a percent less than 100% a number less than the original number =

Solving Percent Problems with Proportions
7.3 Solving Percent Problems with Proportions

Writing Percent Problems as Proportions To understand the proportion method, recall that 30% means the ratio of 30 to 100, or

Writing Percent Problems as Proportions Since the ratio is equal to the ratio , we have the proportion , called the percent proportion.

Percent Proportion always 100 or amount percent base

Symbols and Key Words When we translate percent problems to proportions, the percent can be identified by looking for the symbol % or the word percent. The base usually follows the word of. The amount is the part compared to the whole.

Helpful Hints Part of Proportion How It’s Identified Percent % or percent Base Appears after of Amount Part compared to whole

Solving Percent Proportions for the Amount
Chapter 1 / Whole Numbers and Introduction to Algebra Solving Percent Proportions for the Amount What number is 20% of 8? amount percent base amount percent base

Solving Percent Proportions for the Base
Chapter 1 / Whole Numbers and Introduction to Algebra Solving Percent Proportions for the Base 20 is 40% of what number? amount percent base amount percent base

Solving Percent Proportions for the Percent
What percent of is 8? percent base amount amount percent base Helpful Hint Recall from our percent proportion that this number, p already is a percent. Just keep the number the same and attach a % symbol.

Helpful Hint A ratio in a proportion may be simplified before solving the proportion. The unknown number in both and is 20.

Applications of Percent
7.4 Applications of Percent

Equation Method The freshman class of 450 students is 36% of all students at State College. How many students go to State College? Equation Method State the problem in words, then translate to an equation. In words: is 36% of what number? Translate: 450 = 36% • x Solve: = 0.36x

Proportion Method The freshman class of 450 students is 36% of all students at State College. How many students go to State College? Proportion Equation Method State the problem in words, then translate to a proportion. In words: is 36% of what number? amount percent base Translate and Solve:

Percent Increase Percent Decrease
In each case write the quotient as a percent. Helpful Hint Make sure that this number in the denominator is the original number and not the new number.

Percent and Problem Solving: Sales Tax, Commission, and Discount
7.5 Percent and Problem Solving: Sales Tax, Commission, and Discount

Calculating Sales Tax and Total Price Most states charge a tax on certain items when purchased called a sales tax. A 5% sales tax rate on a purchase of a $10.00 item gives a sales tax of sales tax = 5% of $10 = 0.05 ∙ $10.00 = $0.50

Sales Tax and Total Price The total price to the customer would be purchase price sales tax plus $10.00 + $0.50 = $10.50

Sales Tax and Total Price sales tax = tax rate ∙ purchase price total price = purchase price + sales tax

Calculating Commissions A wage is payment for performing work. An employee who is paid a commission as a wage is paid a percent of his or her total sales. commission = commission rate • sales

Discount and Sale Price` amount of discount = discount rate ∙ original price sale price = original price - amount of discount

Percent and Problem Solving: Interest
7.6 Percent and Problem Solving: Interest

Calculating Simple Interest Interest is money charged for using other people’s money. Money borrowed, loaned, or invested is called the principal amount, or simply principal. The interest rate is the percent used in computing the interest (usually per year). Simple interest is interest computed on the original principal.

Simple Interest simple Interest = Principal • Rate or I = P • R • T where the rate is understood to be per year and time is in years.

Finding the Total Amount of a Loan total amount (paid or received) = principal + interest

Calculating Compound Interest Compound interest is computed on not only the principal, but also on the interest already earned in previous compounding periods. If interest is compounded annually on an investment, this means that interest is added to the principal at the end of each year and next year’s interest is computed on this new amount.

Compound Interest Formula The total amount A in an account is given by where P is the principal, r is the interest rate written as a decimal, t is the length of time in years, and n is the number of times compounded per year.

Finding Total Amounts with Compound Interest total amount = original principal • compound interest factor The compound interest factor comes from the compound interest table found in Appendix C of the textbook.

Chapter 1 Review of Real Numbers.

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