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Numerical and Algebraic Expressions and Equations © Carnegie Learning Sometimes it's hard to tell how a person is feeling when you're not talking to them face to face. People use emoticons in emails and chat messages to show different facial expressions. Each expression shows a different kind of emotion. But you probably already knew that. ; ) 5.1 It Must Be a Sign! Operating with Rational Numbers.................................241 5.2 What’s It Really Saying? Evaluating Algebraic Expressions. ............................... 257 5.3 Express Math Simplifying Expressions Using Distributive Properties................................................ 265 5.4 Reverse Distribution Factoring Algebraic Expressions................................... 271 5.5 Are They the Same or Different? Verifying That Expressions Are Equivalent................... 279 5.6 It is time to justify! Simplifying Algebraic Expressions Using Operations and Their Properties.................................. 287 239 462345_C2_CH05_pp239-302.indd 239 28/08/13 12:04 PM © Carnegie Learning 240 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 240 28/08/13 12:04 PM It Must Be a Sign! Operating with Rational Numbers Learning Goals In this lesson, you will: Model addition and subtraction on a number line. Add and subtract integers. Multiply and divide integers. Sketch a Venn diagram to show relationships between sets of numbers. Operate with rational numbers. Solve real-world problems involving rational numbers. O pposites are all around us. If you move forward two spaces in a board game and then move back in the opposite direction two spaces, you’re right back where you started. When playing tug-of-war, if one team pulls on the rope exactly as hard as the team on the opposite side, no one moves. If an atom has the same number of positively charged protons as it does negatively charged electrons, then the atom has no charge. © Carnegie Learning In what ways have you worked with opposites in mathematics? 5.1 Operating with Rational Numbers • 241 462345_C2_CH05_pp239-302.indd 241 28/08/13 12:04 PM Problem 1 Adding on a Number Line Recall that a number line can be used to model integer addition. When adding a positive integer, move to the right on a number line. When adding a negative integer, move to the left on a number line. Example 1: The number line shows how to determine 3 1 6. 3 Step 1 6 Step 2 –10 –8 –6 –4 –2 0 2 4 6 8 10 Example 2: The number line shows how to determine 3 1 (26). 3 Step 1 –6 –8 –6 –4 –2 0 2 4 6 8 1. Compare the first steps in each example. a. What distance is shown by the first term in each example? b. Describe the graphical representation of the first term. Where does it start and in which direction does it move? 10 Remember that the absolute value of a number is its distance from 0. © Carnegie Learning –10 Step 2 c. What is the absolute value of the first term in each example? 242 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 242 28/08/13 12:04 PM 2. Compare the second steps in each example. a. What distance is shown by the second term in each example? b. Why did the graphical representation for the second terms both start at the endpoints of the first terms but then continue in opposite directions? Explain your reasoning. c. What are the absolute values of the second terms? 3. Consider each expression in the worked examples. Which term in each expression had the greater absolute value? How does the sign of that term relate to the sign of the sum? 4. Use the number line to determine each sum. Show your work. a. 22 1 8 5 © Carnegie Learning –10 –8 –6 –4 –2 0 2 4 6 8 10 –8 –6 –4 –2 0 2 4 6 8 10 –8 –6 –4 –2 0 2 4 6 8 10 b. 2 1 (28) 5 –10 c. 22 1 (28) 5 –10 5.1 Operating with Rational Numbers • 243 462345_C2_CH05_pp239-302.indd 243 28/08/13 12:04 PM d. 2 1 8 5 –10 –8 –6 –4 –2 0 2 5. In Question 4, what patterns do you notice when: a. you are adding two positive numbers? 4 6 8 10 Can you see how knowing the absolute value is important when adding and subtracting signed numbers? b. you are adding two negative numbers? c. you are adding a negative number and a positive number? 