Section 1.1: Introduction to Algebra: Variables and Mathematical ModelsWhen you are done with your homework you should be able to

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2 ELEMENTARY ALGEBRA WORKBOOK/FOR USE WITH ROBERT BLITZER S TEXTBOOK INTRODUCTORY AND INTERMEDIATE ALGEBRA FOR COLLEGE STUDENTS, 3RD ED. Section 1.1: Introduction to Algebra: Variables and Mathematical ModelsWhen you are done with your homework you should be able to Evaluate algebraic expressions Translate English phrases into algebraic expressions Determine whether a number is a solution of an equation Translate English sentences into algebraic equations Evaluate formulas WARM-UP: Perform the indicated operation and simplify EVALUATING ALGEBRAIC EXPRESSIONS We can a that appears in an by a. The is called the. A First Look at Order of Operations 1. Perform all operations, such as. 2. Do all in the in which they occur from to. 1

3 3. Do all and in the in which they from to. Example 1: Find the mistake! Example 2: Evaluate the following algebraic expressions at the given value(s): 1. 2 x 25 x 1, x x 9y 1, x 10, y 4 y x KEY WORDS FOR ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION ADDITION SUBTRACTION MULTIPLICATION DIVISION 2

4 Example 3: Write each English phrase as an algebraic expression. 1. Six more than a number 2. Twelve less a number 3. Two times the sum of a number and five increased by nine EQUATIONS An is a that two are. What symbol does an equation always contain? of an are of the that make the a statement. To determine whether a number is a, that number for the and each side of the equation. If the values on sides of the are the, the is a. Example 4: Determine whether the given number is a solution of the equation. 1. x 17 22; z 30; 8 3

5 Example 5: Write each equation as an English sentence x 7 2. x 2 5 x 4 Example 6: Write each sentence as an equation. 1. The difference between forty and a number is ten. 2. The product of six and a number increased by three is thirty-three. FORMULAS AND MATHEMATICAL MODELS One aim of is to provide a compact, description of the world. These descriptions involve the use of. A is an that expresses a between two or more. The process of finding formulas to describe phenomena is called. Such formulas, together with the assigned to the, are called. Example 7: A bowler s handicap, H, is often found using the following formula: H A, where A denotes the bowler s average score. 1. If your average bowling score is 145, what is your handicap? 2. What would your final score be if you bowled 120 in a game? 4

6 Section 1.2: FRACTIONS IN ALGEBRA When you are done with your homework you should be able to Convert between mixed numbers and improper fractions Write the prime factorization of a composite number Reduce or simplify fractions Multiply fractions Divide fractions Add and subtract fractions with identical denominators Add and subtract fractions with unlike denominators Solve problems involving fractions in algebra WARM-UP: Evaluate the following algebraic expressions at the given value(s): 2. 3x 8 5 x 1, x x 2y 5, x 0, y 2 VOCABULARY Numerator: The or expression that is written the bar. Denominator: The or expression that is written the bar. Natural Numbers: The that we use for. 5

7 Mixed Numbers: A number consists of the of a number and a, expressed the use of an. Improper Fractions: An is a fraction whose is than its. such as. CONVERTING A MIXED NUMBER TO AN IMPROPER FRACTION STEPS 1. the of the by the number and the to this. 2. Place the from step 1 the of the mixed number. 6

8 Example 1: Convert the following mixed numerals to improper fractions CONVERTING FROM AN IMPROPER FRACTION TO A MIXED NUMBER STEPS 1. the into the. Record the (result of the division) and the. 2. Write the number using the following form: Example 2: Convert the following improper fractions to mixed numerals

9 FACTORS AND PRIME FACTORIZATIONS Fractions can be by first the natural numbers that make up the and. To a natural number means to write it as two or more numbers being. VOCABULARY Prime number: A number is a number greater than 1 that has only and as. Composite numbers: A number is a number greater than 1 that is a. EVERY COMPOSITE NUMBER CAN BE EXPRESSED AS THE OF!!! Expressing a number as the of numbers is called the of that composite number. Example 3: Find the prime factorization of the following numbers

10 REDUCING FRACTIONS Two fractions are called if they represent the. Writing a fraction as an with a is called a. A fraction is to its when the and have other than. FUNDAMENTAL PRINCIPLE OF FRACTIONS The of a if both the and are (or ) by the nonzero. STEPS 1. Write the of the and the. 2. the and the by the (the product of all factors common to both). 9

11 Example 4: Reduce each fraction to its lowest terms MULTIPLYING FRACTIONS The of two or more is the of their divided by the of their. Example 5: Multiply and reduce each product to its lowest terms DIVIDING FRACTIONS The of two is the fraction by the of the fraction. 10

12 Example 6: Divide and reduce each quotient to its lowest terms ADDING AND SUBTRACTING FRACTIONS WITH IDENTICAL DENOMINATORS The or of two with is the sum or difference of their over the. and Example 7: Perform the indicated operations ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS The value of a fraction change if the and are by the nonzero. Example 8: Write 5 8 as an equivalent fraction with a denominator of

13 The least common denominator is the number that the numbers in each denominator into. STEPS FOR ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS 1. the fractions as with the. 2. or the, putting this result over the. USING PRIME FACTORIZATIONS TO FIND THE LCD 1. Find the of each. 2. The is obtained by using the number of times each occurs in factorization. Example 9: Perform the indicated operations

14 Example 10: Translate from English to an algebraic expression or equation. Let x represent the variable. 1. A number decreased by one third of itself. 2. The sum of one ninth of a number and one tenth of that number gives 15. APPLICATIONS Shown below is a line from the sheet music for An Irish Lullaby. The time is 2 4, which means that each measure must contain notes that add up to 2. Use vertical 4 lines to divide An Irish Lullaby. 13

