Principles of Algebra

Transcription

1 Principles of lgebra 1 Expressions and Integers 1-1 Evaluating lgebraic Expressions 1-2 Writing lgebraic Expressions 1- Integers and bsolute Value 1-4 dding Integers 1-5 Subtracting Integers 1-6 Multiplying and Dividing Integers 1B Solving Equations 1-7 Solving Equations by dding or Subtracting 1-8 Solving Equations by Multiplying or Dividing LB Model Two-Step Equations 1-9 Solving Two-Step Equations KEYWORD: MT8C Ch1 Firefighters can use algebra to find out how fast a fire is spreading and to create a plan to stop it. 2 Chapter 1

2 Vocabulary Choose the best term from the list to complete each sentence. 1.? is the? of addition. 2. The expressions 4 and 4 are equal by the?.. The expressions 1 (2 ) and (1 2) are equal by the?. 4. Multiplication and? are opposite operations. 5.? and? are commutative. addition ssociative Property Commutative Property division multiplication opposite operation subtraction Complete these exercises to review skills you will need for this chapter. Whole Number Operations Simplify each expression Compare and Order Whole Numbers Order each sequence of numbers from least to greatest ; 11,500; 105; ; 5; 500; ,400; 40,040; 40,400; 44,040 Inverse Operations Rewrite each expression using the inverse operation Order of Operations Simplify each expression (5 2)(5 2) (8 )(8 ) Evaluate Expressions Determine whether the given expressions are equal. 25. (4 7) (2 4) ( ) (50 44) and 4 (7 2) and 2 (4 2) and (2 ) and (4 2) (16 4) (2 ) and (9 4) 2 and 2 ( 4) and 16 (4 4) and (5 2) Principles of lgebra

3 The information below unpacks the standards. The cademic Vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. (Lessons 1- and 1-5) F1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area ). (Lesson 1-2) F1.2 Use the correct order of operations to evaluate algebraic expressions such as (2x 5) 2. (Lesson 1-1) F1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. (Lessons 1-1 and 1-7) F4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. (Lesson 1-9) cademic Vocabulary absolute value the distance a number is from zero on a number line interpret to understand and explain the meaning of variable a symbol, usually a letter, used to show an amount that can change Example: x verbal using words evaluate find the value Example: x for x 2 x 2 5 terminology words used to represent certain items or ideas context the facts or circumstances that surround a situation reasonableness makes sense given the facts or circumstances Chapter Concept You find the absolute value of a number by determining how far it is from zero on a number line. Example: Since is units from 0 on a number line, the absolute value of is. You rewrite a verbal statement using mathematical symbols. Example: three less than a number n You find the value of expressions by performing basic arithmetic operations in a specific order. You learn the names of different mathematical concepts. You perform more than one operation with integers to find the value of a variable that makes an equation true. You also decide if answers make sense for the problems. Standards NS1.1, NS1.2, and F1. are also covered in this chapter. To see these standards unpacked, go to Chapter 2, p. 62 (NS1.2), Chapter, p. 114 (F1.), and Chapter 4, p. 166 (NS1.1). 4 Chapter 1 Principles of lgebra

4 Reading Strategy: Use Your Book for Success Understanding how your textbook is organized will help you locate and use helpful information. s you read through an example problem, pay attention to the margin notes, such as Helpful Hints, Reading Math notes, and Caution notes. These notes will help you understand concepts and avoid common mistakes. Read (4) as 4 to the rd power or 4 cubed. The glossary is found The index is located The Skills Bank is in the back of your textbook. Use it to find definitions and examples of unfamiliar words or properties. at the end of your textbook. Use it to find the page where a particular concept is taught. found in the back of your textbook. These pages review concepts from previous math courses. Try This Use your textbook for the following problems. 1. Use the glossary to find the definition of absolute value. 2. Where can you review the order of operations?. On what page can you find answers to exercises in Chapter 2? 4. Use the index to find the page numbers where algebraic expressions, monomials, and volume of prisms are explained. Principles of lgebra 5

5 1-1 Evaluating lgebraic Expressions California Standards F1.2 Use the correct order of operations to evaluate algebraic expressions such as (2x 5) 2. F1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. Vocabulary expression variable numerical expression algebraic expression evaluate Why learn this? You can evaluate an expression to convert a temperature from degrees Celsius to degrees Fahrenheit. (See Example.) n expression is a mathematical phrase that contains operations, numbers, and/or variables. variable is a letter that represents a value that can change or vary. There are two types of expressions: numerical and algebraic. numerical expression does not contain variables. Numerical Expressions n algebraic expression contains one or more variables. lgebraic Expressions 2 4(5) x 2 4n p r x 4 To evaluate an algebraic expression, substitute a given number for the variable. Then use the order of operations to find the value of the resulting numerical expression. EXMPLE 1 Order of Operations PEMDS: 1. Parentheses 2. Exponents. Multiply and Divide from left to right. 4. dd and Subtract from left to right. See Skills Bank p. SB14. Evaluating lgebraic Expressions with One Variable Evaluate each expression for the given value of the variable. x 5 for x Substitute 11 for x. 16 dd. 2a for a 4 2(4) Substitute 4 for a. 8 Multiply. 11 dd. 4( n) 2 for n 0, 1, 2 n Substitute Parentheses Multiply Subtract 0 4( 0) 2 4() ( 1) 2 4(4) ( 2) 2 4(5) Chapter 1 Principles of lgebra

6 EXMPLE 2 Evaluating lgebraic Expressions with Two Variables Evaluate each expression for the given values of the variables. 5x 2y for x 1 and y 11 5(1) 2(11) Substitute 1 for x and 11 for y Multiply. 87 dd. p(24 q) for p 8 and q 17 8(24 17) Substitute 8 for p and 17 for q. 8(7) Simplify within parentheses. 56 Multiply. EXMPLE Science pplication If c is a temperature in degrees Celsius, then 1.8c 2 can be used to find the temperature in degrees Fahrenheit. Convert each temperature from degrees Celsius to degrees Fahrenheit. freezing point of water: 0 C 1.8c 2 1.8(0) 2 Substitute 0 for c. 0 2 Multiply. 2 dd. 0 C 2 F Water freezes at 2 F. highest recorded temperature in the United States: 57 C 1.8c 2 1.8(57) 2 Substitute 57 for c Multiply dd. 57 C 14.6 F The highest recorded temperature in the United States is 14.6 F. Think and Discuss 1. Give an example of a numerical expression and of an algebraic expression. 2. Tell how to evaluate an algebraic expression for a given value of a variable.. Explain why you cannot find a numerical value for the expression 4x 5y for x. 1-1 Evaluating lgebraic Expressions 7

7 1-1 Exercises California Standards Practice F1.2, F1.4 KEYWORD: MT8C 1-1 See Example 1 See Example 2 See Example See Example 1 See Example 2 See Example Extra Practice See page EP2. GUIDED PRCTICE KEYWORD: MT8C Parent Evaluate each expression for the given value of the variable. 1. x 4 for x a 7 for a 7. 2(4 n) 5 for n 0 Evaluate each expression for the given values of the variables. 4. x 2y for x 8 and y r (12 p) for p 15 and r 9 If c is the number of cups of water needed to make papier-mâché paste, then 1 4 c can be used to find the number of cups of flour needed. Find the number of cups of flour needed for each number of cups of water cups 7. 8 cups 8. 7 cups cups INDEPENDENT PRCTICE Evaluate each expression for the given value of the variable. 10. x 7 for x t 2 for t ( k) 7 for k 0 Evaluate each expression for the given values of the variables. 1. 4x 7y for x 9 and y 14. 4m 2n for m 25 and n 2.5 If c is the number of cups, then 1 c 2 can be used to find the number of pints. Find the number of pints for each of the following cups cups cups cups PRCTICE ND PROBLEM SOLVING Evaluate each expression for the given value of the variable n for n x 4. for x t 5 for t m for m 0 2. a 4 for a g 5 for g y 2 for y y for y (z 9) for z 6 Evaluate each expression for t 0, x 1.5, y 6, and z z y 29. yz z y y x 2. 4(y x). 4( y) 4. (y 6) (2 z) (4 t) 6 7. y( t) 7 8. x y z 9. 10x z y 40. 4(z 5t) 41. 2y 6(x t) 42. 8txz 4. 2z xy 44. rectangular shape has a length-to-width ratio of approximately 5 to. designer can use the expression 1 (5w) to find the length of such a rectangle with a given width w. Find the length of such a rectangle with width 6 inches. 8 Chapter 1 Principles of lgebra

