Inequalities and Equations

Transcription

1 Inequalities and Equations Tightrope walkers often perform at circuses. They have trained to keep their balance while walking across a thin, high rope. Some tightrope walkers use a large pole to help them balance. 9.1 Call to Order Inequalities Opposites Attract to Maintain a Balance Solving One-Step Equations and Inequalities Using Addition and Subtraction Statements of Equality Redux Solving One-Step Equations and Inequalities Using Multiplication and Division there are many ways... Representing Situations in Multiple Ways Measuring Short Using Multiple Representations to Solve Problems Variables and More Variables The Many Uses of Variables in Mathematics Quantities that Change Independent and Dependent Variables _C1_CH09_pp indd /04/14 11:39 AM

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3 Call to Order Inequalities Learning Goals In this lesson, you will: Use inequalities to order the number system. Graph inequalities on the number line. Key Terms inequality graph of an inequality solution set of an inequality ray What happens every morning in your class and usually involves your teacher calling names? If you said roll call, you d be right! So, does your teacher seem to call your classmates names in the same order every morning? Actually, there are a lot of ways for teachers to call roll, but one of the easiest ways is to call roll in alphabetical order. Sometimes teachers will call roll in alphabetical order in ascending order. This means starting at the letter A and moving to the letter Z. Or, teachers will call roll in alphabetical order in descending order, which is the opposite of ascending order. Many people and items are ordered in different ways. When a photographer takes a picture of a group of people, the photographer will usually put the shorter people in the front of the group and the taller people in the back of the group. Mechanics usually arrange their wrenches and sockets in order from smallest to largest. What things do you order? How do you go about ordering items or people and this doesn t mean ordering your brother and sister around to do your chores! 9.1 Inequalities 635

4 Problem 1 Saying So Much with Just One Symbol In the past, you probably used symbols that let you order numbers from least to greatest, or from greatest to least. These symbols are called inequality symbols. An inequality is any mathematical sentence that has an inequality symbol. Symbol Meaning Example, less than 3, 5 3 is less than 5. greater than is greater than 7 # $ less than or equal to greater than or equal to 3 # 9 3 is less than or equal to 9 4 $ 1 4 is greater than or equal to 1 fi not equal to 6 fi 7 6 is not equal to 7 1. For each statement, write the corresponding inequality. a. 7 is less than or equal to 23 b. 56 is greater than 28 c. 2 is not equal to 5 d. 7.6 is less than 8.2 e. 5 3 is greater than Chapter 9 Inequalities and Equations

5 2. Write the meaning of each inequality in words. a. 7.8 fi 23.7 b. 8 1 # c. 3 $ d. 43, e , Write, or. to make each inequality true. a b c d Write # or $ to make each inequality true. a. 1 2 b c d e Inequalities 637

6 For any two numbers a and b, only one of the three statements is true. a, b a. b a 5 b 5. What does this statement mean in terms of the ordering of the number system? If a fi b, then a must be less than b or greater than b. Problem 2 Inequalities and the Number Line A number line is a graphic representation of all numbers. 1. Plot and label each of the numbers shown on the number line. a. 3 b. 2.3 c d e There are five points plotted on the number line shown. Identify the approximate location of each point. a b c d e a. b c. d. e. 638 Chapter 9 Inequalities and Equations

7 3. A point at a is plotted on the number line shown. a 0 a. Plot a point to the right of this point and label it b. Then, write three different inequalities that are true about a and b. b. What can you say about all points to the right of point a on the number line? 4. A point at a is plotted on the number line shown. a 0 a. Plot a point to the left of this point and label it b. Then, write three different inequalities that are true about a and b. b. What can you say about all the points to the left of point a on the number line? 5. Describe the position of all the points on the number line that are: a. greater than a. b. less than a. a Inequalities 639

8 Problem 3 Graphing an Inequality on a Number Line You can use a number line to represent inequalities. The graph of an inequality in one variable is the set of all points on a number line that make the inequality true. The set of all points that make an inequality true is the solution set of the inequality. 1. Look at the two inequalities x. 3 and x $ 3. a. Describe the solution sets for each. b. Analyze the graphs of the two inequalities shown on each number line. x x $ 3 Why does one graph show a see-through point and the other one a black point? Describe each number line representation. c. How does the solution set of the inequality x $ 3 differ from the solution set of x. 3? 640 Chapter 9 Inequalities and Equations

9 2. Look at the two inequalities x, 3 and x # 3. a. Describe the solution sets for each. b. Analyze the graphs of the 2 inequalities shown on each number line. x, x # Describe each number line representation. c. How does the solution set of the inequality x # 3 differ from the solution set of x, 3? 9.1 Inequalities 641

10 The solution to any inequality can be represented on a number line by a ray whose starting point is an open or closed circle. A ray begins at a starting point and goes on forever in one direction. A closed circle means that the starting point is part of the solution set of the inequality. An open circle means that the starting point is not part of the solution set of the inequality. 3. Write the inequality represented by each graph. a b c d Graph the solution set for each inequality. a. x # 14 b. x, c # x d. x e. x fi Chapter 9 Inequalities and Equations