15 Section 1.3: THE REAL NUMBERS When you are done with your homework you should be able to Define the sets that make up the real numbers Graph numbers on a number line Express rational numbers as decimals Classify numbers as belonging to one or more sets of the real numbers Understand and use inequality symbols Find the absolute value of a real number WARM-UP: Perform the indicated operation and simplify: NATURAL NUMBERS AND WHOLE NUMBERS A is a of objects whose contents can be clearly determined. The objects in a set are called the of the set. Natural numbers: The of numbers is Whole numbers: The of numbers is 14

16 INTEGERS AND THE NUMBER LINE The consisting of the numbers,, and the of the numbers is called the set of. Integers: The of is Example 1: Consider the following integers: 3, -3, 5, -5, 0 Graph each integer in the list on the same number line. RATIONAL NUMBERS If two are added, subtracted, or multiplied, the result is always another. Is this true when one integer is divided by another? The set of numbers is the set of all numbers that can be expressed in the form, where and are and is equal to ( ). The integer is called the and the integer is called the. 15

17 Are all integers rational numbers? Example 2: Consider the following rational numbers: 1, 9 2 4, -8, Graph each integer in the list on the same number line. Example 3: Divide RATIONAL NUMBERS AND DECIMALS Any number can be expressed as a. The resulting decimal will either ( ), or it will have a digit or block of digits that. 16

18 IRRATIONAL NUMBERS Any number that can be represented on the line that is a number is called an number. In other words, the set of irrational numbers is the set of numbers whose representations are neither nor. THE SET OF REAL NUMBERS All numbers that can be represented by on the number line are called numbers. THE SETS THAT MAKE UP THE REAL NUMBERS NAME DESCRIPTION EXAMPLES NATURAL NUMBERS WHOLE NUMBERS INTEGERS 17

19 RATIONAL NUMBERS IRRATIONAL NUMBERS Example 4: Consider the following set of numbers: 4 1,8,, 100,0,, List the numbers in the set that are 1. Natural numbers 2. Whole numbers 3. Integers 18

20 4. Rational numbers 5. Irrational numbers 6. Real numbers INEQUALITY SYMBOLS On the real number line, the numbers from to. The or two real numbers is the one farther to the on a number line. The of two real numbers is the one farther to the on a number line. NOTATION Example 5: Insert or between each pair of integers to make the statement true

21 ABSOLUTE VALUE The of a real number, denoted, is the from to on a number line. Is the output of an absolute value expression ever negative? Example 6: Find the absolute value: APPLICATIONS The table below shows the amount spent on ipad apps by Shannon s family during the months of May and July of Name Amount Shannon $48 Morgan $67 Rory $25 Erin $32 Nicole $12 1. Graph the five dollar amounts on a number line. 2. Write the names in order from the least spent on apps to the most spent on apps 20

22 Section 1.4: BASIC RULES OF ALGEBRA When you are done with your homework you should be able to Understand and use the vocabulary of algebraic expressions Use commutative properties Use associative properties Use distributive properties Combine like terms Simplify algebraic expressions WARM-UP: Perform the indicated operation and simplify: VOCABULARY OF ALGEBRAIC EXPRESSIONS Terms: The of an expression are those parts that are by or. A is a, a, or a by one or more. Coefficient: The part of a is called its. What is the coefficient of a term which only has variables? 21

23 Constant term: A term that consists of just a is called a. Like terms: Terms that have the the are called. Are constant terms like terms? Example 1: Consider the following algebraic expression: 12x 9 7x 8 1. How many terms are there in the algebraic expression? 2. What is the coefficient of the first term? 3. List the constant term(s): 4. What are the like terms in the algebraic expression? EQUIVALENT ALGEBRAIC EXPRESSIONS Two expressions that have the value for replacements are called. Example 2: Evaluate the following two algebraic expressions at x x 9 7x x 1 22

24 Constant term: A term that consists of just a is called a. Like terms: Terms that have the the are called. Are constant terms like terms? Example 1: Consider the following algebraic expression: 12x 9 7x 8 5. How many terms are there in the algebraic expression? 6. What is the coefficient of the first term? 7. List the constant term(s): 8. What are the like terms in the algebraic expression? EQUIVALENT ALGEBRAIC EXPRESSIONS Two expressions that have the value for replacements are called. Example 2: Evaluate the following two algebraic expressions at x x 9 7x x 1 23

25 THE COMMUTATIVE PROPERTIES Let a and b represent real numbers, variables, or algebraic expressions. Commutative Property of Addition: Changing when adding does not affect the. Commutative Property of Multiplication: Changing when multiplying does not affect the. Example 3: Use the commutative property to write an algebraic expression equivalent to each of the following: 1. 2x 4 2. x 13 THE ASSOCIATIVE PROPERTIES Let a, b, and c represent real numbers, variables, or algebraic expressions. Associative Property of Addition: Changing when adding does not affect the. 24

26 Associative Property of Multiplication: Changing when multiplying does not affect the. Example 4: Use the associative property to simplify the algebraic expressions: 1. 4x 7 x x THE DISTRIBUTIVE PROPERTY Let a, b, and c represent real numbers, variables, or algebraic expressions. Multiplication over. Example 5: Multiply: 1. 3 x x 25

27 OTHER FORMS OF THE DISTRIBUTIVE PROPERTY PROPERTY MEANING EXAMPLES a b c ab ac a b c d ab ac ad b c a ba ca 26

28 COMBINING LIKE TERMS The property lets us and like terms. Example 6: Combine like terms: 1. 34x x x x 5 2 x 11 3y STEPS FOR SIMPLIFYING ALGEBRAIC EXPRESSIONS 1. Use the property to remove. 2. Rearrange terms and terms using the and properties. As you hone your skills, you ll be doing this step mentally! 27

29 3. Combine terms by combining the of the and keeping the same. APPLICATIONS The percentage of U.S. women, W, who used the internet n years after 2000 can W 2 2n n 2. be modeled by the formula 1. Simplify the formula. 2. Use the simplified form of the mathematical model to find the percentage of U.S. women who used the internet in