8 California Entertainment The photos above were taken in Palo lto as part of Eadweard Muybridge s 1878 study on horses in motion. Muybridge s multi-camera techniques were a landmark in the history of cinematography. 45. Finance bank charges interest on money it loans. Interest is sometimes a fixed amount of the loan. The expression a(1 i) gives the total amount due for a loan of a dollars with interest rate i, where i is written as a decimal. Find the amount due for a loan of $100 with an interest rate of 10%. (Hint: 10% 0.1) 46. Entertainment There are 24 frames, or still shots, in one second of movie footage. To determine the number of frames in a movie, you can use the expression (24)(60)m, where m is the running time in minutes. Using the running time of E.T. the Extra-Terrestrial shown at right, determine how many frames are in the movie. 47. Choose a Strategy basketball league has 288 players and 24 teams, with an equal number of players per team. If the number of teams is reduced by 6 but the total number of players stays the same, there will be? players per team. E.T. the Extra-Terrestrial (1982) has a running time of 115 minutes, or 6900 seconds. 6 more B 4 more C 4 fewer D 6 fewer 48. Write bout It student says that the algebraic expression 5 x 7 can also be written as 5 7x. Is the student correct? Explain. 49. Challenge Can the expressions 2x and x 2 ever have the same value? If so, what must the value of x be? F Multiple Choice What is the value of the expression x 4 for x 2? 4 B 6 C 9 D Multiple Choice bakery charges $7 for a dozen muffins and $2 for a loaf of bread. If a customer bought 2 dozen muffins and 4 loaves of bread, how much did she pay? $22 B $8 C $80 D $ Gridded Response What is the value of 5(x y) for x 19 and y 6? Identify the odd number(s) in each list of numbers. (Previous course) 5. 15, 18, 22, 4, 21, 61, 71, , 114, 122, 411, 117, , 6, 8, 16, 18, 20, 49, 81, , 15, 1, 47, 65, 9, 1,, 4 Find each sum, difference, product, or quotient. (Previous course) Evaluating lgebraic Expressions 9

9 1-2 Writing lgebraic Expressions California Standards F1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area ). Who uses this? dvertisers can write an algebraic expression to represent the cost of airing a commercial a given number of times. (See Example.) To use algebra to solve real-world problems, you may need to translate word phrases into mathematical symbols. Sixty-eight different commercials aired during the 2005 Super Bowl. Word Phrases Expression add 5 to a number sum of a number and 5 n 5 5 more than a number subtract 11 from a number difference of a number and 11 x less than a number multiplied by a number m or m product of and a number 7 divided into a number a 7 or a 7 quotient of a number and 7 EXMPLE 1 ddition is commutative, so order does not matter in Example 1B. 1 12p and 12p 1 are both correct. Subtraction is not commutative, so 1 less than the product of 12 and p is written as 12p 1, not 1 12p. Translating Word Phrases into Math Expressions Write an algebraic expression for each word phrase. 4 times the difference of a number n and 2 4 times the difference of n and 2 4 (n 2) 4(n 2) 1 more than the product of 12 and p 1 more than the product of 12 and p (12 p) 1 12p 1 10 Chapter 1 Principles of lgebra

10 EXMPLE 2 Translating Math Expressions into Word Phrases Write a word phrase for the algebraic expression 4 7b. 4 7 b 4 minus the product of 7 and b 4 minus the product of 7 and b EXMPLE To solve a word problem, first interpret the action you need to perform and then choose the correct operation for that action. Writing and Evaluating Expressions in Word Problems company aired its 0-second commercial during Super Bowl XXXIX at a cost of $2.4 million each time. Write an algebraic expression to determine what the cost would be if the When a word commercial had aired n times. Then evaluate the expression for problem involves 2,, and 4 times. groups of equal size, use multiplication or $2.4 million n Combine n equal amounts of $2.4 million. division. Otherwise, 2.4n Cost in millions of dollars use addition or subtraction. n 2.4n Cost 2 2.4(2) $4.8 million 2.4() $7.2 million 4 2.4(4) $9.6 million Evaluate for n 2,, and 4. EXMPLE 4 Writing a Word Problem from a Math Expression Write a word problem that can be evaluated by the algebraic expression 14,917 m. Then evaluate the expression for m 6. t the beginning of the month, Benny s car had 14,917 miles on the odometer. If Benny drove m miles during the month, how many miles were on the odometer at the end of the month? 14,917 m 14, ,550 Substitute 6 for m. The car had 15,550 miles on the odometer at the end of the month. Think and Discuss 1. Give two words or phrases that can be used to express each operation: addition, subtraction, multiplication, and division. 2. Express 5 7n in words in at least two different ways. 1-2 Writing lgebraic Expressions 11

11 1-2 Exercises California Standards Practice F1.1, F1.2 KEYWORD: MT8C 1-2 See Example 1 See Example 2 See Example See Example 4 See Example 1 See Example 2 See Example See Example 4 Extra Practice See page EP2. GUIDED PRCTICE Write an algebraic expression for each word phrase less than the product of and p more than the product of 2 and u. 16 more than the quotient of d and minus the quotient of u and 2 Write a word phrase for each algebraic expression s y r 8. 29b Mark is going to work for his father s pool cleaning business during the summer. Mark s father will pay him $5 for each pool he helps clean. Write an algebraic expression to determine how much Mark will earn if he cleans n pools. Then evaluate the expression for 15, 25, 5, or 45 pools. 10. Write a word problem that can be evaluated by the algebraic expression x 450. Then evaluate the expression for x 125. INDEPENDENT PRCTICE Write an algebraic expression for each word phrase more than the quotient of 5 and n minus the product of and p less than the product of 78 and j plus the quotient of r and more than the product of 59 and q KEYWORD: MT8C Parent Write a word phrase for each algebraic expression. 5 w t g d community center is trying to raise $1680 to purchase exercise equipment. The center is hoping to receive equal contributions from members of the community. Write an algebraic expression to determine how much will be needed from each person if n people contribute. Then evaluate the expression for 10, 12, 14, or 16 people. 21. Write a word problem that can be evaluated by the algebraic expression 72 r. Then evaluate the expression for r 17. PRCTICE ND PROBLEM SOLVING Write an algebraic expression for each word phrase times the sum of 4 and y 2. the product of 6 and y increased by 9 12 Chapter 1 Principles of lgebra of the sum of 4 and p divided by the sum of and g 26. half the sum of m and less than the product of 1 and y less than m divided by twice the quotient of m and of the difference of p and times the sum of 2 and x

12 Translate each algebraic expression into words b. 8(m 5) 4. 8 x w 6. t age 2, a cat or a dog is considered 24 human years old. Each year after age 2 is equivalent to 4 human years. Let a represent the age of a cat or dog. Fill in the expression [24 (a 2)] so that it represents the age of a cat or dog in human years. Copy the chart and use your expression to complete it. ge (a 2) DO NOT WRITE IN BOOK ge (human years) 7. Reasoning Write two different algebraic expressions for the word phrase the sum of x and What s the Error? student wrote an algebraic expression for 5 less than the quotient of n and as n 5. What error did the student make? 9. Write bout It Paul used addition to solve a word problem about the weekly cost of commuting by toll road for $1.50 each day. Fran solved the same problem by multiplying. They both got the correct answer. How is this possible? 40. Challenge Write an expression for the sum of 1 and twice a number n. If you let n be any number, will the result always be an odd number? Explain. NS1.2, F1.1, F Multiple Choice Which expression means times the difference of y and 4? y 4 B (y 4) C (y 4) D (y 4) 42. Multiple Choice Which expression represents the product of a number n and 2? n 2 B n 2 C n 2 D 2 n 4. Short Response company prints n books at a cost of $9 per book. Write an expression to represent the total cost of printing n books. What is the total cost if 1050 books are printed? Simplify. (Previous course) (20 8) 2 2 Evaluate each expression for the given value of the variable. (Lesson 1-1) 47. 2(4 x) for x (8 x) 2 for x Writing lgebraic Expressions 1

13 1- Integers and bsolute Value California Standards NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. lso covered: NS1.1 Vocabulary integer opposite absolute value Why learn this? You can use integers to represent scores in disc golf. In disc golf, a player throws a disc toward a target, or hole, in as few throws as possible. The expected number of throws to complete an entire course is called par. player s score can be written as an integer to show how many throws he or she is above or below par. If you complete a course in 5 fewer throws than par your score is 5 under par, or 5. If you complete the same course in 4 more throws than par your score is 4 over par, or 4. Integers are the set of whole numbers and their opposites. Opposites are numbers that are the same distance from 0 on a number line, but on opposite sides of Opposites Negative integers Positive integers 0 is neither negative nor positive. EXMPLE 1 Numbers on a number line increase in value as you move from left to right. Sports pplication fter one round of disc golf, the scores are Fred 5, Trevor, Monique 4, and Julie 2. Use,, or to compare Trevor s and Julie s scores. Trevor s score is, and Julie s score is is to the left of. Julie s score is less then Trevor s. Place the scores on a number line. List the golfers in order from the lowest score to the highest. The scores are 5,, 4, and 2. Place the scores on a number line and read them from left to right. In order from the lowest score to the highest, the golfers are Fred, Julie, Trevor, and Monique. 14 Chapter 1 Principles of lgebra