6. The set of rational numbers includes integers, mixed numbers, fractions, and their equivalent decimals. Based on Questions 4 and 5, what patterns do you predict will occur when: b. adding two negative fractions? © Carnegie Learning a. adding two positive mixed numbers? c. adding a negative mixed number to a positive mixed number? 244 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 244 28/08/13 12:04 PM Problem 2 Subtracting on a Number Line Subtraction can mean to “take away” objects from a set. Subtraction can also mean a comparison of two numbers, or the “difference between them.” Just as with integer addition, a number line can be used to model integer subtraction. Cara said, “Subtraction means to back up, or move in the opposite direction. Like in football when a team is penalized or loses yardage, they have to move back.” Analyze Cara’s examples. Example 1: –5 – 2 –5 opposite of 2 –10 –8 –6 –4 –2 0 2 4 6 8 10 First, I moved from 0 to –5, and then I went in the opposite direction of the 2 because I am subtracting. So, I went two units to the left and ended up at –7. So, –5 – 2 = –7. Example 2: –5 – (–2) –5 opposite of –2 –10 –8 –6 –4 –2 0 2 4 6 8 10 In this problem, I went from 0 to –5. Because I am subtracting –2, I went in the opposite direction of the –2, or right two units, and ended up at –3. So, –5 – (–2) = –3. © Carnegie Learning Example 3: 5 – (–2) 5 opposite of –2 –10 –8 –6 –4 –2 0 2 4 6 8 10 1. Read aloud Cara’s examples above to a classmate. Explain the model Cara created in Example 3. Ask your teacher for help if needed. 5.1 Operating with Rational Numbers • 245 462345_C2_CH05_pp239-302.indd 245 28/08/13 12:04 PM Example 4: 5 – 2 5 –10 –8 –6 –4 –2 0 2 opposite of 2 4 6 8 10 2. Explain the model Cara created in Example 4. 3. Consider each expression in Cara’s examples. Which term in each expression had the greater absolute value? How does the sign of that term relate to the sign of the difference? Use Cara's examples for help. 4. Use the number line to complete each number sentence. a. 7 2 (23) 5 –10 –8 –6 –4 –2 0 2 4 6 8 10 –6 –4 –2 0 2 4 6 8 10 –6 –4 –2 0 2 4 6 8 10 –10 –8 c. 22 2 2 5 –10 –8 © Carnegie Learning b. 7 2 3 5 246 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 246 28/08/13 12:04 PM d. 22 2 (22) 5 –10 –8 –6 –4 –2 0 2 4 6 8 10 For a subtraction expression, such as 29 2 (24), Cara’s method is to start at zero and go to 29, and then go four spaces in the opposite direction of 24 to get 25. Dava says, “I see another pattern. Since subtraction is the inverse of addition, you can think of subtraction as adding the opposite number. That matches with Cara’s method of going in the opposite direction.” –9 – (–4) is the same as –9 + –(–4). –9 + –(–4) = –9 + 4 = –5 opposite of –4 = –(–4) –10 –8 –6 –4 –2 0 2 4 6 8 10 An example of Dava’s method is shown. 10 2 (26) 5 10 1 2(26) 5 10 1 6 5 16 © Carnegie Learning 5. Apply Dava’s method to determine each difference. a. 23 2 (23) b.28 2 5 c. 16 2 9 d.10 2 (220) So, I can change any subtraction problem to show addition if I take the opposite of the number that follows the subtraction sign. 5.1 Operating with Rational Numbers • 247 462345_C2_CH05_pp239-302.indd 247 28/08/13 12:04 PM 6. Tell whether these subtraction sentences are always true, sometimes true, or never true. Give examples to explain your thinking. a. positive 2 positive 5 positive b. negative 2 positive 5 negative c. positive 2 negative 5 negative d. negative 2 negative 5 negative 7. If you subtract two negative integers, will the answer be greater than or less than the number you started with? Explain your thinking. 