30 Section 1.5: ADDITION OF REAL NUMBERS When you are done with your 1.5 homework you should be able to Add numbers with a number line Find sums using identity and inverse properties Add numbers without a number line Use addition rules to simplify algebraic expressions Solve applied problems using a series of addition WARM-UP: It is a Sunday during the fall semester. You are watching the Chargers/Raider game. The Chargers are currently on their own 30 yard line. During the first down that the Chargers have possession of the ball, the Chargers complete a pass for a gain of ten yards. On the second down, the Raiders sack the Chargers quarterback, causing a loss of 10 yards. a. Use the image of the football field shown below to model the gain and loss of yardage. b. How would you use signed numbers to represent the ten yard gain? c. How would you use signed numbers to represent the ten yard loss? 29

31 d. What is the net yardage gained? e. How can we use the number line below to model 10 + (-10)? f. Based on the information above, we can conclude that a number and its sum to. IDENTITY AND INVERSE PROPERTIES OF ADDITION PROPERTY MEANING EXAMPLES IDENTITY PROPERTY OF ADDITION INVERSE PROPERTY OF ADDITION 30

32 ADDING INTEGERS NUMBER LINE MODELS POSITIVE/NEGATIVE CHIP MODEL Example 1: Illustrate 4 2 using a. A number line b. The positive/negative chip method c. So the rule for a positive plus a positive is 31

33 Example 2: Illustrate 2 3 using a. A number line b. The positive/negative chip method c. So the rule for a negative plus a negative is Example 3: Illustrate 10 6 using a. A number line b. The positive/negative chip method c. So the rule for a negative plus a positive is 32

34 Example 4: Illustrate 1 7 using a. A number line b. The positive/negative chip method c. So the rule for a positive plus a negative is SIMPLIFYING ALGEBRAIC EXPRESSIONS Example 5: Simplify. b. 6y 22y 5 a. 9 x 4x 33

35 APPLICATIONS Example 6: Example 7: In high school, an elementary algebra class meets five hours a week for nine months. At MiraCosta College, an elementary algebra class meets five hours a week for 4 months. The class at MiraCosta College has how many fewer in-class hours? 34

36 MIXED PRACTICE 1. Fill in the blank. a. 5 is the of -5. b. On a number line, the greater number is to the of the lesser number. c. A number s distance from zero on a number line is the number s. d. Numbers less than zero are called numbers. e. When using an inequality symbol, the arrow points towards the number. 2. Add a. c b d

37 Section 1.6: SUBTRACTION OF REAL NUMBERS When you are done with your homework you should be able to Subtract real numbers Simplify a series of additions and subtractions Use the definition of subtraction to identify terms Use the subtraction definition to simplify algebraic expressions Solve problems involving subtraction WARM-UP: Simplify: x x 2. x 5 325x 1 DEFINITION OF SUBTRACTION For all real numbers a and b, To subtract from, the (or additive inverse) of to. The result of subtraction is called the. 36

38 A PROCEDURE FOR SUBTRACTING REAL NUMBERS 1. Change the subtraction operation to. 2. Change the of the number being. 3.. Example 1: Subtract Example 2: Simplify x 2x x x 5 2 x 11 3y 37

39 APPLICATIONS The line graph below illustrates the value of Sarah s car in dollars from the year 2001 to the year How much was Sarah s car worth in 2005? 2. How much more was Sarah s car worth in 2002? 38

40 Section 1.7: MULTIPLICATION AND DIVISION OF REAL NUMBERS When you are done with your homework you should be able to Multiply real numbers Multiply more than two real numbers Find multiplicative inverses Use the definition of division Divide real numbers Simplify algebraic expressions involving multiplication Determine whether a number is a solution of an equation Use mathematical models involving multiplication and division WARM-UP: 1. Find the value of each expression: a b Write each English phrase as an algebraic expression. Let x represent the number: a. The difference between 9 times a number and -4 times a number b. The quotient of -7 and a number subtracted from the quotient of -12 and a number 39

41 THE PRODUCT OF TWO REAL NUMBERS The of two real numbers with signs is found by their values. The product is. The of two real numbers with the sign is found by their values. The product is. The of zero and any real number is. Example 1: Multiply

42 MULTIPLYING MORE THAN TWO NUMBERS 1. Assuming that no factor is zero, The of an number of numbers is. The of an number of numbers is. 2. If any is, the product is. Example 2: Multiply THE MEANING OF DIVISION The result of the real number by the nonzero real number is called the of and. We can write this as or. We can define division in terms of by using inverse or. 41

43 Example 3: Find the multiplicative inverse of each number DEFINITION OF DIVISION If a and b are real numbers and b is not equal to zero, then the of and is defined as The of two real numbers is the of the number and the of the number. Example 4: Divide using the definition of division

44 THE QUOTIENT OF TWO REAL NUMBERS The of two real numbers with signs is found by their values. The quotient is. The of two real numbers with the sign is found by their values. The quotient is. Division of any real number by is. Any nonzero number divided into is. Example 5: Divide

45 ADDITIONAL PROPERTIES OF MULTIPLICATION PROPERTY MEANING EXAMPLES IDENTITY PROPERTY OF MULTIPLICATION INVERSE PROPERTY OF MULTIPLICATION MULTIPLICATION PROPERTY OF -1 DOUBLE NEGATIVE PROPERTY 44

46 NEGATIVE SIGNS AND PARENTHESIS If a sign precedes parentheses, the parentheses and the of within the parentheses. Example 6: Simplify x y 1 14 y 2 APPLICATIONS 5 Use the formula C F 32 to express each Fahrenheit temperature, F, as its 9 equivalent Celsius temperature, C F 2. 5 F 45