14 EXMPLE 2 Ordering Integers Write the integers 7, 4, and in order from least to greatest. Graph the integers on a number line. Then read them from left to right The integers in order from least to greatest are 4,, and 7. number s absolute value is its distance from 0 on a number line. bsolute value of a nonzero number is always positive because distance is always positive. The absolute value of 4 is written as 4. Opposites have the same absolute value. 4 units 4 units EXMPLE Simplifying bsolute-value Expressions Simplify each expression. 6 6 units is 6 units from 0, so Subtract first: ; is 6 units from 0. 5 is 5 units from 0. 1 Then find the absolute value: 0 is 0 units from 0. Find the absolute values first: 9 is 9 units from 0. 7 is 7 units from 0. Then add. Think and Discuss 1. Explain the steps you would take to simplify Explain whether an absolute value is ever negative. 1- Integers and bsolute Value 15

15 1- Exercises NS1.1, California Standards Practice NS2.5 KEYWORD: MT8C 1- See Example 1 See Example 2 See Example GUIDED PRCTICE KEYWORD: MT8C Parent 1. fter the first round of the 2005 Masters golf tournament, scores relative to par were Tiger Woods 2, Vijay Singh 4, Phil Mickelson 2, and Justin Leonard 5. Use,, or to compare Vijay Singh s and Phil Mickelson s scores. Then list the golfers in order from the lowest score to the highest. Write the integers in order from least to greatest. 2. 5, 2,. 17, 6, , 21, 14 5., 7, 0 Simplify each expression See Example 1 See Example 2 See Example INDEPENDENT PRCTICE 14. During a very cold week, the temperature in Philadelphia was 7 F on Monday, 4 F on Tuesday, 2 F on Wednesday, and F on Thursday. Use,, or to compare the temperatures on Wednesday and Thursday. Then list the days in order from the coldest to the warmest. Write the integers in order from least to greatest , 5, , 11, , 0, , 2, 1 Simplify each expression Extra Practice See page EP2. PRCTICE ND PROBLEM SOLVING Compare. Write,, or Chapter 1 Principles of lgebra Write the integers in order from least to greatest , 16, , 1, , 5, 25 Simplify each expression Reasoning Two integers and B are graphed on a number line. If is less than B, is always less than B? Explain your answer.

16 Science Simplify each expression The Ice Hotel in Jukkasjärvi, Sweden, is rebuilt every winter from 0,000 tons of snow and 4000 tons of ice. 5. Science The table shows the lowest recorded temperatures for each continent. Write the continents in order from the lowest recorded temperature to the highest recorded temperature. 54. Chemistry The boiling point of nitrogen is 196 C. The boiling point of oxygen is 18 C. Which element has the greater boiling point? Explain your answer. 55. Reasoning Write rules for using absolute value to compare two integers. Be sure to take all of the possible combinations into account. Lowest Recorded Temperatures Continent Temperature frica 11 F ntarctica 129 F sia 90 F ustralia 9 F Europe 67 F North merica 81 F South merica 27 F 56. Write bout It Explain why there is no number that can replace n to make the equation n 1 true. 57. Challenge List the integers that can replace n to make the statement 8 n 5 true. NS1.1, NS2.5, F1.1, F Multiple Choice Which set of integers is in order from greatest to least? 10, 8, 5 B 8, 5, 10 C 5, 8, 10 D 10, 5, Multiple Choice Which integer is between 4 and 2? 0 B C 4 D Multiple Choice Which expression has the greatest value? 12 B 15 C 18 D 21 Evaluate each expression for a, b 2.5, and c 24. (Lesson 1-1) 61. c a (a 2b) 64. bc a Write an algebraic expression for each word phrase. (Lesson 1-2) more than the product of 7 and a number t 66. pizzeria delivered p pizzas on Thursday. On Friday, it delivered more than twice the number of pizzas delivered on Thursday. Write an expression to show the number of pizzas delivered on Friday. 1- Integers and bsolute Value 17

17 1-4 dding Integers California Standards NS1.2 dd, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. F1. Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Vocabulary additive inverse Why learn this? You can track your daily calorie count by adding integers. (See Example 4.) Suppose you eat a piece of grapefruit that contains 5 calories. Then you walk 2 laps around a track and burn 5 calories. Using opposites, you can represent this situation as 5 (5) 0. When you add two opposites, or additive inverses, the sum is zero. You can model integer addition on a number line. EXMPLE 1 To add a positive number, move to the right. To add a negative number, move to the left. Using a Number Line to dd Integers Use a number line to find each sum. (7) You finish at 4, so (7) 4. 2 (5) You finish at 7, so 2 (5) 7. Start at 0. Move right units. From, move left 7 units. Start at 0. Move left 2 units. From 2, move left 5 units. nother way to add integers is to use absolute value. If the signs are the same find the sum of the absolute values. Use the same sign as the integers. DDING TWO INTEGERS If the signs are different find the difference of the absolute values. Use the sign of the integer with the greater absolute value. 18 Chapter 1 Principles of lgebra

18 EXMPLE 2 Using bsolute Value to dd Integers dd. 4 (6) 4 (6) Think: Find the sum of 4 and Same sign; use the sign of the integers. 8 (9) 8 (9) Think: Find the difference of 8 and ; use the sign of Think: Find the difference of 5 and ; use the sign of 11. EXMPLE Evaluating Expressions with Integers Evaluate b 11 for b 6. b Replace b with 6. Think: Find the difference of 11 and ; use the sign of 11. When finding the sum of three or more integers, you may want to group opposites, or additive inverses. You may also want to group positive integers together and negative integers together. EXMPLE Monday Morning Calories Oatmeal Toast w/jam 8 fl oz juice Calories burned Walked six laps Swam six laps Jogged two laps Health pplication Melanie wants to check her calorie count after breakfast and exercise. Use information from the journal entry to find her total. Use positive for calories eaten and negative for calories burned (110) (40) (65) ( ) (40) (65) Group opposites (40) (65) (145 62) (40 65) Group integers with same signs. 207 (105) dd integers within each group ; use the sign of 207. Melanie s calorie count after breakfast and exercise is 102 calories. Think and Discuss 1. Compare the sums of 10 (22) and Describe how to add the following expressions on a number line: 9 (1) and 1 9. Then compare the sums. 1-4 dding Integers 19 RD PRINT

19 1-4 Exercises California Standards Practice NS1.2, F1. KEYWORD: MT8C 1-4 See Example 1 See Example 2 See Example GUIDED PRCTICE KEYWORD: MT8C Parent Use a number line to find each sum (4) (2) dd () (8) Evaluate each expression for the given value of the variable. 9. t 16 for t m 7 for m p (5) for p 5 See Example Lee opens a checking account. In the first month, he makes two deposits and writes three checks, as shown at right. Find what his balance is at the end of the month. (Hint: Checks count as negative amounts.) Checks Deposits $14 $600 $56 $225 $02 See Example 1 See Example 2 See Example INDEPENDENT PRCTICE Use a number line to find each sum (7) (9) dd (7) (8) (5) (15) (12) Evaluate each expression for the given value of the variable. 25. q 1 for q x 21 for x z (7) for z 16 See Example On Monday morning, a mechanic has no cars in her shop. The table at right shows the number of cars dropped off and picked up each day. Find the total number of cars left in her shop on Friday. Cars Dropped Cars Picked Off Up Monday 8 4 Tuesday 11 6 Wednesday 9 12 Thursday 14 9 Friday 7 6 Extra Practice See page EP2. PRCTICE ND PROBLEM SOLVING Write an addition equation for each number line diagram Chapter 1 Principles of lgebra

20 Economics Find each sum () (22) (67) (9) 8. 1 (1) Evaluate each expression for the given value of the variable. 9. c 17 for c k (12) for k b (6) for b r for r w for w n (8) for n 5 The number one category of imported goods in the United States is industrial supplies, including petroleum and petroleum products. In 2004, this category accounted for over $412 million of imports. 45. Economics Refer to the Exports Imports data at right about U.S. international trade for Goods $807,584,000,000 $1,47,768,000,000 the year Consider Services $8,55,000,000 $290,095,000,000 values of exports as positive quantities and values of imports as negative quantities. a. What was the total of U.S. exports in 2004? b. What was the total of U.S. imports in 2004? c. The sum of exports and imports is called the balance of trade. pproximate the 2004 U.S. balance of trade to the nearest billion dollars. Source: U.S. Census Bureau 46. What s the Error? student evaluated 4 d for d 6 and gave an answer of 2. What might the student have done wrong? Give the correct answer. 47. Write bout It Explain the different ways it is possible to add two integers and get a negative answer. 48. Challenge What is the sum of () ()... when there are 10 terms? 19 terms? 24 terms? 25 terms? Explain any patterns that you find. NS1.2, NS2.5, F Multiple Choice Which of the following is the value of 7 h when h 5? 22 B 8 C 8 D Gridded Response Evaluate the expression 5 y for y 8. Evaluate each expression for the given values of the variables. (Lesson 1-1) 51. 2x y for x 8 and y s t for s 7 and t 12 Simplify each expression. (Lesson 1-) dding Integers 21