11 Talk the Talk 1. Explain the meaning of each sentence in words. Then, define a variable and write a mathematical statement to represent each statement. Finally, sketch a graph of each inequality. a. The maximum load for an elevator is 2900 lbs. "Maximum" means that the weight can't go over that amount. b. A car can seat up to 8 passengers. c. No persons under the age of 18 are permitted. d. You must be at least 13 years old to join. Be prepared to share your solutions and methods. 9.1 Inequalities 643

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13 Opposites Attract to Maintain a Balance Solving One-Step Equations and Inequalities Using Addition and Subtraction Learning Goals In this lesson, you will: Use models to represent one-step equations. Use inverse operations to solve one-step equations. Solve one-step inequalities using addition and subtraction. Key Terms one-step equation solution inverse operations Properties of Equality for Addition and Subtraction Properties of Inequalities for Addition and Subtraction You ve certainly seen parallel lines before. Railroad tracks look like parallel lines. The opposite sides of a straight street form parallel lines. Even a very important symbol in mathematics looks like parallel lines: the equals sign ( ). Did you know there is a reason for why an equals sign looks the way it does? In 1557, mathematician Robert Recorde first used parallel line segments to represent equality because he didn t want to keep writing the phrase is equal to and, as he explained, no two things can be more equal than parallel lines. What does equality mean in mathematics? How can you determine whether two or more things are equal? 9.2 Solving One-Step Equations and Inequalities Using Addition and Subtraction 645

14 Problem 1 Maintaining Balance Each representation shows a balance. Determine what will balance 1 rectangle in each. Adjustments can be made in each pan as long as the balance is maintained. Then, describe your strategies. You might want to get your algebra tiles out. 1. a. Strategies: b. What will balance one rectangle? 646 Chapter 9 Inequalities and Equations

15 2. a. Strategies: b. What will balance one rectangle? 3. Describe the general strategy you used to maintain balance in Questions 1 and Generalize the strategies for maintaining balance by completing each sentence. a. To maintain balance when you subtract a quantity from one side, you must b. To maintain balance when you add a quantity to one side, you must 9.2 Solving One-Step Equations and Inequalities Using Addition and Subtraction 647

16 Problem 2 One Step at a Time 1. Write an equation that represents each pan balance. These are the same pan balances that you analyzed for Question 1 and Question 2 in Problem 1. Use the variable x to represent, and count the units to determine the number they represent together. Then, describe how the strategies you used to determine what balanced one rectangle can apply to an equation. In other words, what balances x? a. b. 648 Chapter 9 Inequalities and Equations

17 You just wrote and solved one-step equations. Previously, you wrote an equation by setting two expressions equal to each other. You solve an equation by determining what value will replace the variable to make the equation true. If you can solve an equation using only one operation, this equation is called a one-step equation. To determine if your value is correct, substitute the value for the variable in the original equation. If the equation is true, or remains balanced, then you correctly solved the equation. 2. Check each of your solutions to Question 1, part (a) and part (b), by substituting your value for x into the original equation you wrote. Show your work. You just determined solutions to your equations. A solution to an equation is any value for a variable that makes the equation true. 3. State the operations in each equation you wrote for Question 1, and the operation you used to determine the value of x. Describe how they relate to each other. To solve an equation, you must isolate the variable by performing inverse operations. Inverse operations are pairs of operations that undo each other. 4. State the inverse operation for each stated operation. a. addition b. subtraction To isolate the variable means to get the variable by itself on one side of the equation. 9.2 Solving One-Step Equations and Inequalities Using Addition and Subtraction 649

18 Problem 3 Solving Equations Example 1 Example 2 a b 2 8 The answers are the same. What is different about the two methods? Method 1: Method 1: a a a b b b Method 2: Method 2: a a a b b b 1. Analyze each example and the different methods used to solve each equation. a. Describe the inverse operation used in each example and explain why. Example 1: Example 2: b. Describe the difference in strategy between Method 1 and Method 2 for Example 1. c. What property states that a a and b b? d. The final step in each method shows the variable isolated. What is the coefficient of each variable? 650 Chapter 9 Inequalities and Equations

19 2. Consider the equations shown. State the inverse operation needed to isolate the variable. Then, solve the equation. Make sure you show your work. Finally, check to see if the value of your solution maintains balance in the original equation. a. m b. 5 5 x 2 8 c. b d x Solving One-Step Equations and Inequalities Using Addition and Subtraction 651

20 e a f. 7 5 y g. w h c Don't forget to check your answers! 652 Chapter 9 Inequalities and Equations

21 3. Determine if each solution is true. Explain your reasoning. a. Is x 5 25 a solution to the equation x ? b. Is x 5 16 a solution to the equation x ? c. Is x a solution to the equation x ? d. Is x a solution to the equation x ? 9.2 Solving One-Step Equations and Inequalities Using Addition and Subtraction 653

22 Problem 4 Solving Inequalities What happens when each side of an inequality is added or subtracted by the same number? Consider the relationship between the two numbers 3 and 6. Since 3 is to the left of 6, you know that 3, Perform each operation to the numbers 3 and 6. Then, plot the new values on the number line. Finally, write a corresponding inequality statement. a. Add 1 2 to each number b. Add 2 to each number c. Add 3 to each number d. Subtract 1 2 from each number e. Subtract 2 from each number f. Subtract 3 from each number Chapter 9 Inequalities and Equations