47 Section 1.8: EXPONENTS AND ORDER OF OPERATIONS When you are done with your homework you should be able to Evaluate exponential expressions Simplify algebraic expressions with exponents Use the order of operations agreement Evaluate mathematical models WARM-UP: 1. Determine whether the given number is a solution of the equation. 5m 1 3m 2 ; Write a numerical expression for each phrase. Then simplify the numerical expression. a. 14 added to the product of 4 and -10 b. The quotient of -18 and the sum of -15 and 12 46

48 DEFINITION OF A NATURAL NUMBER EXPONENT If b is a real number and n is a natural number, is read the of or to the power. The expression is called an. Example 1: Evaluate ORDER OF OPERATIONS 1. Perform all within symbols 2. Evaluate all expressions. 3. Do all and in the order in which they occur, working from to. 4. Finally, do all and using one of the following procedures: Work from to and do additions and subtractions in the in which they occur. 47

49 or Rewrite subtractions as of. Combine and numbers separately, and then these results. Example 2: Simplify Example 3: Simplify each algebraic expression. 1. 6x 18x x x y 4 48

50 APPLICATIONS In Palo Alto, CA, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers). The 200x mathematical model C describes the cost, C, in tens of thousands of 100 x dollars, for removing x percent of the contaminants. 1. Find the cost, in tens of thousands of dollars, for removing 50% of the contaminants. 2. Find the cost, in tens of thousands of dollars, for removing 60% of the contaminants. 3. Describe what is happening to the cost of the cleanup as the percentage of contaminant removed increases. 49

51 Section 2.1: THE ADDITION PROPERTY OF EQUALITY When you are done with your homework you should be able to Identify linear equations in one variable Use the addition property of equality to solve equations Solve applied problems using formulas WARM-UP: Simplify: LINEAR EQUATIONS IN ONE VARIABLE In Chapter 1, we learned that an is a statement that two expressions are.we determined whether a given number is an equation s by substituting that number for each occurrence of the. When the resulted in a true statement, that was a. When the substituted number resulted in a statement, that number was a. 50

52 VOCABULARY Solving an equation: The of finding the (or ) that make the equation a statement. These numbers are called the or of the equation, and we say that they the equation. DEFINITION OF A LINEAR EQUATION IN ONE VARIABLE A in is an equation that can be written in the form where,, and are real numbers, and. Example 1: Give three examples of a linear equation in one variable Example 2: Give two examples of a nonlinear equation in one variable

53 VOCABULARY Equivalent equations: Equations that have the solution are. THE ADDITION PROPERTY OF EQUALITY The real number or expression may be to sides of an without changing the equation s. That is, Example 3: Solve the following equations. Check your solutions. 1. y x z x 5 4x 9 x 6. x x

54 ADDING AND SUBTRACTING VARIABLE TERMS ON BOTH SIDES OF AN EQUATION Our goal is to all the terms on one side of the equation. We can use the of to do this. APPLICATIONS 1. The cost, C, of an item (the price paid by a retailer) plus the markup, M, on that item (the retailer s profit) equals the selling price, S, of the item. The formula is C M S. The selling price of a television is $650. If the cost to the retailer for the television is $520, find the markup. 2. What is the difference between solving an equation such as 5y 3 4y and simplifying an algebraic expression such as 5y 3 4y 8? 53

55 Section 2.2: THE MULTIPLICATION PROPERTY OF EQUALITY When you are done with your homework you should be able to Use the multiplication property of equality to solve equations Solve equations in the form of x c Use the addition and multiplication properties to solve equations Solve applied problems using formulas WARM-UP: Solve: 2. x x 1. 5z 12 z THE MULTIPLICATION PROPERTY OF EQUALITY The real number or expression may sides of an without changing the. That is, Example 1: Solve the following equations. Check your solutions. 1. 5z x 54

56 2. 51 y 5. 6z 3 z x 3x y 6 3y 6 APPLICATIONS The formula n M models your distance, M, from a lightning strike in a 5 thunderstorm if it takes n seconds to hear thunder after seeing the lightning. If you are three miles away from the lightning flash, how long will it take the sound of thunder to reach you? 55

57 Section 2.3: SOLVING LINEAR EQUATIONS When you are done with your homework you should be able to Solve linear equations Solve linear equations containing fractions Identify equations with no solution or infinitely many solutions Solve applied problems using formulas WARM-UP: Solve: 1. 12z x 7x 24 A STEP-BY-STEP PROCEDURE FOR SOLVING LINEAR EQUATIONS 1. the on each side. 2. Collect all the terms on one side and all the terms on the other side. 3. the and. 4. the proposed solution in the equation. Example 1: Solve the following equations. Check your solutions. 4. x 1. z 34 10z 2 10z x 30 56

58 x 5. 2 x 15 3x 6 4x 9x x 4 x 9 2x x 1 4x 6 LINEAR EQUATIONS WITH FRACTIONS Equations are to solve when they do not contain. To remove fractions, we can sides of the equation by the of any fractions in the equation. Remember the is the number that all will into. This is often called an equation of. 57

59 Example 2: Solve the following equations. Clear the fractions first. Check your solutions. x y z 1 z x 2 x

60 RECOGNIZING INCONSISTENT EQUATIONS AND IDENTITIES If you attempt to an equation with or one that is for real number, you will the. An equation with results in a statement, such as. An that is for real number results in a statement, such as. Example 3: Solve the following equations. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. Check your solutions. 1. x 2 5 2x x 2x 3 x x x x 5 3x 7 2 x 1 59

61 APPLICATIONS The formula 5d p 15 describes the pressure of sea water, p, in pounds per 11 square foot, at a depth of d feet below the surface. 1. The record depth for breath-held diving, by Francisco Ferreras (Cuba) off Grand Bahama Island, on November 14, 1993, involved pressure of 201 pounds per square foot. To what depth did Francisco descend on this venture? (He was underwater for 2 minutes and 9 seconds!) 2. At what depth is the pressure 20 pounds per square foot? 60