21 1-5 Subtracting Integers California Standards NS1.2 dd, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. Why learn this? You can represent the differences between the drops and climbs of a roller coaster by subtracting integers. (See Example.) In Lesson 1-4 you modeled integer addition on a number line. You can also model integer subtraction on a number line. The model below shows how to find Start at 0. Move right 8 units. From 8, move left 10 units. When you subtract a positive integer, the difference is less than the original number. Therefore, you move to the left on the number line. To subtract a negative integer, you move to the right. You can also subtract an integer by adding its opposite. You can then use the rules for addition of integers. SUBTRCTING TWO INTEGERS Words Numbers lgebra To subtract an integer, 7 (7) a b a (b) add its opposite. 5 (8) 5 8 a (b) a b EXMPLE 1 Subtracting Integers Subtract (7) dd the opposite of Same sign; add; use the sign of the integers. 2 (4) 2 (4) 2 4 dd the opposite of 4. 6 Same sign; add; use the sign of the integers. 1 (5) 1 (5) 1 5 dd the opposite of 5. 8 Unlike signs; subtract; 1 5; use the sign of Chapter 1 Principles of lgebra

22 EXMPLE 2 Evaluating Expressions with Integers Evaluate each expression for the given value of the variable. 4 s for s 9 4 s 4 (9) Substitute 9 for s. 4 9 dd the opposite of ; use the sign of 9. x for x 5 x 5 Substitute 5 for x. (5) dd the opposite of 5. 8 Same sign; use the sign of the integers. 6 t 5 for t 4 6 t 5 6 (4) 5 Substitute 4 for t dd the opposite of The absolute value of 10 is dd. EXMPLE rchitecture pplication The roller coaster Desperado has a maximum height of 209 feet and a maximum drop of 225 feet. How far underground does the roller coaster go? Desperado Roller Coaster 209 ft 209 (225) Subtract the drop from the height. 225 ft 209 (225) dd the opposite of ; use the sign of 225. Ground level, 0 ft? ft Desperado goes 16 feet underground. Think and Discuss 1. Explain why 10 (10) does not equal Describe the answer that you get when you subtract a greater number from a lesser number. Then give an example. 1-5 Subtracting Integers 2

23 1-5 Exercises California Standards Practice NS1.2, NS2.5 KEYWORD: MT8C 1-5 See Example 1 See Example 2 See Example See Example 1 See Example 2 See Example Extra Practice See page EP. GUIDED PRCTICE Subtract (6). 8 (4) (6) Evaluate each expression for the given value of the variable h for h m for m 5 7. k for k The temperature rose from 4 F to 45 F in Spearfish, South Dakota, on January 22, 194, in only 2 minutes! By how many degrees did the temperature change? INDEPENDENT PRCTICE Subtract (9) (6) (2) Evaluate each expression for the given value of the variable b for b q for q f 7 for f submarine cruising at 27 m below sea level, or 27 m, descends 14 m. What is its new depth? PRCTICE ND PROBLEM SOLVING Write a subtraction equation for each number line diagram KEYWORD: MT8C Parent Perform the given operations (11) (27) (6) (4) (9) Evaluate each expression for the given value of the variable. 25. x 16 for x t for t y for y s (22) for s r 11 9 for r p for p Estimation roller coaster starts with a 160-foot climb and then plunges 228 feet down a canyon wall. It then climbs a gradual 72 feet before a steep climb of 189 feet. pproximately how far is the coaster above or below its starting point? 24 Chapter 1 Principles of lgebra

24 Social Studies Use the timeline to answer the questions. Use negative numbers for years B.C.E. ssume that there was a year 0 (there wasn t) and that there have been no major changes to the calendar (there have been). 2. How long was the Greco-Roman era, when Greece and Rome ruled Egypt?. Which was a longer period of time: from the Great Pyramid to Cleopatra, or from Cleopatra to the present? By how many years? 4. Queen Neferteri ruled Egypt about 2900 years before the Turks ruled. In what year did she rule? 5. There are 1846 years between which two events on this timeline? 6. Write bout It What is it about years B.C.E. that make negative numbers a good choice for representing them? 7. Challenge How would your calculations differ if you took into account the fact that there was no year 0? KEYWORD: MT8C Egypt NS1.2, NS2.5, F Multiple Choice Which of the following is equivalent to 7 ()? 7 B 7 C 10 D 4 9. Gridded Response Subtract: 4 (12). Write an algebraic expression to evaluate the word problem. (Lesson 1-2) 40. Tate bought a compact disc for $ The sales tax on the disc was t dollars. What was the total cost including sales tax? Evaluate each expression for m. (Lesson 1-4) 41. m m (5) 4. 9 m 44. m 1-5 Subtracting Integers 25

25 1-6 Multiplying and Dividing Integers California Standards NS1.2 dd, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Who uses this? Football players can multiply integers to represent the total change in yards during several plays. (See Example.) If a football team loses 10 yards in each of plays, the total change in yards can be represented by (10). You can write this product as repeated addition. (10) 10 (10) (10) 0 Notice that a positive integer multiplied by a negative integer gives a negative product. You already know that the product of two positive integers is a positive integer. The pattern shown at right can help you understand that the product of two negative integers is also a positive integer. (10) 0 2(10) 20 1(10) 10 0(10) 0 1(10) 10 2(10) 20 (10) MULTIPLYING ND DIVIDING TWO INTEGERS If the signs are the same, the sign of the answer is positive. If the signs are different, the sign of the answer is negative. EXMPLE 1 The sum of two negative integers is always negative, but the product or quotient of two negative integers is always positive. Multiplying and Dividing Integers Multiply or divide. 5(8) Signs are different Signs are the same. 40 nswer is negative. 5 nswer is positive. 2 4(2)() Multiply two integers. Signs are different. 8 4(2)() Signs are different. 4 nswer is negative. 8() nswer is negative. 8() Signs are the same. 24 nswer is positive. 26 Chapter 1 Principles of lgebra

26 EXMPLE 2 Using the Order of Operations with Integers Simplify. (2 8) (2 8) Subtract inside the parentheses. (2 8) dd the opposite of 8 inside the parentheses. (6) Think: The signs are the same. 18 The answer is positive. 4(7) 2 4(7) (14 6) 2(14 6) 2(20) 40 Multiply first. Think: The signs are different; answer is negative. Same sign; use the sign of the integers. dd inside the parentheses. Think: The signs are different. The answer is negative. EXMPLE Sports pplication football team runs 10 plays. On 6 plays, it has a gain of 4 yards each. On 4 plays, it has a loss of 5 yards each. Each gain in yards can be represented by a positive integer, and each loss can be represented by a negative integer. Find the total net change in yards. 6(4) 4(5) dd the losses to the gains. 24 (20) Multiply. 4 dd. The team gained 4 yards. Think and Discuss 1. List all possible multiplication and division statements for the integers with absolute values of 5, 6, and 0. For example, Compare the sign of the product of two negative integers with the sign of the sum of two negative integers.. Suppose the product of two integers is positive. What do you know about the signs of the integers? 1-6 Multiplying and Dividing Integers 27

27 1-6 Exercises California Standards Practice NS1.2, F1.2 KEYWORD: MT8C 1-6 See Example 1 See Example 2 See Example See Example 1 See Example 2 See Example Extra Practice See page EP. GUIDED PRCTICE Multiply or divide. 1. 8(4) (4)() 4. 8 Simplify. 5. 7(5 12) 6. 4() (5 9) 8. 11(7 ) Business n investor buys shares of stock and stock B. Stock loses $8 per share, and stock B gains $5 per share. Given the number of shares, how much does the investor lose or gain? 9. stock : 20 shares, stock B: 5 shares 10. stock : 0 shares, stock B: 20 shares INDEPENDENT PRCTICE Multiply or divide (7) (7) (4)(2) (9) 17. 5(7)(4) (21) Simplify (9 14) 20. 1(2) (8 11) (5 8) Banking student puts $50 in the bank each time he makes a deposit. He takes $20 each time he makes a withdrawal. Given the number of transactions, what is the net change in the student s account? 2. deposits: 4, withdrawals: deposits:, withdrawals: 8 PRCTICE ND PROBLEM SOLVING 25. Science Ocean tides are the result of the gravitational force between the sun, the moon, and the earth. When ocean tides occur, the earth s crust also moves. This is called an earth tide. The formula for the height of an earth tide is y x, where x is the height of the ocean tide. Fill in the table. KEYWORD: MT8C Parent Ocean Tide x Earth Tide Height (x) Height ( y) Perform the given operations (6) (7) ()(5) ()(2). 1 8(6) Chapter 1 Principles of lgebra