23 2. When you add the same number to each side of the inequality or subtract the same number from each side of the inequality, what do you notice about the resulting inequality symbol? 3. Explain why Simone is correct. Simone No matter what number I add to or subtract from both sides of the inequality, the relationship between the two sides of the inequality stays the same: 3 < a < 6 + a 3 - a < 6 - a 4. Consider the inequality x a. Write an inequality to describe the possible values of x. b. What could you do to both sides of the original inequality to determine your answer to part (a)? 5. Suppose you have the inequality x Determine the possible values of x. Explain your reasoning. 9.2 Solving One-Step Equations and Inequalities Using Addition and Subtraction 655

24 6. Mike is 5 years older than his brother Jim. For each question, write and solve an equation or inequality to describe Jim s possible ages. Then, graph the solution on the number line. a. How old will Jim be when Mike is at least 25 years old? b. How old will Jim be when Mike is younger than 30 years old? c. How old will Jim be when Mike is 29 years old? 7. Solve each inequality and graph the solution set on the number line. Then choose a value from your solution set to check your work. a. 13, x 1 11 b x $ Chapter 9 Inequalities and Equations

25 c. x, d. x 2 3 # Choose one of the inequalities from Question 7 and write a real-world situation that can be modeled by the algebraic statement. Talk the Talk The Properties of Equality allow you to balance and solve equations involving any number. Properties of Equality For all numbers a, b, and c, Addition Property of Equality If a 5 b, then a 1 c 5 b 1 c. Subtraction Property of Equality If a 5 b, then a 2 c 5 b 2 c. 1. Describe in your own words what the Properties of Equality represent. 2. What does it mean to solve a one-step equation? 3. Describe how to solve any one-step equation. 9.2 Solving One-Step Equations and Inequalities Using Addition and Subtraction 657

26 4. How do you check to see if a value is the solution to an equation? 5. Given the solution x 5 12, write two different equations using the Properties of Equality. The Properties of Inequality allow you to balance and solve inequalities involving any number. Properties of Inequality Addition Property of Inequalities Subtraction Property of Inequalities For all numbers a, b, and c, If a, b, then a 1 c, b 1 c. If a. b, then a 1 c. b 1 c. If a, b, then a 2 c, b 2 c. If a. b, then a 1 c. b 1 c. 6. Describe in your own words what the Properties of Inequalities represent. These properties also hold true for # and $. 7. What does it mean to solve a one-step inequality? 8. Describe how to solve any one-step inequality. How do you check to see if a value is a solution to an inequality? Be prepared to share your solutions and methods. 658 Chapter 9 Inequalities and Equations

27 Statements of Equality Redux Solving One-Step Equations and Inequalities with Multiplication and Division Learning Goals In this lesson, you will: Use models to represent one-step equations. Use inverse operations to solve one-step equations. Solve inequalities using multiplication and division of positive numbers. Key Term Properties of Equality for Multiplication and Division Properties of Inequalities for Multiplication and Division, when c. 0 In 1997, Arulanantham Suresh Joachim set a world record for balancing on one foot: 76 hours and 40 minutes. That s slightly more than 3 days! How long do you think you could balance on one foot? Don t try it out now, because you have some more to learn about balancing in mathematics. What other examples of balancing are there in mathematics? 9.3 Solving One-Step Equations and Inequalities with Multiplication and Division 659

28 Problem 1 Maintaining Balance Each representation shows a balance. Determine what will balance 1 rectangle in each. Adjustments can be made in each pan as long as the balance is maintained. Describe your strategies. 1. a. Strategies: b. What will balance one rectangle? 660 Chapter 9 Inequalities and Equations

29 2. a. Strategies: b. What will balance one rectangle? 3. Describe the general strategy you used to maintain balance in Questions 1 and Generalize the strategies for maintaining balance by completing each sentence. a. To maintain balance when you multiply a quantity by one side, you must b. To maintain balance when you divide a quantity by one side, you must 9.3 Solving One-Step Equations and Inequalities with Multiplication and Division 661

30 Problem 2 One Step at a Time 1. Write an equation that represents each pan balance. These are the same pan balances that you analyzed for Question 1 and Question 2 in Problem 1. Let represent the variable x, and let represent one unit. Then, describe how the strategies you used to determine what balanced one rectangle can apply to an equation. In other words, what balances x? a. b. 662 Chapter 9 Inequalities and Equations

31 2. Check each of your solutions to Question 1, parts (a) through (b) by substituting your value for x back into the original equation you wrote. Show your work. 3. State the operations in each equation you wrote for Question 1, parts (a) through (b) and the operation you used to determine the value of x. Describe how they relate to each other. As you learned previously, to solve an equation, you must isolate the variable by performing inverse operations. 4. State the inverse operation for each stated operation. a. addition b. subtraction c. multiplication d. division 9.3 Solving One-Step Equations and Inequalities with Multiplication and Division 663