62 Section 2.4: FORMULAS AND PERCENTS When you are done with your homework you should be able to Solve a formula for a variable Express a percent as a decimal Express a decimal as a percent Use the percent formula Solve applied problems involving percent change WARM-UP: Solve: B P 26 SOLVING A FORMULA FOR ONE OF ITS VARIABLES Solving a formula for a variable means the so that the is on one side of the equation. To solve a formula for one of its variables, treat that as if it were the only in the. PERIMETER The of a figure is the of the of its. Perimeter is measured in units, such as,,, or. 61

63 PERIMETER OF A RECTANGLE The perimeter,, of a rectangle with length and width is given by the formula SQUARE UNITS A unit is a, each of whose sides is unit in length. The of a figure is the number of it takes to fill the interior of the figure. AREA OF A RECTANGLE The area,, of a rectangle with length and width is given by the formula Example 1: Solve the following formulas for the specified variable. 1. d rt; t 2. P C MC; C 62

64 Example 2: Consider a rectangle which has an area of 15 square feet and a width of 3 feet. 1. Find the length. 2. Find the perimeter. BASICS OF PERCENTS are the result of numbers as of. The word means. PERCENT NOTATION means. STEPS FOR EXPRESSING A PERCENT AS A DECIMAL NUMBER 1. Move the point places to the. 2. Remove the sign. Example 3: Express each percent as a decimal % % 63

65 STEPS FOR EXPRESSING A DECIMAL NUMBER AS A PERCENT 1. Move the point places to the. 2. Attach a sign. Example 4: Express each decimal as a percent A FORMULA INVOLVING PERCENT are useful in comparing two. To the number to the number using a percent, the following formula is used: Example 5: Solve. 1. What is 12% of 50? 2. 6 is 30% of what? is what percent of 20? 64

66 PERCENT INCREASE PERCENT DECREASE APPLICATIONS 1. The average, or mean, A, of four exam grades, x, y, z, and w, is given by the formula A x y z w 4. a. Solve the formula for w. b. Use the formula in part (a) to solve this problem: On your first three exams, your grades are 76%, 78%, and 79%: x 76, y 78, and z 79. What must you get on the fourth exam to have an average of 80%? 65

67 2. A charity has raised $225,000, with a goal of raising $500,000. What percent of the goal has been raised? 3. Suppose that the local sales tax rate is 7% and you buy a graphing calculator for $96. a. How much tax is due? b. What is the calculator s total cost? 4. The price of a color printer is reduced by 30% of its original price. When it still does not sell, its price is reduced by 20% of the reduced price. The salesperson informs you that there has been a total reduction of 50%. Is the salesperson using percentages properly? If not, what is the actual percent reduction from the original price? 66

68 Section 2.5: AN INTRODUCTION TO PROBLEM SOLVING When you are done with your homework you should be able to Translate English phrases into algebraic expressions Solve algebraic word problems using linear equations WARM-UP: Solve: A fax machine regularly sells for $380. The sale price is $266. Find the percent decrease in the machine s price. STEPS FOR SOLVING WORD PROBLEMS 1. Analysis: READ the problem. Then, the problem again!!! Draw a and/or make a. Identify and name all known and unknown. 2. Translate to Mathese: Write an equation that translates, or, the conditions of the problem. 3. Solve: the equation. Then your solution. 4. Conclusion: Write your result, in. 67

69 Example 1: Solve the following word problems. 1. The sum of a number and 28 is 245. Find the number. 2. Three times the sum of five and a number is 48. Find the number. 3. Eight subtracted from six times a number is 298. Find the number. 68

70 4. If the quotient of three times a number and four is decreased by three, the result is nine. Find the number. 5. A car rental agency charges $180 per week plus $0.25 per mile to rent a car. How many miles can you travel in one week for $395? 69

71 6. A basketball court is a rectangle with a perimeter of 86 meters. The length is 13 meters more than the width. Find the width and length of the basketball court. 7. This year s salary, $42,074, is a 9% increase over last year s salary. What was last year s salary? 70

72 8. A repair bill on a sailboat came to $1603, including $532 for parts and the remainder for labor. If the cost of labor is $35 per hour, how many hours of labor did it take to repair the sailboat? 71

73 Section 2.6: PROBLEM SOLVING IN GEOMETRY When you are done with your homework you should be able to Solve problems using formulas for perimeter and area Solve problems using formulas for a circle s area and circumference Solve problems using formulas for volume Solve problems involving the angles of a triangle Solve problems involving complementary and supplementary angles WARM-UP: Solve: After a 30% reduction, you purchase a DVD player for $98. What was the selling price before the reduction? COMMON FORMULAS FOR PERIMETER AND AREA 72

74 Example 1: Solve. 1. A triangle has a base of 6 feet and an area of 30 square feet. Find the triangle s height. 2. A rectangle has a width of 46 cm and a perimeter of 208 cm. What is the rectangle s length? 3. Find the area of the trapezoid. 5 m 4 m 4.5 m 8 m 73

75 GEOMETRIC FORMULAS FOR CIRCUMFERENCE AND AREA OF A CIRCLE A is the set of all in the equally distant from a given point, its. A (plural ),, is a line from the to any point on the. For a given circle, radii have the same. A,, is a segment through the whose endpoints both lie on the. For a given circle, all have the length. In any circle, the length of a is the length of a and the length of a is the length of a. Area Circumference 74

76 Example 2: Solve. 1. Find the area and circumference of a circle which has a diameter of 40 feet. 2. Which one of the following is a better buy: a large pizza with a 16-inch diameter for $12 or two small pizzas, each with a 10-inch diameter, for $12? GEOMETRIC FORMULAS FOR VOLUME refers to the amount of occupied by a - figure. To measure this space, we use units. 75

77 Example 3: Solve. 1. Solve the formula for the volume of a cone for h. 76

78 2. A cylinder with radius 2 inches and height 3 inches has its radius quadrupled. How many times greater is the volume of the larger cylinder than the smaller cylinder? 3. Find the volume of a shoebox with dimensions 6 in x 12 in x 5 in. THE ANGLES OF TRIANGLES An, symbolized by, is made up of two that have a common. The common endpoint is called the. The two rays that form the angle are called its. 77