9. What happens when a negative number is subtracted from zero? © Carnegie Learning 8. What happens when a positive number is subtracted from zero? 248 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 248 28/08/13 12:04 PM 10. The set of rational numbers includes integers, mixed numbers, fractions, and their equivalent decimals. Decide whether each statement is true or false. Explain your reasoning. a. Subtracting a positive rational number from a positive rational number always results in a positive answer. b. Subtracting a positive rational number from negative rational number always results in a negative answer. c. Subtracting a negative rational number from a positive rational number always results in a negative answer. d. Subtracting a negative rational number from a negative rational number always © Carnegie Learning results in a negative answer. 5.1 Operating with Rational Numbers • 249 462345_C2_CH05_pp239-302.indd 249 28/08/13 12:04 PM 11. Just by looking at the problem, how do you know if the sum of two rational numbers is positive, negative, or zero? 12. How are addition and subtraction of rational numbers related? Problem 3 Multiplying and Dividing Integers When you multiply integers, you can think of multiplication as repeated addition. Example 1 Consider the expression 3 3 (24). As repeated addition, it means (24) 1 (24) 1 (24) 5 212. You can think of 3 3 (24) as three groups of (24). – – – – – – – – – – – – (–4) –15 –12 – 5 –10 (–4) –5 – – – – – – – – – – © Carnegie Learning – (–4) 0 5 10 15 250 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 250 28/08/13 12:04 PM Example 2 Consider the expression: 4 3 (23). You can think of this as four sets of (23), or (23) 1 (23) 1 (23) 1 (23) 5 212. – – – – – – – – 5 – – – – – – – – – – – – – – (–3) –15 – – –12 (–3) (–3) –10 (–3) –5 0 5 10 15 Example 3 Consider the expression: (23) 3 (24). You know that 3 3 (24) means “three groups of (24)” and that 23 means “the opposite of 3.” So, (23) 3 (24) means “the opposite of 3 groups of (24).” © Carnegie Learning Opposite of (–4) Opposite of (–4) + + + + + + + + + + + + + + 5 + + + + + + + + + + Opposite of (–4) (+4) –15 –10 –5 0 (+4) 5 (+4) 10 12 15 5.1 Operating with Rational Numbers • 251 462345_C2_CH05_pp239-302.indd 251 28/08/13 12:04 PM 1. What is the sign of the product of two integers when: a. they are both positive? b. they are both negative? c. one is positive and one is negative? d. one is zero? 2. What is the sign of the quotient of two integers when: a. they are both positive? b. they are both negative? c. one is positive and one is negative? d. the dividend is zero? e. the divisor is zero? your reasoning. © Carnegie Learning 3. How do the answers to Question 2 compare to the answers to Question 1? Explain 252 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 252 28/08/13 12:04 PM The set of integers is a subset of the rational numbers. The set of rational numbers also includes mixed numbers, fractions, and their decimal equivalents. The rules to determine the sign of a product or a quotient of integers also applies to mixed numbers, fractions, and their decimal equivalents. Remember that the set of rational numbers are all numbers that can be written as __ a , b where a and b are integers and b 0. 4. Sketch a Venn diagram to show all of the sets and subsets of numbers that you know. 5. Use your Venn diagram to decide whether each statement is true or false. Explain your reasoning. a. An integer is sometimes a rational number. b. A rational number is always an integer. 6. How will you determine the sign of a product if there are: © Carnegie Learning a. an even number of negative rational numbers? b. an odd number of negative rational numbers? 5.1 Operating with Rational Numbers • 253 462345_C2_CH05_pp239-302.indd 253 28/08/13 12:04 PM 7. How will you determine the sign of a quotient if there are: a. an even number of negative rational numbers? b. an odd number of negative rational numbers? 8. Determine each product or quotient. 4 2 3 2 __ a. __ 5 3 254 b. _____ 218 1 d. 26.3 3 3 __ 3 e. 16.5 4 26.6 7 f. 2 ___ 3 ___ 10 10 7 88 g. 2 ___ 44 5 1 h. 2 __ 4 2 __ 5 6 © Carnegie Learning c. 27 4 49 254 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 254 28/08/13 12:04 PM Problem 4 Rational Number Operations in Context 1. Melissa has a total of $56 in her checking account. She uses her check card to make a purchase of $27.42. How much money does Melissa have left in her checking account? 