32 Problem 3 Solving Equations Example 1 Example 2 8c d 4 Method 1: Method 1: 8c d c 5 6 c d d Looks like there is more than one way to solve these equations. What's different about each method? Method 2: Method 2: 1 8 8c c 5 6 c d d 8 5 d 1. Analyze each example and the different methods used to solve each equation. a. Describe the inverse operation used in each example and explain why. Example 1: Example 2: b. How are Method 1 and Method 2 in Example 1 similar? c. Describe the difference in strategy between Method 1 and Method 2 for Example 2. d. What property states that 1c 5 c and 1d 5 d? e. The final step in each method shows the variable isolated. What is the coefficient of each variable? 664 Chapter 9 Inequalities and Equations

33 2. Consider the equations shown. State the inverse operation needed to isolate the variable. Then, solve the equation. Make sure that you show your work. Finally, check to see if the value of your solution maintains balance in the original equation. a. n b. 3y 5 18 c. n d. 3 y Solving One-Step Equations and Inequalities with Multiplication and Division 665

34 e. y f. 3.14y g. y Don't forget to check your solutions. h x 666 Chapter 9 Inequalities and Equations

35 3. Determine if each solution is true. Explain your reasoning. a. Is p 5 12 a solution to the equation 9p 5 108? b. Is n 5 4 a solution to the equation n 5 24? 6 c. Is p 5 18 a solution to the equation 3p 5 54? 9.3 Solving One-Step Equations and Inequalities with Multiplication and Division 667

36 Problem 4 Solving Inequalities What happens when each side of an inequality is multiplied or divided by the same positive number? Consider the inequality 3, 6. Remember, the representation for division can include fraction notation Perform each operation to the numbers 3 and 6. Then, plot the new values on the number line. Finally, write a corresponding inequality statement. a. Multiply each number by ( 1 2 ) 6 ( 1 2 ) b. Multiply each number by 2. 3(2) 6(2) c. Multiply each number by 3. 3(3) 6(3) d. Divide each number by e. Divide each number by f. Divide each number by Chapter 9 Inequalities and Equations

37 2. When you multiply the same positive number to each side of the inequality or divide the same positive number from each side of the inequality, what do you notice about the resulting inequality symbol? 3. Explain why Robin is correct. Robin No matter what positive number I multiply or divide from both sides of the inequality, the relationship between the two sides of the inequality stays the same: 3 < 6 3 x a < 6 x a 3 a < 6 a 4. Consider the inequality 2x, 6(2). a. Write an inequality to describe the possible values of x. b. What could you do to both sides of the original inequality to determine your answer to part (a)? 5. Suppose you have the inequality 2x, 6. Determine the possible values of x. Explain your reasoning. 9.3 Solving One-Step Equations and Inequalities with Multiplication and Division 669

38 6. Michelle is 3 times as old as her sister Beth. For each question, write and solve an equation or inequality to describe Beth s possible ages. Then, graph the solution on the number line. a. How old will Beth be when Michelle is at least 27 years old? b. How old will Beth be when Michelle is younger than 30 years old? c. How old will Beth be when Michelle is 42 years old? 7. Solve each inequality and graph the solution on the number line. a. 3, x 4 8 b. 10x $ 45 c. x # Choose one of the inequalities from Question 7 and write a real-world situation that can be modeled by the algebraic statement. 670 Chapter 9 Inequalities and Equations

39 Talk the Talk The Properties of Equality allow you to balance and solve equations involving any number. Properties of Equality For all numbers a, b, and c, Addition Property of Equality If a 5 b, then a 1 c 5 b 1 c. Subtraction Property of Equality If a 5 b, then a 2 c 5 b 2 c. Multiplication Property of Equality Division Property of Equality If a 5 b, then ac 5 bc. If a 5 b, and c fi 0, then a c 5 b c. 1. Describe in your own words what the Properties of Equality represent. 2. What does it mean to solve a one-step equation? 3. Describe how to solve any one-step equation. How do you check to see if a value is the solution to an equation? 4. When you are solving, the convention is to not include the numerical coefficient of 1 in your final answer. Explain which property is used to justify this convention. 9.3 Solving One-Step Equations and Inequalities with Multiplication and Division 671

40 5. Given the solution x 5 12, write two different equations using the Multiplication and Division Properties of Equality. The Properties of Inequality allow you to balance and solve inequalities involving any number. Properties of Inequality For all numbers a, b, and c, Addition Property of Inequalities Subtraction Property of Inequalities Multiplication Property of Inequalities Division Property of Inequalities If a, b, then a 1 c, b 1 c. If a. b, then a 1 c. b 1 c. If a, b, then a 2 c, b 2 c. If a. b, then a 2 c. b 2 c. If a, b, then a? c, b? c, for c. 0. If a. b, then a? c. b? c, for c. 0. If a, b, then a c, b c, for c. 0. If a. b, then a c. b c, for c Describe in your own words what the Properties of Inequalities represent. These properties also hold true for # and $. 7. What does it mean to solve a one-step inequality? 8. Describe how to solve any one-step inequality. How do you check to see if a value is a solution to an inequality? Be prepared to share your solutions and methods. 672 Chapter 9 Inequalities and Equations