79 One way to angles is in, symbolized by a small, raised. There are in a circle. is of a complete rotation. THE ANGLES OF A TRIANGLE The of the of the three angles of triangle is. COMPLEMENTARY AND SUPPLEMENTARY ANGLES Two angles with measures having a of are called angles. Two angles with measures having a of are called. 78

80 Example 4: Solve. 1. One angle of a triangle is three times as large as another. The measure of the third angle is 40º more than that of the smallest angle. Find the measure of each angle. 2. Find the measure of the complement of each angle. a. 56º b. 89.5º 3. Find the measure of the supplement of each angle. a. 177º b. 0.2º 4. Find the measure of the angle described. The measure of the angle s supplement is 52º more than twice that of its complement. 79

81 Example 5: Find the area of the shaded region. 2 in 12 in 80

82 Section 2.7: SOLVING LINEAR INEQUALITIES When you are done with your homework you should be able to Graph the solutions of an inequality on a number line Use interval notation Understand properties used to solve linear inequalities Solve linear inequalities Identify inequalities with no solution of infinitely many solutions Solve problems using linear inequalities WARM-UP: Solve: Find the volume of a sphere with diameter 11 meters. VOCABULARY Linear inequality in one variable: An inequality in the form,,, or is a linear inequality in one variable. means, means or, means, and means or. 81

83 Solving an inequality: The of finding the of that will make the inequality a statement. These numbers are called the solutions of the, and we say they satisfy the. The of solutions is called the solution set of the inequality. GRAPHS OF INEQUALITIES There are solutions to the inequality x 5. In other words, the solution set for this inequality is all numbers which are. Can we list all these numbers? What does the graph of the solution set look like? Hmmmm Graphs of to are shown on a by shading representing numbers that are.,, indicate that are and,, indicate that are. Example 1: Graph the solutions of each inequality. a. x 6 82

84 b. 3 x 2 c. 3 x 6 2 SOLUTION SETS OF INEQUALITIES INEQUALITY INTERVAL NOTATION SET-BUILDER NOTATION GRAPH x a x a x b x b a x b a x b a x b a x b PARENTHESIS ARE ALWAYS USED WITH OR!!! 83

85 PROPERTIES OF INEQUALITIES PROPERTY THE ADDITION PROPERTY OF INEQUALITY If, then. If, then. THE PROPERTY IN WORDS EXAMPLE THE POSITIVE MULTIPLICATION PROPERTY OF INEQUALITY If and is positive, then. If and is positive, then. THE NEGATIVE PROPERTY OF INEQUALITY If and is negative, then. If and is negative, then. 84

86 STEPS FOR SOLVING A LINEAR INEQUALITY 1. Simplify the on each side. 2. Use the property of to collect all the terms on one side and all the terms on the other side. 3. Use the property of to the and. the of the when or both sides by a number. 4. Express the set in or - notation, and the solution set on a line. Example 2: Solve each inequality and graph the solution. a. x

87 b. 5x 8 2x 7 c. x 4 1 3x 6 RECOGNIZING INEQUALITIES WITH NO SOLUTION OR INFINITELY MANY SOLUTIONS If you attempt to solve an inequality with or one that is for number, you will the. An inequality with results in a statement, such as. The solution set is or, the set, and the is an number line. 86

88 An inequality that is for number results in a statement, such as. The solution set is or, and the graph is a line. Example 3: Solve each inequality and graph the solution. a. x x 1 5x 2 x 7 3x b. 87

89 APPLICATION On three examinations, you have grades of 88, 78, and 86. There is still a final examination, which counts as one grade. 1. In order to get an A, your average must be at least 90. If you get 100 on the final, compute your average and determine if an A in the course is possible. 2. To earn a B in the course, you must have a final average of at least 80. What must you get on the final to earn a B in the course? 88

90 Section 3.1: GRAPHING LINEAR EQUATIONS IN TWO VARIABLES When you are done with your homework you should be able to Plot ordered pairs in the rectangular coordinate system Find coordinates of points in the rectangular coordinate system Determine whether an ordered pair is a solution of an equation Find solutions of an equation in two variables Use point plotting to graph linear equations Use graphs of linear equations to solve problems WARM-UP: 1. Find the volume of a box with dimensions ½ ft by 3 ft by 8 ft. 2. Solve the following inequalities and graph the solution sets. x 6 3x 5 a. b. 2x 1 2x 89

91 POINTS AND ORDERED PAIRS The idea of visualizing equations as geometric figures was developed by the French philosopher and mathematician. This idea is the system or the coordinate system. The rectangular coordinate system consists of lines that at right at their points. The horizontal number line is the and the vertical number line is the. The point of intersection is a called the. Positive numbers are to the and the origin. Negative numbers are to the and the origin. The divide the into regions, called. The points located on the are in any quadrant. Each in the rectangular coordinate system to an of real numbers,. The number in each pair, called the, denotes the and from the along the. The second number, called the, denoted the distance along a to the or along the itself. 90

92 Example 1: Plot the following ordered pairs. (2,5) (-3,7) (-2,-4) 2,5, 3, 7, 2, 4 SOLUTIONS OF EQUATIONS IN TWO VARIABLES A of an in variables, and, is an of real numbers with the following property: When the is substituted for and the is substituted for in the equation, we obtain a statement. Example 2: Determine whether each of the given points is a solution of the equation 8x y 1. a. 0,1 b. 1,3 c. 2, 15 91

93 Example 3: Find three solutions of 2y x 1. GRAPHING LINEAR EQUATIONS IN THE FORM y mx b The of the is the of all whose satisfy the equation. STEPS FOR USING THE POINT-PLOTTING METHOD FOR GRAPHING AN EQUATION IN TWO VARIABLES 1. Find several that are of the equation. 2. Plot these ordered pairs as in the coordinate system. 3. the points with a curve or, depending on the type of equation. Example 3: Graph the following equations by plotting points. a. y 2x x y 2x x, y 92