2. Marco is an extreme biker. He is participating in an all-terrain biking event. During the 1 miles during 1 miles. He rides a total of 31 __ first leg of the event, he rides a total of 26 __ 5 2 the second leg. How many total miles did Marco ride during the biking event? 3. Serena plays the trumpet in her school’s marching band. She recently sold popcorn as part of a fundraiser for the band’s next field trip. On Monday, she collected $45. She collected $55.75 on Tuesday, and on Wednesday she collected $82.50. What is the total amount of money that Serena collected for the fundraiser during the three days? 4. A bus has a total of 14 passengers (not including the driver). At the next bus stop, 8 people get on the bus. At the stop after that, 11 people get off of the bus. How © Carnegie Learning many passengers remain on the bus? 5.1 Operating with Rational Numbers • 255 462345_C2_CH05_pp239-302.indd 255 28/08/13 12:04 PM © Carnegie Learning 1 square feet. The length of the 5. The area of a rectangular playground is 325 __ 2 playground is 21 feet. What is the width of the playground? Be prepared to share your solutions and methods. 256 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 256 28/08/13 12:04 PM What’s It Really Saying? Evaluating Algebraic Expressions Learning Goal Key Terms In this lesson, you will: variable algebraic expression Evaluate algebraic expressions. evaluate an algebraic expression D o you have all your ducks in a row? That’s just a drop in the bucket! That’s a piece of cake! What do each of these statements have in common? Well, they are all idioms. Idioms are expressions that have meanings which are completely different from their literal meanings. For example, the “ducks in a row” idiom refers to asking if someone is organized and ready to start a task. A person who uses this idiom is not literally asking if you have ducks that are all lined up. For people just learning a language, idioms can be very challenging to understand. Usually if someone struggles with an idiom’s meaning, a person will say “that’s just an expression,” and explain its meaning in a different way. Can you think of © Carnegie Learning other idioms? What does your idiom mean? 5.2 Evaluating Algebraic Expressions • 257 462345_C2_CH05_pp239-302.indd 257 28/08/13 12:04 PM Problem 1 Game Day Special You volunteer to help out in the concession stand at your middle-school football game. You must create a poster to display the Game Day Special: a hot dog, a bag of chips, and a drink for $3.75. 1. Complete the poster by multiplying the number of specials by the cost of the special. Game Day Special 1 hot dog, 1 bag of chips, and a drink for $3.75 Number of Specials Cost 1 2 3 4 5 6 In algebra, a variable is a letter or symbol that is used to represent a quantity. An algebraic expression is a mathematical phrase that has at least one variable, and it can contain numbers and operation symbols. Whenever you perform the same mathematical process over and over again, an algebraic expression is often used to represent the situation. 2. What algebraic expression did you use to represent the total cost on your poster? 3. The cheerleading coach wants to purchase a Game Day Special for every student on the squad. Use your algebraic expression to calculate the total cost of purchasing 18 Game Day Specials. © Carnegie Learning Let s represent the number of Game Day Specials. 258 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 258 28/08/13 12:04 PM Problem 2 Planning a Graduation Party Your aunt is planning to host your cousin’s high school graduation party at Lattanzi’s Restaurant and Reception Hall. Lattanzi’s has a flyer that describes the Deluxe Graduation Reception. Deluxe Graduation Reception Includes: One salad (chef or Cæsar) One entree (chicken, beef, or seafood) Two side dishes One dessert Fee: $105 for the reception hall plus $40 per guest 1. Write an algebraic expression to determine the cost of the graduation party. Let g represent the number of guests attending the party. So, an equation has an