28 Science noplogaster cornuta, often called a fangtooth or ogrefish, is a predatory fish that reaches a maximum length of 15 cm. It can be found in tropical and temperate waters at 16,000 ft. Evaluate the expressions for the given value of the variable. 4. 4t 5 for t 5. x 2 for x (s 9) for s 1 7. r for r Science The ocean floor is extremely uneven. It includes underwater mountains, ridges, and extremely deep areas called trenches. To the nearest foot, find the average depth of the trenches shown y 11 t for t 6 9. for y Reasoning football team runs 11 plays. There are plays that result in a loss of 2 yards each and 8 plays that result in a gain of 4 yards each. To find the total yards gained, rt simplifies the expression (2) 8(4). Bella first finds the total yards lost, 6, and the total yards gained, 2. Then she subtracts 6 from 2. Compare these two methods. 42. Choose a Strategy P is the set of positive factors of 0, and Q is the set of negative factors of 18. If x is a member of set P and y is a member of set Q, what is the greatest possible value of x y? (Sea level) 0 Depths of Ocean Trenches 540 B 180 C 90 D 1 4. Write bout It If you know that the product of two integers is negative, what can you say about the two integers? Give examples. 44. Challenge visiting player is positioned on his own 0-yard line. The player loses yards after a gain of 12 yards. What is the player s position if 4 yards are lost on the next two plays? Depth (ft) 20,000 25,000 0,000 5,000 40,000 Bonin 2,788 Kuril 1,988 Mariana 5,840 Yap 27,976 NS1.2, F Multiple Choice What is the product of 7 and 10? 70 B 17 C D Short Response Brenda donates part of her salary to the local children s hospital each month by having $15 taken out of her monthly paycheck. Write an integer to represent the amount taken out of each paycheck. Find an integer to represent the change in the amount of money in Brenda s paychecks after 1.5 years. Write an algebraic expression for each word phrase. (Lesson 1-2) 47. j decreased by twice b less less than y Find each sum or difference. (Lessons 1-4 and 1-5) (4) 52. (6) (259) 1-6 Multiplying and Dividing Integers 29

29 Quiz for Lessons 1-1 Through Evaluating lgebraic Expressions Evaluate each expression for the given values of the variables. 1. 5x 6y for x 8 and y (r 7t) for r 80 and t Writing lgebraic Expressions Write an algebraic expression for each word phrase.. one-sixth the sum of r and plus the product of 16 and m Write a word phrase for each algebraic expression. 5. 7y (18 t) 7. x Integers and bsolute Value Write the integers in order from least to greatest , 25, 18, , 8, 9, 1 p Simplify each expression dding Integers Evaluate each expression for the given value of the variable. 14. p 14 for p w (9) for w In Loma, Montana, on January 15, 1972, the temperature increased 10 degrees in a 24-hour period. If the lowest temperature on that day was 54 F, what was the highest temperature? 1-5 Subtracting Integers Subtract (8) (5) (16) The point of highest elevation in the United States is on Mount McKinley, laska, at 20,20 feet. The point of lowest elevation is in Death Valley, California, at 282 feet. What is the difference in the elevations? 1-6 Multiplying and Dividing Integers Multiply or divide. 22. (8)(6) (2)(5)(6) 0 Chapter 1 Principles of lgebra

30 Solve California Standards MR1.1 nalyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. lso covered: NS1.2 Choose an operation: ddition or Subtraction To decide whether to add or subtract to solve a word problem, you need to determine what action is taking place in the problem. If you are combining numbers or putting numbers together, you need to add. If you are taking away or finding out how far apart two numbers are, you need to subtract. ction Operation Illustration Combining or putting together dd Removing or taking away Subtract Finding the difference Subtract Jan has 10 red marbles. Joe gives her more. How many marbles does Jan have now? The action is combining marbles. dd 10 and. Determine the action in each problem. Use the actions to restate the problem. Then give the operation that must be used to solve the problem. 1 Lake Superior is the largest of the Great Lakes and contains approximately 000 mi of water. Lake Michigan is the second largest Great Lake by volume and contains approximately 1180 mi of water. pproximately how much more water is in Lake Superior? Lake Superior Upper Peninsula W N S E 2 The average temperature in Homer, laska, is approximately 5 F in July and approximately 24 F in December. How much hotter is the average temperature in Homer in July than in December? Einar has $18 to spend on his friend s birthday presents. He buys one present that costs $12. How much does he have left to spend? Lake Michigan Lower Peninsula Lake Huron Michigan Lake Erie L. Ontario 4 Dinah got 87 points on her first test and 9 points on her second test. What is her point total for the first two tests? Focus on Problem Solving 1

31 1-7 Solving Equations by dding or Subtracting California Standards Preparation for F4.0 Students solve simple linear equations and inequalities over the rational numbers. F1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. Vocabulary equation inverse operations Why learn this? You can use an equation to represent the forces acting on an object. (See Example.) 8 11 n equation is a mathematical statement that uses an equal sign to show that two expressions have the same value. ll of these are equations r x To solve an equation that contains a variable, find the value of the variable that makes the equation true. This value of the variable is called the solution of the equation. EXMPLE 1 The phrase subtraction undoes addition can be understood with this example: If you start with and add 4, you can get back to by subtracting Determining Whether a Number Is a Solution of an Equation Determine which value of x is a solution of the equation. x 7 1; x 12 or 20 Substitute each value for x in the equation. x ? 1 Substitute 12 for x. x ? 1 Substitute 20 for x. 5? 1 So 12 is not a solution. 1? 1 So 20 is a solution. dding and subtracting by the same number are inverse operations. Inverse operations undo each other. To solve an equation, use inverse operations to isolate the variable. In other words, get the variable alone on one side of the equal sign. The properties of equality allow you to perform inverse operations. These properties show that you can perform the same operation on both sides of an equation. You can use the properties of equality along with the Identity Property of ddition to solve addition and subtraction equations. IDENTITY PROPERTY OF DDITION Words Numbers lgebra The sum of any number and zero is that number a 0 a 2 Chapter 1 Principles of lgebra

32 EXMPLE 2 Solving Equations Using ddition and Subtraction Properties Solve. 6 t 28 6 t t 22 t 22 Since 6 is added to t, subtract 6 from both sides to undo the addition. Identity Property of ddition: 0 t t Check 6 t ? 28 To check your solution, substitute 22 for t in the 28? 28 original equation. m 8 14 m Since 8 is subtracted from m, add 8 to both sides to undo the subtraction. m 0 6 m 6 Identity Property of ddition: m 0 m Check m ? 14? 14 To check your solution, substitute 6 for m in the original equation. 15 w (14) 15 w (14) Since 14 is added to w, subtract 15 (14) w (14) (14) 14 from both sides to undo the 29 w 0 addition. 29 w Identity Property of ddition w 29 Definition of Equality PROPERTIES OF EQULITY Words Numbers lgebra ddition Property of Equality You can add the same number to both sides of an equation, and the statement will still be true. Subtraction Property of Equality You can subtract the same number from both sides of an equation, and the statement will still be true If x y, then x z y z. If x y, then x z y z. 1-7 Solving Equations by dding or Subtracting

33 EXMPLE PROBLEM SOLVING PPLICTION Net force is the sum of all forces acting on an object. Expressed in newtons (N), it tells you in which direction and how quickly the object will move. If two dogs are playing tug-of-war, and the dog on the right pulls with a force of 12 N, what force is the dog on the left exerting on the rope if the net force is 2 N? 1. 1 Understand the Problem Force is measured in newtons (N). The number of newtons tells the size of the force and the sign tells its direction. Positive is to the right and up, and negative is to the left and down. The answer is the force that the left dog exerts on the rope. List the important information: The dog on the right pulls with a force of 12 N. The net force is 2 N. Show the relationship of the information: net force left dog s force right dog s force 2. 2 Make a Plan Write an equation and solve it. Let f represent the left dog s force on the rope, and use the equation model. Math Builders For more on writing equations from words, see the Graph and Equation Builder on page MB2.. Solve 2 f f 2 Subtract 12 from both sides. The left dog is exerting a force of 10 newtons on the rope. f Look Back The problem states that the net force is 2 N, which means that the dog on the right must be pulling with more force. The absolute value of the left dog s force is less than the absolute value of the right dog s force, 10 12, so the answer is reasonable. Think and Discuss 1. Explain what the result would be in the tug-of-war match in Example if the dog on the left pulled with a force of 7 N and the dog on the right pulled with a force of 6 N. 2. Describe the steps to solve y Chapter 1 Principles of lgebra

34 1-7 See Example 1 Exercises GUIDED PRCTICE California Standards Practice NS1.2, F1.4, Preparation for F4.0 KEYWORD: MT8C 1-7 KEYWORD: MT8C Parent Determine which value of x is a solution of each equation. 1. x 6 18; x 10, 12, or x 7 14; x 2, 7, or 21 See Example 2 Solve.. m t 1 5. p (1) q (25) t p (41) See Example 9. team of mountain climbers descended 600 feet to a camp that was at an altitude of 12,05 feet. t what altitude did they start? See Example 1 See Example 2 INDEPENDENT PRCTICE Determine which value of x is a solution for each equation. 10. x 14 8; x 6, 22, or x 2 55; x 15, 28, or 2 Solve w (8) 1. m t z (22) p (10) h (8) See Example 18. Olivia owns 4 CDs. This is 15 more CDs than ngela owns. How many CDs does ngela own? Extra Practice See page EP. PRCTICE ND PROBLEM SOLVING Solve. Check your answer t h m (9) 22. m h t t (86) 26. m x p n (14) w 1. 8 t c w Social Studies In 1990, the population of Cheyenne, Wyoming, was 7,142. By 2000, the population had increased to 81,607. Write and solve an equation to find n, the increase in Cheyenne s population from 1990 to stronomy Mercury s surface temperature has a range of 600 C. This range is the broadest of any planet in the solar system. Given that the lowest temperature on Mercury s surface is 17 C, write and solve an equation to find the highest temperature. 1-7 Solving Equations by dding or Subtracting 5