94 b. y 3x 9 x y 3x 9 x, y c. y 2 x 3 5 x 2 y x 3 x, y 5 93

95 COMPARING GRAPHS OF LINEAR EQUATIONS If the value of does not change, The graph of is the graph of shifted units when is a positive number. The graph of is the graph of shifted units when is a positive number. APPLICATION In 1960, per capita fish consumption was 10 pounds. This increased by approximately 0.15 pound per year from 1960 through These conditions can be described by the mathematical model F 0.15n 10, where F is per capita fish consumption n years after a. Let n = 0, 10, 20, 30, and 40. Make a table of values showing five solutions of the equation. n F 0.15n 10 n, F 94

96 b. Graph the formula in a rectangular coordinate system. c. Use the graph to estimate per capita fish consumption in d. Use the formula to project per capita fish consumption in

97 Section 3.2: GRAPHING LINEAR EQUATIONS USING INTERCEPTS When you are done with your homework you should be able to Use a graph to identify intercepts Graph a linear equation in two variables using intercepts Graph horizontal or vertical lines WARM-UP: Graph the following equations by plotting points. a. y x x y x x, y 2 b. y x 7 3 x 2 y x 7 x, y 3 96

98 INTERCEPTS An of a graph is the of a point where the graph the. The corresponding to an is always!!! A of a graph is the of a point where the graph the. The corresponding to a is always!!! Example 1: Use the graph to identify the a. x-intercept b. y-intercept GRAPHING USING INTERCEPTS An equation of the form, where,, and are integers, is called the form of a line. 97

99 STEPS FOR USING INTERCEPTS TO GRAPH Ax By C 1. Find the. Let and solve for. 2. Find the. Let and solve for. 3. Find a checkpoint, a ordered-pair. 4. Graph the equation by drawing a through the points. Example 2: Graph using intercepts and a checkpoint. a. x y 6 x x y 6 x, y b. 3x 2y 7 x 3x 2y 7 x, y 98

100 EQUATIONS OF HORIZONTAL AND VERTICAL LINES We know that the graph of any equation of the form is a as long as and are not both. What happens if or, but not both, is zero? HORIZONTAL AND VERTICAL LINES The graph of is a line. The is. The graph of is a line. The is. 99

101 Example 3: Graph. a. y 8 b. 12x 60 Example 4: Write an equation for each graph. a. b. 100

102 APPLICATION A new car worth $24,000 is depreciating in value by $3000 per year. The mathematical model y 3000x describes the car s value, y, in dollars, after x years. a. Find the x-intercept. Describe what this means in terms of the car s value. b. Find the y-intercept. Describe what this means in terms of the car s value. c. Use the intercepts to graph the linear equation. d. Use your graph to estimate the car s value after five years. 101

103 Section 3.3: SLOPE When you are done with your homework you should be able to Compute a line s slope Use slope to show that lines are parallel Use slope to show that lines are perpendicular Calculate rate of change in applied situations WARM-UP: Graph each equation. a. y 2 0 b. 2x 3y 9 x 2x 3y 9 x, y 102

104 THE SLOPE OF A LINE Mathematicians have developed a useful of the of a line, called the of the line. Slope compares the change (the ) to the change (the ) when moving from one point to another along the line. DEFINITION OF SLOPE The of the line through the distinct points and is where. It is common to use the letter to represent the slope of a line. This letter is used because it is the first letter of the French verb monter, meaning to rise, or to ascend. Example 1: Find the slope of the line passing through each pair of points: a. 1, 4 and 3, 6 b , and,

105 Example 2: Use the graph to find the slope of the line POSSIBILITIES FOR A LINE S SLOPE POSITIVE SLOPE NEGATIVE SLOPE ZERO SLOPE UNDEFINED SLOPE 104

106 SLOPE AND PARALLEL LINES Two lines that lie in the same plane are. If two lines do not, the of the change to the change is the for each. Because two parallel lines have the same, they must have the same. 1. If two nonvertical lines are, then they have the same. 2. If two distinct nonvertical lines have the same, then they are. 3. Two distinct vertical lines, each with slope, are. SLOPE AND PERPENDICULAR LINES Two lines that at a ( ) are said to be. 1. If two nonvertical lines are, then the of their is. 2. If the of the of two lines is, then the lines are. 105

107 3. A line having slope is to a vertical line having slope. Example 3: Determine whether the lines through each pair of points are parallel, perpendicular, or neither. a. 2, 15 and 0, 3 ; 12, 6 and 6,3 b. 2, 7 and 3,13 ; 1, 9 and 5,15 c. 1, 11 and 0, 5 ; 0, 8 and 12, 6 106

108 APPLICATION Construction laws are very specific when it comes to access ramps for the disabled. Every vertical rise of 1 foot requires a horizontal run of 12 feet. What is the grade of such a ramp? Round to the nearest tenth of a percent. 107

109 Section 3.4: THE SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE When you are done with your homework you should be able to Find a line s slope and y-intercept from its equation Graph lines in slope-intercept form Use slope and y-intercept to graph Ax By C Use slope and y-intercept to model data WARM-UP: Graph each equation. a. 4x 8y 2 0 x 4x 8y 2 0 x, y b. The line which passes through the points 1,2 and 3,0. 108

110 SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE The - form of the of a nonvertical line with slope and is Example 1: Find the slope and the y-intercept of the line with the given equation: a. y 4x 1 b. 6x y 1 c. 5 y x 2 7 d. x 2 y 3 3 Example 2: Use the graph to find the equation of the line in slope-intercept form. 109