equals sign. An expression does not. 2. Determine the cost of the party for each number of attendees. Show your work. a. 8 guests attend © Carnegie Learning b. 10 guests attend c. 12 guests attend 5.2 Evaluating Algebraic Expressions • 259 462345_C2_CH05_pp239-302.indd 259 28/08/13 12:04 PM Problem 3 Evaluating Expressions In Problems 1 and 2, you worked with two expressions, 3.75s and (105 1 40g). You evaluated those expressions for different values of the variable. To evaluate an algebraic expression, you replace each variable in the expression with a number or numerical expression and then perform all possible mathematical operations. 1. Evaluate each algebraic expression. a. x 2 7 ● for x 5 28 ● for x 5 211 ● for x 5 16 Use parentheses to show multiplication like -6(-3). b. 26y ● for y 5 23 ● for y 5 0 ● for y 5 7 c. 3b 2 5 ● for b 5 22 ● for b 5 3 ● for b 5 9 ● for n 5 25 ● for n 5 0 ● for n 5 4 © Carnegie Learning d. 21.6 1 5.3n 260 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 260 28/08/13 12:04 PM Sometimes, it is more convenient to use a table to record the results when evaluating the same expression with multiple values. 2. Complete each table. a. h 2h 7 2 1 8 7 b. a 12 10 0 4 a __ 1 6 4 c. x x2 5 1 3 © Carnegie Learning 6 2 d. y 5 1 0 15 1 2 __ __ y 1 3 5 5 5.2 Evaluating Algebraic Expressions • 261 462345_C2_CH05_pp239-302.indd 261 28/08/13 12:04 PM Problem 4 Evaluating Algebraic Expressions Using Given Values 1. Evaluate each algebraic expression for x 5 2, 23, 0.5, and 22__ 1 . 3 a. 23x b. 5x 1 10 c. 6 2 3x d. 8x 1 75 © Carnegie Learning Using tables may help you evaluate these expressions. 262 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 262 28/08/13 12:04 PM 1 . 2. Evaluate each algebraic expression for x 5 27, 5, 1.5, and 21 __ 6 a. 5x b. 2x 1 3x c. 8x 2 3x © Carnegie Learning I'm noticing something similar about all of these expressions. What is it? 5.2 Evaluating Algebraic Expressions • 263 462345_C2_CH05_pp239-302.indd 263 28/08/13 12:04 PM 5 . 3. Evaluate each algebraic expression for x 5 23.76 and 221 __ 6 a.2.67x 2 31.85 3 x 1 56 __ 3 b. 11 __ 4 8 Talk the Talk 1. Describe your basic strategy for evaluating any algebraic expression. © Carnegie Learning 2. How are tables helpful when evaluating expressions? Be prepared to share your solutions and methods. 264 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 264 28/08/13 12:04 PM Express Math Simplifying Expressions Using Distributive Properties Learning Goals Key Terms In this lesson, you will: Write and use the distributive properties. Use distributive properties Distributive Property of Multiplication over Addition Distributive Property of Multiplication over Subtraction Distributive Property of Division over Addition Distributive Property of Division over Subtraction to simplify expressions. I t once started out with camping out the night before the sale. Then, it evolved to handing out wrist bands to prevent camping out. Now, it’s all about the Internet. What do these three activities have in common? For concerts, movie premieres, and highly-anticipated sporting events, the distribution and sale of tickets have changed with computer technology. Generally, hopeful ticket buyers log into a Web site and hope to get a chance to buy tickets. What are other ways to distribute tickets? What are other things that © Carnegie Learning routinely get distributed to people? 5.3 Simplifying Expressions Using Distributive Properties • 265 462345_C2_CH05_pp239-302.indd 265 28/08/13 12:04 PM Problem 1 Fastest Math in the Wild West Dominique and Sarah are checking each other’s math homework. Sarah Sarah is able to correctly multiply tells Dominique that she has a quick 7 by 230 in her head. She explains her and easy way to multiply a one-digit method to Dominique. number by any other number in her head. Dominique decides to test Sarah by asking her to multiply 7 by 230. 1. Calculate the product 7 3 230. Sarah 230 3 7 First, break 230 into the sum of 200 and 30. Then, multiply 7 x 200 to get a product of 1400 and 7 x 30 to get a product of 210. Finally, add 1400 and 210 together to get 1610. 