35 Determine which value of the variable is a solution of the equation. 6. d 4 24; d 6, 20, or k (1) 27; k 40, 45, or d 17 6; d 19, 17, or k 4; k 1, 7, or s; s 20, 26, or g; g 58, 25, Science n ion is a charged particle. Each proton in an ion has a charge of 1 and each electron has a charge of 1. The ion charge is the electron charge plus the proton charge. Write and solve an equation to find the electron charge for each ion. Hydrogen sulfate ion (HSO 4 ) Name of Ion Proton Charge Electron Charge Ion Charge luminum ion (l ) 1 Hydroxide ion (OH ) 9 1 Oxide ion (O 2 ) 8 2 Sodium ion (Na ) What s the Error? student evaluated the expression 7 () and came up with the answer 10. What did the student do wrong? 44. Write bout It Write a set of rules to use when solving addition and subtraction equations. 45. Challenge Explain how you could solve for h in the equation 14 h 8 using algebra. Then find the value of h. NS1.2, Prep for F Multiple Choice Which value of x is the solution of the equation x (5) 8? x x 11 x 1 x Multiple Choice Len bought a pair of $12 flip-flops and a shirt. He paid $0 in all. Which equation can you use to find the price p he paid for the shirt? B 12 p 0 12 p 0 0 p 12 p 12 0 B 48. Gridded Response What value of x is the solution of the equation x 2 19? dd. (Lesson 1-4) (9) (22) (75) C C D D Multiply or divide. (Lesson 1-6) (8) (7) (1) 6 Chapter 1 Principles of lgebra

36 1-8 Solving Equations by Multiplying or Dividing California Standards Preparation for F4.0 Students solve simple linear equations and inequalities over the rational numbers. lso covered: F1. Vocabulary Identity Property of Multiplication Division Property of Equality Why learn this? You can write a multiplication equation to solve problems about person-hours. (See Example.) Just as with addition and subtraction equations, you can use an identity property to solve multiplication and division equations. IDENTITY PROPERTY OF MULTIPLICTION Words Numbers lgebra The product of any number and one is that number a 1 a You can use the properties of equality along with the Identity Property of Multiplication to solve multiplication and division equations. DIVISION PROPERTY OF EQULITY Words Numbers lgebra You can divide both sides of an equation by the same nonzero number, and the statement will still be true If x y and z 0, then x z y z. EXMPLE 1 Solving Equations Using Division Solve. 8x 2 8x 2 8 x Since x is multiplied by 8, divide both sides by 8 to undo the multiplication. 1x 4 x 4 7y 91 7y 91 7y y 1 y 1 Identity Property of Multiplication: 1 x x Since y is multiplied by 7, divide both sides by 7 to undo the multiplication. Identity Property of Multiplication: 1 y y 1-8 Solving Equations by Multiplying or Dividing 7

37 MULTIPLICTION PROPERTY OF EQULITY Words Numbers lgebra You can multiply both sides of an 2 6 If x y, then equation by the same number, zx zy. and the statement will still be true EXMPLE 2 Solving Equations Using Multiplication h Solve 6. h 6 h 6 1h 18 h 18 Since h is divided by, multiply both sides by to undo the division. Identity Property of Multiplication: 1 h h EXMPLE person-hour is a unit of work. It represents the amount of work that can be completed by one person in one hour. Sports pplication It takes 96 person-hours to prepare a baseball stadium for a playoff game. The stadium s manager assigns 6 people to the job. Each person works the same number of hours. How many hours does each person work? Let x represent the number of hours each person works. number of people 6 number of hours each person works total number of person-hours x 96 6x 96 6 x x 16 x 16 Write the equation. Since x is multiplied by 6, divide both sides by 6 to undo the multiplication. Identity Property of Multiplication: 1 x x Each person works for 16 hours. Think and Discuss 1. Explain the steps you would use to solve k Tell how the Multiplication and Division Properties of Equality are similar to the ddition and Subtraction Properties of Equality. 8 Chapter 1 Principles of lgebra

38 1-8 Exercises California Standards Practice F1.1, Preparation for F4.0 KEYWORD: MT8C 1-8 See Example 1 See Example 2 See Example See Example 1 See Example 2 See Example Extra Practice See page EP. GUIDED PRCTICE Solve. 1. 4x y q m 125 l h m t Business It takes 270 person-days to prepare one issue of a magazine for publication. Each week, it takes 5 days to prepare an issue of the magazine. ssuming each person works the same number of days, how many people work on the magazine? INDEPENDENT PRCTICE Solve. 10. d x g y p n b u n h a j 8 12 y d t 5 60 p kilowatt-hour is one kilowatt of power that is supplied for one hour. The lighting for a -hour concert uses a total of 210 kilowatt-hours of electricity. How many kilowatts of electricity do the lights require? PRCTICE ND PROBLEM SOLVING Solve x y y m 9 1. k x b n x g r x 6 b KEYWORD: MT8C Parent m a y Nutrition One serving of milk contains 8 grams of protein, and one serving of steak contains 2 grams of protein. Write and solve an equation to find the number of servings of milk n needed to get the same amount of protein as there is in one serving of steak. x 44. Reasoning Will the solution of 11 be greater than 11 or less than 5 11? Explain how you know. 45. Multi-Step Joy earns $8 per hour at an after-school job. Each month she earns $128. How many hours does she work each month? fter six months, she gets a $2 per hour raise. How much money does she earn per month now? 1-8 Solving Equations by Multiplying or Dividing 9

39 Translate each sentence into an equation. Then solve the equation. 46. The quotient of a number d and 4 is. 47. The sum of 7 and a number n is The product of a number b and 5 is Twelve is the difference of a number q and Zero is the product of 8 and a number t. 51. Recreation While on vacation, Milo drove his car a total of 70 miles. This was 5 times as many miles as he drives in a normal week. How many miles does Milo drive in a normal week? 52. Reasoning In the equation ax 12, a is an integer. Is it possible to choose a value of a so that the solution of the equation is x 0? Why or why not? 5. Music Jason wants to buy the trumpet advertised in the classified ads. He has saved $156. Using the information from the ad, write and solve an equation to find how much more money he needs to buy the trumpet. 54. Write a Problem Write a problem that can be solved by using the equation 15x 75. What is the solution of the problem? 55. Write bout It Explain how to estimate the solution of m Challenge During a winter storm, the temperature dropped F every hour. The overall change in the temperature was 18 F. Write and solve an equation to determine how long the storm lasted. NS1.2, Prep for F Multiple Choice Solve the equation 7x 42. x 49 B x 5 C x 6 D x Gridded Response On a game show, Paul missed q questions, each worth 100 points. Paul received a total of 900 points. How many questions did he miss? Subtract. (Lesson 1-5) (7) (9) Solve each equation. (Lesson 1-7) 6. 4 x x x x Chapter 1 Principles of lgebra

40 Model Two-Step Equations 1-9 Use with Lesson 1-9 KEY = 1 = 1 = variable REMEMBER = 0 You can perform the same operation with the same numbers on both sides of an equation without changing the value of the equation. You can use algebra tiles to model and solve two-step equations. To solve a two-step equation, you use two different operations. KEYWORD: MT8C Lab1 California Standards F4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. 1 ctivity Use algebra tiles to model and solve s s 4 10 Two steps are needed to solve this equation. Step 1: Remove 4 yellow tiles from each side. Step 2: Divide each side into equal groups. Substitute to check: s 6 s 2 s 4 10 (2) 4? ? 10 10? 10 Lab continued on next page. 1-9 Hands-On Lab 41

41 2 Use algebra tiles to model and solve 2r r 4 6 Step 1: Since 4 is being added to 2r, add 4 red tiles to both sides and remove the zero pairs on the left side. Step 2: Divide each side into 2 equal groups. dd 4 to both sides. 2r 10 r 5 Substitute to check: 2r 4 6 2(5) 4? ? 6 6? 6 Think and Discuss 1. Why can you add zero pairs to one side of an equation without having to add them to the other side as well? 2. Show how you could have modeled to check your solution for each equation. Try This Use algebra tiles to model and solve each of the following equations. 1. 2x p 9. 5r n b a m h Gerry walked dogs five days a week and got paid the same amount each day. One week his boss added on a $15 bonus. That week Gerry earned $90. What was his daily salary? 42 Chapter 1 Principles of lgebra

42 1-9 Solving Two-Step Equations California Standards F4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. lso covered: F1.1 Who uses this? Two-step equations can help pet owners decide how much pet food they should purchase. (See Example 4.) Two-step equations contain two operations. For example, the equation 6x 2 10 contains multiplication and subtraction. Multiplication 6x 2 10 Subtraction EXMPLE 1 Translating Sentences into Two-Step Equations Translate each sentence into an equation. more than the product of 5 and a number y is 17. more than the product of 5 and a number y is y 17 5y 17 The quotient of a number n and 4, minus 8, is 42. The quotient of a number n and 4, minus 8, is 42. [n (4)] 8 42 n Math Builders For more on solving two-step equations, see the Step-by-Step Solution Builder on page MB4. Equations that contain two operations require two steps to solve. Identify the operations and the order in which they are applied to the variable. Then use inverse operations to undo them in the reverse order. Operations in the Equation 1 First x is multiplied by To Solve dd 2 to both sides of the equation. Then 2 is subtracted. 2 Then divide both sides by Solving Two-Step Equations 4