41 There Are Many Ways... Representing Situations in Multiple Ways Learning Goals In this lesson, you will: Represent two quantities that change in words, symbols, tables, and graphs. Solve one-step equations. People are constantly confronted with problems in their lives. How many bags of fertilizer will you need for your lawn? How much paint is needed to paint a room? After viewing a graph of sales over the last year, what predictions can be made for next year? After looking at a pattern of brick for a walkway, how can you decide how many of each type of brick to order? These are all examples of problems that could be solved more efficiently using mathematics. The real power of mathematics is in providing people with the ability to model and solve problems more efficiently and accurately. Can you think of other examples where mathematics would be useful? 9.4 Representing Situations in Multiple Ways 673

42 Problem 1 Buying on the Internet A site on the Internet sells closeout items and charges a flat fee of $2.95 for shipping. 1. How much would the total order cost if an item costs: a. $26.45? b. $16.95? 2. Explain how you calculated your answers. 3. Define variables for the cost of an item and the total cost of the order. 4. Write an equation that models the relationship between these variables. 5. Use your equation to calculate the total cost of an order given the item cost. a. $100 b. $45.25 c. $ Chapter 9 Inequalities and Equations

43 6. Use your equation to calculate the cost of an item given the total cost. Then, check to see if the value of your solution maintains balance in the original equation. a. $125 b. $37.45 Don't forget about all the estimation strategies you have learned! Does your answer make sense in terms of the problem situation? c. $ Representing Situations in Multiple Ways 675

44 7. Complete the table with your answers from Question 1 through Question 6. Cost of an Item (dollars) Total Cost with Shipping (dollars) Use the table to complete the graph of the total cost versus the cost of an item. y Total Cost (dollars) x Cost of an Item (dollars) 676 Chapter 9 Inequalities and Equations

45 9. Your graph represents the cost of each item and the total cost with shipping from your table of values. What pattern do you notice? Would it make sense to connect the points on this graph? In this situation, the cost of an item can be a fractional value. So, it would make sense to connect the points to show other ordered pairs that make the equation true. Problem 2 Working for that Paycheck! Your friend got a job working at the local hardware store making $6.76 per hour. 1. How much would your friend earn if she worked: a. 5 hours? b hours? 2. Explain how you calculated your answers. 3. Define variables for the number of hours worked, and for the amount earned. 4. Write an equation that models the relationship between these variables. 9.4 Representing Situations in Multiple Ways 677

46 5. Use your equation to calculate the earnings given her hours worked. a. 6 hours If I know the number of hours worked, what do I do to that number to calculate the amount earned? b. 5 hours and 30 minutes c. 10 hours and 15 minutes 678 Chapter 9 Inequalities and Equations

47 6. Use your equation to calculate the number of hours she worked given her total pay. Then, check to see if the value of your solutions maintains balance in the original equation. a. $169 b. $ c. $ Representing Situations in Multiple Ways 679

48 7. Complete the table using your answers to Question 1 through Question 6. Time Worked (hours) Earnings (dollars) Use the table to complete the graph of the money earned versus the hours worked. Earnings (dollars) y x Time Worked (hours) 9. Would it make sense to connect the points on this graph? Explain why or why not. Be prepared to share your solutions and methods. 680 Chapter 9 Inequalities and Equations

49 Measuring Short Using Multiple Representations to Solve Problems Learning Goal In this lesson, you will: Use multiple representations (words, symbols, tables, and graphs) to solve problems. How do you get your news? Do you listen to the radio? Watch TV? Read the newspaper? Check sites online? There are many ways to get the same news story. The information you might hear on the radio in a 30 second news blurb could also be talked about on an hour long television special. That same story may also be mentioned in a brief article in the newspaper or perhaps there is a whole web site devoted to it online. Even though this news story is presented in different ways, the basic facts are still the same. Can you think of different ways we represent the same information in mathematics? 9.5 Using Multiple Representations to Solve Problems 681

50 Problem 1 Broken Yardstick Jayme and Liliana need to measure some pictures so they can buy picture frames. They looked for something to use to measure the pictures, but they could only find a broken yardstick. The yardstick was missing the first inches. They both thought about how to use this yardstick. Liliana said that all they had to do was measure the pictures and then subtract inches from each measurement. 1. Is Liliana correct? Explain your reasoning. 2. They measured the first picture s length to be 11 inches. What was the actual length? 3. They measured the first picture s width to be 9 1 inches. What was the actual width? 2 4. Define variables for a measurement with the broken yardstick and the actual measurement. 5. Write an equation that models the relationship between these variables. 682 Chapter 9 Inequalities and Equations

51 6. Use your equation to calculate the actual measurement if the measurement taken with the broken yardstick is: a inches. b. 21 inches. c inches. 9.5 Using Multiple Representations to Solve Problems 683

52 7. Use your equation to calculate the measurement taken with the broken yardstick given the actual measurement. Then, check your solution using the original equation. a. 12 inches. b inches. c inches. 684 Chapter 9 Inequalities and Equations