2. Write an expression that shows the mathematical steps Sarah performed to calculate the product. Dominique makes the connection between Sarah’s quick mental calculation and calculating the area of a rectangle. She draws the models shown to Dominique e as Calculating 230 x 7 is the sam tangle by determining the area of a rec width. multiplying the length by the tangle into But I can also divide the rec culate the two smaller rectangles and cal then add area of each rectangle. I can al. the two areas to get the tot © Carnegie Learning represent Sarah’s thinking in a different way. 230 7 7 200 30 1400 210 1400 + 210 = 1610 266 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 266 28/08/13 12:04 PM 3. First, use Dominique’s method and sketch a model for each. Then, write an expression that shows Sarah’s method and calculate. a. 9(48) b. 6(73) I can draw at least two different models to determine 4(460). c. 4(460) © Carnegie Learning Sarah’s and Dominique’s methods are both examples of the Distributive Property of Multiplication over Addition, which states that if a, b, and c are any real numbers, then a ( b 1 c) 5 a b 1 a c. Including the Distributive Property of Multiplication over Addition, there are a total of four different forms of the Distributive Property. Another Distributive Property is the Distributive Property of Multiplication over Subtraction, which states that if a, b, and c are any real numbers, then a ( b 2 c) 5 a b 2 a c. 5.3 Simplifying Expressions Using Distributive Properties • 267 462345_C2_CH05_pp239-302.indd 267 28/08/13 12:04 PM The Distributive Property also holds true for division over addition and division over subtraction as well. The Distributive Property of Division over Addition states that if a, b, and c are real a b b c . 5 __ c 1 __ numbers and c fi 0, then ______ a 1 c The Distributive Property of Division over Subtraction states that if a, b, and c are real a b a 2 b c . 5 __ c 2 __ numbers and c fi 0, then ______ c 4. Draw a model for each expression, and then simplify. a. 6(x 1 9) c. 2(4a 1 1) b. 7(2b 2 5) x 1 15 d. _______ 5 Dividing by 5 is the same as multiplying by what number? 5. Use one of the Distributive Properties to rewrite each expression in a. 3y(4y 1 2) b. 12( x 1 3) c. 24a(3b 2 5) d. 27y(2y 2 3x 1 9) 6m 1 12 e. ________ 22 22 2 4x f. ________ 2 © Carnegie Learning an equivalent form. 268 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 268 28/08/13 12:04 PM Problem 2 Simplifying and Evaluating 1. Simplify each expression. Show your work. a. 26(3x 1 (24y)) b. 24(23x 2 8) 2 34 27.2 2 6.4x c. ____________ 20.8 ( )( ) ( )( ) © Carnegie Learning 1 3__ 1 22 __ 1 1 1 22 __ d.22 __ 2 4 2 4 ( ) 1 1 5y 27 __ 2 ___________ e. 1 2 __ 2 5.3 Simplifying Expressions Using Distributive Properties • 269 462345_C2_CH05_pp239-302.indd 269 28/08/13 12:04 PM 2. Evaluate each expression for the given value. Then, use properties to simplify the original expression. Finally, evaluate the simplified expression. 2 a. 2x (23x 1 7) for x 5 21 __ 3 4.2x 27 b. ________ for x 5 1.26 1.4 © Carnegie Learning c. Which form—simplified or not simplified—did you prefer to evaluate? Why? Be prepared to share your solutions and methods. 270 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 270 28/08/13 12:04 PM Reverse Distribution Factoring Algebraic Expressions Learning Goals Key Terms In this lesson, you will: factor common factor greatest common Use the distributive properties to factor expressions. Combine like terms to simplify expressions. factor (GCF) coefficient like terms combining like terms M any beginning drivers have difficulty with driving in reverse. They think that they must turn the wheel in the opposite direction of where they want the back end to go. But actually, the reverse, is true. To turn the back end of the car to the left, turn the steering wheel to the left. To turn the back end to the right, turn the wheel to the right. Even after mastering reversing, most people would have difficulty driving that way all the time. But