43 EXMPLE 2 Reverse the order of operations when solving equations that have more than one operation. Solving Two-Step Equations Using Division Solve. 2x 1 7 Step 1: 2x Check 2x 1 7 2(4) 1 7? ? 7 2x 8 Step 2: 2 x x 4 Note that x is multiplied by 2. Then 1 is added. Work backward: Since 1 is added to 2x, subtract 1 from both sides. Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. To check your solution, substitute 4 for x in the original equation. 4 is a solution. 18 7n 18 7n 21 7n 21 7n 7 7 n Since is subtracted from 7n, add to both sides to undo the subtraction. Since n is multiplied by 7, divide both sides by 7 to undo the multiplication. EXMPLE Solving Two-Step Equations Using Multiplication Solve. 5 k 1 6 Step 1: 5 k k 8 6 Step 2: (6) k (6)8 6 k 48 Note that k is divided by 6. Then 5 is added. Work backward: Since 5 is added to k 6, subtract 5 from both sides. Since k is divided by 6, multiply both sides by 6 to undo the division. 1 9 v v v v (9)11 (9) 9 99 v Since 10 is subtracted from 9 v, add 10 to both sides to undo the subtraction. Since v is divided by 9, multiply both sides by 9 to undo the division. 44 Chapter 1 Principles of lgebra

44 EXMPLE 4 Reasoning Consumer Math pplication Tyrell is ordering cat food online. Shipping is $12 no matter how many cases he orders. Each case of cat food costs $16. Tyrell has $10 to spend. How many cases of cat food can he order? Let c represent the number of cases that Tyrell can order. That means Tyrell can spend $16c plus the cost of shipping. cost of shipping cost of cat food total cost $12 16c $ c c c c 7.75 Subtract 12 from both sides. Divide both sides by 16. Since you cannot order part of a case, Tyrell can order 7 cases. Think and Discuss 1. Explain how you decide which operation to use first when solving a two-step equation. 2. Describe the steps you would follow to solve 5 7x Exercises F1.1, California Standards Practice F4.1 KEYWORD: MT8C 1-9 See Example 1 GUIDED PRCTICE Translate each sentence into an equation more than 9 times a number x is The quotient of a number t and 7, plus 2, is 4. KEYWORD: MT8C Parent See Example 2 See Example Solve.. 4x s 5 5. n p y w 8 20 Solve m x 9 7 w x r z Solving Two-Step Equations 45

45 See Example 4 See Example 1 See Example 2 See Example See Example 4 Extra Practice See page EP. 15. Consumer Math Web site that rents DVDs charges a one-time fee of $7 to become a member of the site. It costs $ to rent each DVD. Diana joined the site and spent a total of $61. How many DVDs did she rent? INDEPENDENT PRCTICE Translate each sentence into an equation less than the quotient of a number b and 12 is more than the product of 4 and a number z is The difference of twice a number n and 9 is 17. Solve y x t w q m g y z 11 8 Solve w r 8 2 y x w n 8 5 t g x Bob s uto Shop charges $75 to replace the timing belt in a car. The belt costs $50 and the labor to install the belt costs $65 per hour. How many hours does it take to install the timing belt? PRCTICE ND PROBLEM SOLVING Translate each equation into words. Then solve the equation. x w y Solve and check m p j 12 9 x z 5 r Translate each sentence into an equation. Then solve the equation. 47. Three less than the quotient of a number x and 9 is Thirteen plus the product of 15 and a number a is If 5 is decreased by times a number m, the result is Hobbies Mike has 60 baseball cards in his collection. Each week he buys a pack of cards to add to the collection. Each pack contains the same number of cards. fter 12 weeks, Mike has 156 cards. Write and solve an equation to find the number of cards in a pack. 46 Chapter 1 Principles of lgebra

46 Social Studies In 1956, during President Eisenhower s term, construction began on the United States interstate highway system. The original plan was for 42,000 miles of highways to be completed within 16 years. It actually took 7 years to complete. The last part, Interstate 105 in Los ngeles, was completed in Write and solve an equation to show how many miles m needed to be completed per year for 42,000 miles of highways to be built in 16 years. 52. Interstate 5 runs north and south from Laredo, Texas, to Duluth, Minnesota, covering 1568 miles. There are 505 miles of I-5 in Texas and 262 miles in Minnesota. Write and solve an equation to find m, the number of miles of I-5 that are not in either state. 5. portion of I-476 in Pennsylvania, known as the Blue Route, is about 22 miles long. The length of the Blue Route is about one-sixth the total length of I-476. Write and solve an equation to calculate the length of I-476 in miles m. 54. Challenge Interstate 80 extends from California to New Jersey. t right are the number of miles of Interstate 80 in each state the highway passes through. a.? has 14 more miles than?. b.? has 174 fewer miles than?. Number of I-80 Miles State Miles California 195 Nevada 410 Utah 197 Wyoming 401 Nebraska 455 Iowa 01 Illinois 16 Indiana 167 Ohio 26 Pennsylvania 14 New Jersey 68 F Multiple Choice Solve the equation 4 5 t 8. t 60 B t 40 C t 20 D t Multiple Choice For which equation is m a solution? 5m 6 21 B m 1 1 C 2 m 11 D 12 m 15 Multiply or divide. (Lesson 1-6) 57. 4(11) (5) Solve. (Lesson 1-8) y m p g Solving Two-Step Equations 47

47 Quiz for Lessons 1-7 Through Solving Equations by dding or Subtracting Solve. 1. p w (9) 14. t (14) k p y (6) The approximate surface temperature of Pluto, the coldest planet, is 91 F. This is approximately 1255 degrees cooler than the approximate surface temperature of Venus, the hottest planet. What is the approximate surface temperature of Venus? 1-8 Solving Equations by Multiplying or Dividing Solve. x y x y r d 12. 5p y hmed s baseball card collection consists of 228 cards. This is 4 times as many cards as Ming has. How many baseball cards are in Ming s collection? 17. It takes 72 person-hours to clean an office building each night. crew of 12 people cleans the building and each person works the same number of hours. How many hours does each person work? 1-9 Solving Two-Step Equations Solve. y 18. x n k y x 2 x k n Richard is collecting bottle caps for a school fundraiser. He has collected 5 caps toward his goal of 500 caps. If he collects 15 bottle caps per week, how many more weeks will it take before he reaches his goal? 28. One week Ella worked 6 hours as a prep chef. Her earnings for the week totaled $210, which included $0 in tips that she made as a waitress. How much money does Ella make per hour? 48 Chapter 1 Principles of lgebra

48 Have a Ball physical education class is playing a variation of basketball. When a team makes a basket from inside the three-point line, the team gets a Climb (C), or 2 points. When a team makes a basket from outside the three-point line, the other team gets a Slide (S), or 1 point. 1. During the first 20 minutes of the game, a team gets the following: C, C, S, S, C, S, C, C, and S. Simplify the expression 2 2 (1) (1) 2 (1) 2 2 (1) to determine the team s score. 2. The points scored by two teams during a game are shown in the table. Which team won the game? What was the difference in the teams scores?. Diego s team gets Climbs and 2 Slides, but not necessarily in that order. Find his team s score by substituting S 1 and C 2 in the expression C 2S. 4. fter four consecutive baskets are made, Leann s team s score is 8. fter the next basket is made, the team s score is 6. Write and solve an equation for the last made basket. 5. Daryl s team finishes the game with a score of 12. If his team scored 9 times, how many Climbs did the team get? 6. Is it possible to finish with a score of 2 after five baskets are made? Explain your reasoning. Game Results Team 1 Team 2 C C S C S C S S S S C S C S C S Concept Connection 49

49 Math Magic You can guess what your friends are thinking by learning to operate your way into their minds! For example, try this math magic trick. Think of a number. Multiply the number by 8, divide by 2, add 5, and then subtract 4 times the original number. No matter what number you choose, the answer will always be 5. Try another number and see. You can use what you know about variables to prove it. Here s how: What you say: What the person thinks: What the math is: Step 1: Pick any number. 6 (for example) n Step 2: Multiply by 8. 8(6) 48 8n Step : Divide by n 2 4n Step 4: dd n 5 Step 5: Subtract 4 times the 29 4(6) n 5 4n 5 original number. Invent your own math magic trick that has at least five steps. Show an example using numbers and variables. Try it on a friend! Crazy Crzzy Cubes This game, called The Great Tantalizer around 1900, was reintroduced in the 1960s as Instant Insanity. Make four cubes with paper and tape, numbering each side as shown The goal is to line up the cubes so that 1, 2,, and 4 can be seen along the top, bottom, front, and back of the row of cubes. They can be in any order, and the numbers do not have to be right-side up. 50 Chapter 1 Principles of lgebra

50 Materials sheet of decorative paper (8 1 2 by 11 in.) ruler pencil scissors glue markers PROJECT Note-Taking Taking Shape _ 5 8 in. _ 5 8 in. Make this notebook to help you organize examples of algebraic expressions. Directions 1 Hold the sheet of paper horizontally. Make two vertical lines 5 8 inches from each end of the sheet. Figure 2 Fold the sheet in half lengthwise. Then cut it in half by cutting along the fold. Figure B B On one half of the sheet, cut out rectangles and B. On the other half, cut out rectangles C and D. Figure B C D Rectangle : 4 in. by 5 8 in. Rectangle B: in. by 5 8 in. Rectangle C: in. by 5 8 in. Rectangle D: in. by 5 8 in. C 4 Place the piece with the taller rectangular panels on top of the piece with the shorter rectangular panels. Glue the middle sections of the two pieces together. Figure C 5 Fold the four panels into the center, starting with the tallest panel and working your way down to the shortest. Taking Note of the Math Write ddition, Subtraction, Multiplication, and Division on the tabs at the top of each panel. Use the space below the name of each operation to list examples of verbal, numerical, and algebraic expressions.