53 8. Complete the table using your answers from Question 1 through Question 7. Measurement with Broken Yardstick (in.) Actual Measurement (in.) Use the table to complete the graph of actual measurement versus the measurement taken with the broken yardstick. y Actual Measurement (in.) x Measurement with the Broken Yardstick (in.) 9.5 Using Multiple Representations to Solve Problems 685

54 10. Would it make sense to connect the points on this graph? Explain why or why not. Problem 2 Running Henry is biking to get ready for football season. He records his distances and times after each ride in a table. Distance Biked (km) Time (hours) kilometers is about 3.1 miles Assuming Henry bikes at the same average rate, how long would it take him to bike: a. 68 kilometers? b. 42 kilometers? 2. Explain how you calculated your answers. Then, complete the last two rows of the table. 686 Chapter 9 Inequalities and Equations

55 3. Define variables for the distance biked and for the time. 4. Write an equation that models the relationship between these variables. 5. Use your equation to calculate the time it would take Henry to bike: a. 50 kilometers. b. 60 kilometers. c kilometers. 9.5 Using Multiple Representations to Solve Problems 687

56 6. Use your equation to calculate how far Henry could bike given each amount of time. Then, check your solution using the original equation. a. 45 minutes b. 2 hours c hours 688 Chapter 9 Inequalities and Equations

57 7. Use the table and your calculations in Question 1 through Question 6 to complete the graph of the time in minutes Henry bikes versus the distance he bikes in kilometers. When would it not make sense to connect points on a graph? 5 y Distance (km) Time (hours) x 8. Would it make sense to connect the points on this graph? Explain why or why not. Talk the Talk 1. Write a real-world situation that can be modeled by each expression. a. x 1 2 Be sure to define the variable when you write each real-world situation. 9.5 Using Multiple Representations to Solve Problems 689

58 b. n 5 2. Write a real-world situation that can be modeled by each equation. a. 1.25b 5 10 b a Write a real-world situation that can be modeled by each inequality. a. y b. 1 2 c. 5 Be prepared to share your solutions and methods. 690 Chapter 9 Inequalities and Equations

59 Variables and More Variables The Many Uses of Variables in Mathematics Learning Goal In this lesson, you will: Examine the many different uses of variables in mathematics. Key Term homonyms In English there are many words that have the same spelling and the same pronunciation, but have different meanings. For example, the word left can mean a direction, as in goes to the left. Left can also be the past tense of the word leave, as in he just left. Words like this are called homonyms. Can you think of other homonyms? 9.6 The Many Uses of Variables in Mathematics 691

60 Problem 1 A Little History The Unknown One of the first artifacts showing the use of mathematics is a cuneiform tablet. This tablet, from about 1800 b.c., in Babylonia, illustrates mathematical relationships that are still being studied and learned today. The Babylonians were the first to write equations that were full sentences, for example, some quantity plus one equals two. The Babylonians were also the first to use a symbol or word to represent an unknown quantity. This early algebra was the dominant form of algebra up through 1600 a.d. Thus, we have one meaning, or use, for a variable as an unknown quantity. One way variables are used is in solving equations. 1. Solve each equation for the unknown quantity. Then, check the value of your solution. a. x b x 3 In equations, variables represent the unknown value. c. x d. 7.5b Chapter 9 Inequalities and Equations

61 2. Omar owns 345 more chickens than Henry. If Omar owns 467 chickens, how many does Henry own? a. Write an equation for this problem situation. b. Solve your equation to answer the original question. Then, check the value of your solution. c. If possible, write another equation that can be used for this problem. d. Does writing an equation help you solve this problem? Explain your reasoning. 3. There is an unknown number such that when it is multiplied by 5 and the product is added to 200, the answer is 265. What is the number? Hint: Remember, you can undo the process. Explain how you calculated your answer. 9.6 The Many Uses of Variables in Mathematics 693

62 Problem 2 Variables That Represent All Numbers 1. Use the variables a, b, and c to state each property. a. Commutative Property of Addition b. Associative Property of Multiplication c. Distributive Property of Multiplication over Addition 2. How are the variables in this Problem used differently than in Problem 1? 3. What is true for the equations in these properties and the values of the variables in these properties that is not true for the equations in Problem 1? 694 Chapter 9 Inequalities and Equations

63 Problem 3 Variables in Formulas 1. Complete the table to calculate the perimeter of each rectangle. Length (units) Width (units) Perimeter of the Rectangle (units) Remember, perimeter means the distance around a figure How did you calculate the perimeter? 3. Write the formula for the perimeter of a rectangle. Define your variables. The formula for converting a temperature in Celsius to a temperature in Fahrenheit is F 5 9 C 1 32, where C is the temperature in Celsius, and F is the temperature 5 in Fahrenheit. 4. Complete the table for the given temperatures in Celsius. Temperature in Celsius 100 C 0 C Temperature in Fahrenheit 25 C 9.6 The Many Uses of Variables in Mathematics 695