not Rajagopal Kawendar. In 2010, Kawendar set a world © Carnegie Learning record for driving in reverse—over 600 miles at about 40 miles per hour! 5.4 Factoring Algebraic Expressions • 271 462345_C2_CH05_pp239-302.indd 271 28/08/13 12:04 PM Problem 1 Factoring You can use the Distributive Property in reverse. Consider the expression: 7(26) 1 7(14) Since both 26 and 14 are being multiplied by the same number, 7, the Distributive Property says you can add the multiplicands together first, and then multiply their sum by 7 just once. 7(26) 1 7(14) 5 7(26 1 14) You have factored the original expression. To factor an expression means to rewrite the expression as a product of factors. The number 7 is a common factor of both 7(26) and 7(14). A common factor is a number or an algebraic expression that is a factor of two or more numbers or algebraic expressions. 1. Factor each expression using a Distributive Property. a. 4(33) 2 4(28) b. 16(17) 1 16(13) The Distributive Properties can also be used in reverse to factor algebraic expressions. For example, the expression 3x 1 15 can be written as 3( x) 1 3(5), or 3( x 1 5). The factor, 3, is the greatest common factor to both terms. The greatest common factor (GCF) is the largest factor that two or more numbers or terms have in common. When factoring algebraic expressions, you can factor out the greatest common factor from all the terms. Consider the expression 12x 1 42. The greatest common factor of 12x and 42 is 6. Therefore, you can rewrite the It is important to pay attention to negative numbers. When factoring an expression that contains a negative leading coefficient, or first term, it is preferred to factor out the negative sign. A coefficient is the number that is multiplied by a variable in an © Carnegie Learning expression as 6(2x 1 7). algebraic expression. 272 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 272 28/08/13 12:04 PM How can you check to make sure you factored correctly? Look at the expression 22x 1 8. You can think about the greatest common factor as being the coefficient of 22. 22x 1 8 5 (22)x 1 (22)(24) 5 22(x 4) 2. Rewrite each expression by factoring out the greatest common factor. a. 7x 1 14 b. 9x 2 27 c. 10y 2 25 d. 8n 1 28 e. 3x2 2 21x f. 24a2 1 18a g. 15mn 2 35n h. 23x 2 27 © Carnegie Learning i. 26x 1 30 So, when you factor out a negative number all the signs will change. 5.4 Factoring Algebraic Expressions • 273 462345_C2_CH05_pp239-302.indd 273 28/08/13 12:05 PM Problem 2 Using the Distributive Properties to Simplify So far, the Distributive Properties have provided ways to rewrite given algebraic expressions in equivalent forms. You can also use the Distributive Properties to simplify algebraic expressions. Consider the algebraic expression 5x 1 11x. 1. What factors do the terms have in common? 2. Rewrite 5x 1 11x using the Distributive Property. 3. Simplify your expression. The terms 5x and 11x are called like terms, meaning that their variable portions are the same. When you add 5x and 11x together, you are combining like terms. 4. Simplify each expression by combining like terms. a. 5ab 1 22ab b. 32x2 2 44x2 5. Simplify each algebraic expression by combining like terms. If the expression is a. 6x 1 9x b. 213y 2 34y c. 14mn 2 19mn d. 8mn 2 5m e. 6x2 1 12x2 2 7x2 f. 6x2 1 12x2 2 7x g. 23z 2 8z 2 7 © Carnegie Learning already simplified, state how you know. h. 5x 1 5y 274 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 274 28/08/13 12:05 PM Problem 3 More Factoring and Evaluating 1. Factor each expression. a. 224x 1 16y 5 b. 24.4 2 1.21z 5 c. 227x 2 33 5 d. 22x 2 9y 5 e. 4x 1 (25xy) 2 3x 5 2. Evaluate each expression for the given value. Then factor the expression and evaluate the factored expression for the given value. b. 30x 2 140 for x 5 5.63 © Carnegie Learning 1 a. 24x 1 16 for x 5 2 __ 2 c. Which form—simplified or not simplified—did you prefer to evaluate? Why? 5.4 Factoring Algebraic Expressions • 275 462345_C2_CH05_pp239-302.indd 275 28/08/13 12:05 PM Problem 4 Combining Like Terms and Evaluating 1. Simplify each expression