51 Vocabulary absolute value ddition Property of Equality additive inverse algebraic expression Division Property of Equality equation evaluate expression Identity Property of ddition Identity Property of Multiplication integer inverse operations numerical expression opposite Subtraction Property of Equality variable Complete the sentences below with vocabulary words from the list above. 1. (n)? is a statement that two expressions have the same value. 2.? is another word for additive inverse.. The? of is. 1-1 Evaluating lgebraic Expressions (pp. 6 9) F1.2, F1.4 EXMPLE Evaluate 4x 9y for x 2 and y 5. 4x 9y 4(2) 9(5) Substitute 2 for x and 5 for y Multiply. 5 dd. EXERCISES Evaluate each expression. 4. 9a 7b for a 7 and b m n for m 10 and n r 19s for r 8 and s Writing lgebraic Expressions (pp. 10 1) F1.1 EXMPLE Write an algebraic expression for the word phrase 2 less than a number n. n 2 Write as subtraction. Write a word phrase for 25 1t. 25 plus the product of 1 and t EXERCISES Write an algebraic expression for each phrase. 7. twice the sum of k and more than the product of 4 and t Write a word phrase for each algebraic expression. 9. 5b s y r 8 52 Chapter 1 Principles of lgebra

52 1- Integers and bsolute Value (pp ) NS1.1, NS2.5 EXMPLE Simplify the expression and 6 Subtract. EXERCISES Simplify each expression dding Integers (pp ) NS1.2, F1. EXMPLE dd. 8 2 Find the difference of 8 and ; use the sign of the 8. Evaluate. 4 a for a 7 4 (7) Substitute. 11 Same sign EXERCISES dd (9) (7) () (5) (8) Evaluate. 24. k 11 for k m for m Subtracting Integers (pp ) NS1.2, NS2.5 EXMPLE Subtract. (5) 5 dd the opposite of ; use the sign of the 5. Evaluate. 9 d for d Substitute. 9 (2) dd the opposite of Same sign EXERCISES Subtract (9) (5) (2) Evaluate h for h z for z Multiplying and Dividing Integers (pp ) NS1.2 EXMPLE Multiply or divide. 4(9) The signs are different. 6 The answer is negative. The signs are the same. 11 The answer is positive. EXERCISES Multiply or divide ( 5) (1) ()(5) (5) 25 Study Guide: Review 5

53 1-7 Solving Equations by dding or Subtracting (pp. 2 6) EXMPLE Solve. x Subtract 7 from both sides. x 0 5 x 5 Identity Property of Zero y 1.5 y y 4.5 dd to both sides. Identity Property of Zero EXERCISES F1.4, Prep for Solve and check. 40. z t k x 2 1 F4.0 Write an equation and solve. 44. polar bear weighs 715 lb, which is 585 lb less than a sea cow. How much does the sea cow weigh? 45. The Mojave Desert, at 15,000 mi 2, is 11,700 mi 2 larger than Death Valley. What is the area of Death Valley? 1-8 Solving Equations by Multiplying or Dividing (pp. 7 40) EXMPLE EXERCISES Prep for F4.0 Solve. 4h 24 4 h Divide both sides by 4. 1h h 6 1 h h 4 t t 4 16 Multiply both sides by 4. 1t t 64 1 t t Solve and check g k p w 12 4 y z The Lewis family drove 25 mi toward their destination. This was 1 of the total distance. What was the total distance? 5. Luz will pay a total of $960 on her car loan. Her monthly payment is $90. For how many months is the loan? 1-9 Solving Two-Step Equations (pp. 4 47) F4.1 EXMPLE Solve 6 k 5. 6 k k 1 () k ()(1) k Subtract 6 from both sides. Multiply both sides by. EXERCISES Solve. y x n x n y home computer repair service charges $68 for the first hour and $48 for each additional hour of service. If a repair bill is $212, how many additional hours did the repair take? 54 Chapter 1 Principles of lgebra

54 Evaluate each expression for the given value of the variable p for p t 7 for t x (2) for x y 27 for y 9 Write an algebraic expression for each word phrase more than the product of and y less than the quotient of x and times the sum of 7 and h divided by the difference of t and 9 Write each set of integers in order from least to greatest. 9. 7, 7, 2,, 0, , 45, 1, 100, , 7, 54, 41, 7 Simplify each expression Perform the given operations (12) (22) 18. (20) (4) (5) (1) (6) (5) 2. 2(21 17) 24. (15 ) (4) 25. (54 6) (1) 26. (16 4) The temperature on a winter day increased 7 F. If the beginning temperature was 9 F, what was the temperature after the increase? Solve. 28. y z p 1. w 9 t q g p cme Sporting Products manufactures 216 tennis balls a day. Each container holds balls. How many tennis ball containers are needed daily? 7. It takes 105 person-hours to set up new computers in an office. crew sets up the computers and each person works the same number of hours. It takes 7 hours to complete the job. How many people are on the crew? Solve. 8. 9x x n 4 y y n round-trip ticket to a soccer camp costs $250. If the total cost for the ticket and 5 days at camp is $715, what is the cost to attend the camp per day? 45. Group rates at Putt-Putt Pfun include charges of $4 per person and entry to the game room at $10 per group. How many people are in a group that pays $74 for putt-putt and games? Chapter 1 Test 55

55 Multiple Choice: Eliminate nswer Choices With some multiple-choice test items, you can use logical reasoning or estimation to eliminate some of the answer choices. Test writers often create the incorrect choices, called distracters, using common student errors. Which expression represents 4 times the sum of x and 8? B 4 (x 8) C 4 x 8 4 (x 8) D 4 (x 8) Read the question. Then try to eliminate some of the answer choices. Use logical reasoning. Times means to multiply, and sum means to add. You can eliminate any option without a multiplication symbol and an addition symbol. You can eliminate B and D. The sum of x and 8 is being multiplied by 4, so you need to add before you multiply. Because multiplication comes before addition in the order of operations, x 8 should be in parentheses. The correct answer is. Which value for k is a solution of the equation k.5 12? B k 8.5 C k 42 k 15.5 k 47 Read the question. Then try to eliminate some of the answer choices. Use estimation. D You can eliminate C and D immediately because they are too large. Estimate by rounding.5 to 4. If x 47, then This is not even close to 12. Similarly, if x 42, then , which is also too large to be correct. Choice is called a distracter because it was created using a common student error, subtracting.5 from 12 instead of adding.5 to 12. Therefore, is also incorrect. The correct answer is B. 56 Chapter 1 Principles of lgebra

56 Read each test item and answer the questions that follow. Item The table shows average high temperatures for Nome, laska. Which answer choice lists the months in order from coolest to warmest? B C D Even if the answer you calculated is an answer choice, it may not be the correct answer. It could be a distracter. lways check your answers! Month Jan Feb Mar pr Temperature ( C) May 6 Jun 12 Jul 15 ug 1 Sep 9 Oct 1 Nov Dec 5 9 Jul, ug, Jun, Sep, May, Oct, pr, Nov, Mar, Dec, Feb, Jan Jul, Jun, ug, Jan, Feb, Sep, Dec, Mar, May, Nov, pr, Oct Jan, pr, Jun, Jul, Sep, Nov, Feb, Mar, May, Jul, Sep, Nov Jan, Feb, Dec, Mar, Nov, pr, Oct, May, Sep, Jun, ug, Jul 1. Which two choices can you eliminate by using logic? Explain your reasoning. 2. What common error does choice represent? Item B Which value for p is a solution of the equation p ? B p 0.2 C p 20.2 p 9.8 p 78. Which choices can you eliminate by using estimation? Explain your reasoning. 4. What common error does choice C represent? Item C x Solve 20 8 for x. 4 B x 112 C x 12 x 112 x Which two choices can you eliminate by using logic? 6. What common error does choice C represent? Item D Which word phrase can be translated into the algebraic expression 2x 6? B C D six more than twice a number the sum of twice a number and six twice the difference of a number and six six less than twice a number 7. Can you eliminate any of the choices immediately by using logic? Explain your reasoning. 8. Describe how you can determine the correct answer from the remaining choices. D D Strategies for Success 57