64 5. How are variables used differently in these formulas than in the previous two problems? Problem 4 Variables That Vary Sherilyn is a bicyclist training for a long-distance bike race. She usually rides her bike at the rate of 16 miles per hour. 1. If Sherilyn maintains her rate, how far would she cycle in: a. 4 hours? b hours? c. 10 hours and 15 minutes? 2. Define variables for the time she cycles and her distance. 3. Write an equation that models the relationship between these variables. 4. Use your equation to calculate the distance Sherilyn would bike in: a. 7 3 hours. b. 3.5 hours Chapter 9 Inequalities and Equations

65 5. If Sherilyn maintains her rate, write and solve your equation to calculate how long it would take Sherilyn to bike: a. 100 miles. b. 50 miles. 6. Complete the table using your answers from Question 1 through Question 5. Time (hours) Distance (miles) hours 15 minutes The Many Uses of Variables in Mathematics 697

66 7. Use the table to complete the graph of the time, in hours, that Sherilyn bikes versus the distance she bikes. y Distance (miles) Time (hours) x 8. How are variables used differently in this problem than in the previous problems in this lesson? Talk the Talk The concept of variable is a foundation concept in the study of mathematics. 1. Complete the graphic organizer. Show examples of the different ways variables are used in the study of mathematics. Be prepared to share your solutions and methods. 698 Chapter 9 Inequalities and Equations

67 unknowns all numbers in mathematical properties Example: Example: Variables Example: Example: particular quantities in formulas Quantities that vary within a problem situation 9.6 The Many Uses of Variables in Mathematics 699

68 700 Chapter 9 Inequalities and Equations

69 Quantities That Change Independent and Dependent Variables Learning Goal In this lesson, you will: Identify and define independent and dependent variables and quantities. Key Terms dependent quantity independent quantity independent variable dependent variable Sometimes you get to make choices and other times you do not. Sometimes making one decision depends on an earlier decision. If you go to a carnival and decide to pay the ride-all-day price versus paying the admission price and then paying for each ride, what decisions do you still need to make and what decisions are already made for you? Can you think of other decisions that you make that then determine other decisions? 9.7 Independent and Dependent Variables 701

70 Problem 1 Quantities That Change In this chapter, you have solved and analyzed problems where quantities changed or varied. Let s consider another example, but this time let s think about how one quantity depends on another quantity. Dawson just purchased a new diesel-powered car that averages 41 miles to the gallon. 1. How far does this car travel on: a. 10 gallons of fuel? b. a full tank of fuel, 13.9 gallons? 2. There are two quantities that are changing in this problem situation. Name the quantities that are changing. 3. Does the value of one quantity depend on the value of the other? 4. Define variables for each quantity. 5. Write an equation for the relationship between these variables. When one quantity depends on another in a problem situation, it is said to be the dependent quantity. The quantity on which it depends is called the independent quantity. The variable that represents the independent quantity is called the independent variable, and the variable that represents the dependent quantity is called the dependent variable. 6. Identify the independent and dependent variables in this situation. 702 Chapter 9 Inequalities and Equations

71 7. Use your equation to calculate the fuel needed to travel: a miles. b. 100 miles. 8. Complete the table using your values from Question 1 and Question 7. Independent Quantity Dependent Quantity Quantity Name Distance Unit of Measure gallons Variable Independent and Dependent Variables 703

72 Problem 2 Sometimes One, Sometimes the Other A store makes 20% profit on each item they sell. 1. Determine the store s profit in selling an item for: a. $ b. $ Profit is the extra money for selling items over and above the cost of the items. c. $ Name the two quantities that are changing. 3. Describe which value depends on the other. Let c represent the cost of the items in dollars, and let p represent the profit in dollars. 4. Write an equation for the relationship between these variables. 5. Identify the independent and dependent variables in this situation. 704 Chapter 9 Inequalities and Equations

73 6. Complete the table using your answers from Questions 1 and 7. Independent Quantity Dependent Quantity Quantity Name Profit Unit of Measure dollars Variable Use this table to complete the graph. y Profit (dollars) x Cost (dollars) 9.7 Independent and Dependent Variables 705

74 Problem 3 Looking at Problem Situations in a Different Way Let s think about the problem situation in a different way. Suppose you are operating this business and you know how much profit you want to make on each item. 1. How much should an item cost if you want to make: a. $7.50 profit? b. $10 profit? c. $19.99 profit? 2. Name the two quantities that are changing. 3. Describe which value depends on the other. Let p be equal to the profit, and let c be equal to the cost of the item. 4. Write an equation for the relationship between these variables. 706 Chapter 9 Inequalities and Equations

75 5. Identify the independent and dependent variables in this situation. 6. Complete the table using your answers from Question 1. Independent Quantity Dependent Quantity Quantity Name Profit Unit of Measure Variable Use this table to complete the graph. y Cost ($) x Profit ($) 9.7 Independent and Dependent Variables 707

76 Talk the Talk The situations in Problems 2 and 3 were similar, but were presented in two different ways. Problem 2 Problem 3 The profit depends on the cost of the item. p 5 0.2c The cost of the item depends on the profit. c 5 p Solve p 5 0.2c for c. 2. Solve c 5 p for p What do you notice about the two solutions? 4. How does examining this same situation from different perspectives affect the independent and dependent variables? 5. What can you conclude about the designation of a variable as independent or dependent? Go back and look at the two graphs in Question 7 for both Problem 2 and 3. The graphs were labeled differently depending on how you defined the independent and dependent quantities. They are similar because they both represent the independent and dependent on the same axis. Dependent y Independent x 708 Chapter 9 Inequalities and Equations