111 GRAPHING BY USING y mx b SLOPE AND Y-INTERCEPT 1. Plot the point containing the on the axis. This is the point. 2. Obtain a second using the,. Write as a, and use over, starting at the. 3. Use a to draw a through the two. Draw at the of the line to show that the line continues in both directions. Example 3: Graph using the slope and y-intercept. a. y 5x 3 110

112 b. 10x 5y 25 c. x 2y 3 111

113 d. y x 1 e. 6 y x

114 APPLICATION Write an equation in the form of y mx b of the line that is described. 1. The y-intercept is -4 and the line is parallel to the line whose equation is 2x y The line falls from left to right. It passes through the origin and a second point with opposite x- and y-coordinates. 113

115 Section 3.5: THE POINT-SLOPE FORM OF THE EQUATION OF A LINE When you are done with your homework you should be able to Use the point-slope form to write equations of a line Find slopes and equations of parallel and perpendicular lines Write linear equations that model data and make predictions WARM-UP: 1. Simplify x 2 2. Graph the equation using the slope and y-intercept. x y

116 POINT-SLOPE FORM We can use the of a line to obtain another useful form of the line s equation. Consider a nonvertical line that has slope and contains the point. Now let represent any other on the. Keep in mind that the point is and is in position. The point is. POINT-SLOPE FORM OF THE EQUATION OF A LINE The - form of the of a nonvertical line with slope that passes through the point is Example 1: Write the point-slope form of the equation of the line with the given slope that passes through the given point. a. m 2; 5, 11 b. 5 1 m ;,

117 c. m 0; 21,5 d. m undefined; 0, 0 Example 2: Use the graph to find two equations of the line in point-slope form Now write the slope-intercept form:

118 EQUATIONS OF LINES FORM Standard Form WHAT YOU SHOULD KNOW Graph equations in this form using and a. y x b Graph equations in this form as lines with as the. a Graph equations in this form as lines with as the. Slope-Intercept Form Graph equations in this form using the, and the slope,. *Start with this form when writing a equation if you know a line s and. Point-Slope Form Start with this form when writing a linear equation if you know the of the line and a on the NOT containing the OR points on the line, of which contains the. Calculate the using 117

119 PARALLEL AND PERPENDICULAR LINES Recall that parallel lines have the and perpendicular lines have which are. Example 3: Use the given conditions to write an equation for each line in pointslope form and slope-intercept form. a. Passing through 2, 7 and parallel to the line whose equation is y 5x 4. b. Passing through 4,2 and perpendicular to the line whose equation is 1 y x 7. 3 c. Passing through 5, 9 and parallel to the line whose equation is x 7 y

120 Section 4.1: SOLVING SYSTEMS OF LINEAR EQUATIONS BY GRAPHING When you are done with your homework you should be able to Decide whether an ordered pair is a solution of a linear system Solve systems of linear equations by graphing Use graphing to identify systems with no solution or infinitely many solutions Use graphs of linear systems to solve problems WARM-UP: 1. Determine if the given number or ordered pair is a solution to the given equation. 18 a. 5x 3 21; b. x 2y 0; 4, Graph the line which passes through the points (0,1) and (-5,3). 119

121 SYSTEMS OF LINEAR EQUATIONS AND THEIR SOLUTIONS We have seen that all in the form are straight when graphed. such equations are called a of or a. A to a system of two equations in two is an that equations in the. Example 1: Determine whether the given ordered pair is a solution of the system. a. 2, 5 6x 2y 2 3 x y 11 b. 10,7 6 x 5y 25 4x 15 y 13 SOLVING LINEAR SYSTEMS BY GRAPHING The of a of two linear equations in variables can be found by of the in the rectangular system. For a system with solution, the of the point of give the solution. 120

122 STEPS FOR SOLVING SYSTEMS OF TWO LINEAR EQUATIONS IN TWO VARIABLES, x AND y, BY GRAPHING 1. Graph the first. 2. the second equation on the set of. 3. If the representing the graphs at a, determine the of this point of intersection. The is the of the. 4. the in equations. Example 2: Use the graph below to find the solution of the system of linear equations. 121

123 Example 3: Solve each system by graphing. Use set notation to express solution sets. a. x x y 2 y 4 b. y 3x 4 y 2x 1 c. x y 6 y 3 122

124 LINEAR SYSTEMS HAVING NO SOLUTION OR INFINITELY MANY SOLUTIONS We have seen that a of linear equations in variables represents a of. The lines either at point, are, or are. Thus, there are possibilities for the of solutions to a system of two linear equations. THE NUMBER OF SOLUTIONS TO A SYSTEM OF TWO LINEAR EQUATIONS NUMBER OF SOLUTIONS Exactly ordered pair solution. WHAT THIS MEANS GRAPHICALLY The two lines at point. This is a system. Solution The two lines are. This is an system. many solutions The two lines are. This is a system with equations. 123

125 Example 4: Solve each system by graphing. If there is no solution or infinitely many solutions, so state. Use set notation to express solution sets. a. x y 4 2x 2 y 8 b. y 3x 1 y 3x 2 c. 2x y 0 y 2x 124

126 APPLICATION A band plans to record a demo. Studio A rents for $100 plus $50 per hour. Studio B rents for $50 plus $75 per hour. The total cost, y, in dollars, of renting the studios for x hours can be modeled by the linear system y y 50x x 50 a. Use graphing to solve the system. Extend the x-axis from 0 to 4 and let each tick mark represent 1 unit (one hour in a recording studio). Extend the y-axis from 0 to 400 and let each tick mark represent 100 units (a rental cost of $100). b. Interpret the coordinates of the solution in practical terms. 125

127 When you are done with your 4.2 homework you should be able to Solve linear systems by the substitution method Use the substitution method to identify systems with no solution or infinitely many solutions Solve problems using the substitution method WARM-UP: 1. Solve. 5x 3 2x 7 x Solve the following system of linear equations by graphing. State whether the system is consistent or inconsistent. For those systems that are consistent, state whether the equations are dependent or independent. y 4x 6 y 2x 126