77 6. Read each situation and analyze the corresponding table of values. Identify the independent and dependent quantities in each. Then, write an equation that models the relationship between these quantities. a. The total profit, t, made on cutting lawns and the profit, p, made by each person is represented in the table shown. Total Profit Made (dollars) Profit Made by Each Person (dollars) b. The number of boxes of cookies, n, and the total profit, t, is represented in the table shown. Boxes of Cookies Sold Total Profit (dollars) Independent and Dependent Variables 709

78 c. The number of tiles, r, required to complete a job and the number of tiles ordered, t, is represented in the table shown. Number of Tiles Required Number of Tiles Ordered Be prepared to share your solutions and methods. 710 Chapter 9 Inequalities and Equations

79 Chapter 9 Summary Key Terms inequality (9.1) graph of an inequality (9.1) solution set of an inequality (9.1) ray (9.1) one-step equation (9.2) solution (9.2) inverse operations (9.2) homonyms (9.6) dependent quantity (9.7) independent quantity (9.7) independent variable (9.7) dependent variable (9.7) Properties Properties of Equality for Addition and Subtraction (9.2) Properties of Inequalities for Addition and Subtraction (9.2) Properties of Equality for Multiplication and Division (9.3) Properties of Inequalities for Multiplication and Division, c. 0 (9.3) Graphing Inequalities on the Number Line An inequality is any mathematical sentence that has an inequality symbol. The solution to any inequality can be represented on a number line by a ray whose starting point is an open or closed circle. A ray begins at a starting point and goes on forever in one direction. A closed circle means that the starting point is part of the solution set of the inequality. An open circle means that the starting point is not a part of the solution set of the inequality. The graph of an inequality in one variable is the set of all points on a number line that make the inequality true. This set of points is the solution set of the inequality. Example Graph the solution set for each inequality. Your brain is a hard-working machine. A balanced lifestyle of plenty of sleep, exercise, and healthy food will keep it that way. x > x Chapter 9 Summary 711

80 Using Models to Represent One-Step Equations Equations are mathematical statements that declare that two expressions are equal. Similarly, a balanced scale shows that two quantities are equivalent. Scales can be used as models to represent equations. Example Write an equation that represents the given pan balance. Use the variable x to represent and use numbers to represent each group of units. Then, solve the equation. 8 5 x x x In the pan balance, one is equivalent to 2 units. Using Inverse Operations to Solve One-Step Equations To solve an equation means to determine what value or values will replace the variable to make the equation true. If you can solve an equation using only one operation, the equation is called a one-step equation. A solution to an equation is any value for a variable that makes the equation true. To solve an equation, you must isolate the variable using inverse operations. Inverse operations are operations that undo each other. Addition is the inverse operation of subtraction, and subtraction is the inverse operation of addition. Example In the given equation, state the inverse operation needed to isolate the variable. Then, solve the equation. Check to see if the value of your solution maintains balance in the original equation t 2 9 The inverse operation would be to add 9 to both sides t 2 9 Check: t t Chapter 9 Inequalities and Equations

81 Solving One-Step Inequalities Using Addition and Subtraction To solve an inequality means to determine what value or values will replace the variable to make the inequality true. A solution set to an inequality is any value or values for a variable that makes the inequality true. To solve an inequality, you must isolate the variable using inverse operations. The inequality symbol remains the same when you add or subtract the same value to both sides of the inequality. Example In the given inequality, state the inverse operation needed to isolate the variable. Then, solve the inequality and graph the solution set on the number line. Check to see if the values of your solution set make the inequality true. x The inverse operation would be to add 6 to both sides. x Check: x ? 10 x Using Inverse Operations to Solve One-Step Equations As you learned in Lesson 9.2, to solve an equation means to determine what value or values will replace the variable to make the equation true. To solve an equation, you must isolate the variable using inverse operations. Inverse operations are operations that undo each other. Multiplication is the inverse operation of division, and division is the inverse operation of multiplication. Example In the given equation, state the inverse operation needed to isolate the variable. Then, solve the equation. Check to see if the value of your solution maintains balance in the original equation. 5.2x The inverse operation would be to divide both sides by x Check 5.2x x Chapter 9 Summary 713

82 Solving One-Step Inequalities using Multiplication and Division To solve an inequality means to determine what value or values will replace the variable to make the inequality true. A solution set to an inequality is any value or values for a variable that makes the inequality true. To solve an inequality, you must isolate the variable using inverse operations. The inequality symbol remains the same when you multiply or divide the same positive value to both sides of the inequality. Example In the given inequality, state the inverse operation needed to isolate the variable. Then, solve the inequality and graph the solution set on the number line. Check to see if the values of your solution set make the inequality true. x 4. 3 The inverse operation would be to multiply 4 to both sides. x. 3 Check: 4 x 4 (4). 3(4) 20 4.? 3 x Chapter 9 Inequalities and Equations