by combining like terms. a. 30x 2 140 2 23x 5 b. 25(22x 2 13) 2 7x 5 c. 24x 2 5(2x 2 y) 2 3y 5 d. 7.6x 2 3.2(3.1x 2 2.4) 5 ( ) 3 4x 2 2 __ 2 x 2 1 __ 1 5 e. 3 __ 7 3 4 expression and evaluate the simplified expression for the given value. a. 25x 2 12 1 3x for x 5 2.4 © Carnegie Learning 2. Evaluate each expression for the given value. Then combine the like terms in each 276 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 276 28/08/13 12:05 PM ( ) 1 x 2 1 __ 2 for x = 21 __ 1 2 6x 1 2 __ b. 22 __ 3 5 4 2 © Carnegie Learning Why doesn't it change the answer when I simplify first? Be prepared to share your solutions and methods. 5.4 Factoring Algebraic Expressions • 277 462345_C2_CH05_pp239-302.indd 277 28/08/13 12:05 PM © Carnegie Learning 278 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 278 28/08/13 12:05 PM Are They the Same or Different? Verifying That Expressions Are Equivalent Learning Goals In this lesson, you will: Simplify algebraic expressions. Verify that algebraic expressions are equivalent by graphing, simplifying, and evaluating expressions. B art and Lisa are competing to see who can get the highest grades. But they are in different classes. In the first week, Lisa took a quiz and got 9 out of 10 correct for a 90%. Bart took a test and got 70 out of 100 for a 70%. Looks like Lisa won the first week! The next week, Lisa took a test and got 35 out of 100 correct for a 35%. Bart took a quiz and got 2 out of 10 correct for a 20%. Lisa won the second week also! Over the two weeks, it looks like Lisa was the winner. But look at the total number of questions and the total each of them got correct: Lisa answered a total of 110 questions and got a total of 34 correct for about a 31%. Bart answered a total of © Carnegie Learning 110 questions and got a total of 72 correct for a 65%! Is Bart the real winner? This surprising result is known as Simpson’s Paradox. Can you see how it works? 5.5 Verifying That Expressions Are Equivalent • 279 462345_C2_CH05_pp239-302.indd 279 28/08/13 12:05 PM Problem 1 Are They Equivalent? Consider this equation: 24(3.2x 2 5.4) 5 12.8x 1 21.6 Keegan says that to tell if the expressions in an equation So, equivalent is not the same as equal ? What's the difference? are equivalent (not just equal), you just need to evaluate each expression for the same value of x. 1. Evaluate the expression on each side of the equals sign for x 5 2. 2. Are these expressions equivalent? Explain your reasoning. Jasmine t two expressions are not Keegan’s method can prove tha that they are equivalent. equivalent, but it can’t prove 3. Explain why Jasmine is correct. Provide an example of two expressions that verify Kaitlyn t, two expressions are equivalen There is a way to prove that to try and turn one expression not just equal: use properties algebraically into the other. © Carnegie Learning your answer. 280 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 280 11/04/14 5:10 PM 4. Reconsider again the equation 24(3.2x 2 5.4) 5 12.8x 1 21.6. Use the distributive properties to simplify the left side and to factor the right side to try to determine if these expressions are equivalent. 5. Are the expressions equivalent? Explain your reasoning. 6. Will Kaitlyn’s method always work? Explain your reasoning. Jason verify the equivalence of There is another method to then if their graphs are the same, expressions: graph them, and ivalent. the expressions must be equ 7. Graph each expression on your graphics calculator or other graphing technology. Sketch the graphs. If you are using your graphing calculator, don't forget to set your bounds. © Carnegie Learning y 36 32 28 24 20 16 12 8 4 –12 –10 –8 –6 –4 –2 –4 –8 –12 –16 –20 –24 –28 –32 –36 2 4 6 8 10 12 x 5.5 Verifying That Expressions Are Equivalent • 281 462345_C2_CH05_pp239-302.indd 281 28/08/13 12:05 PM 8. Are the expressions equivalent? Explain your reasoning. 9. Which method is more efficient for this problem? Explain your reasoning. Problem 2 Are These Equivalent? Determine whether the expressions are equivalent using the specified method. 1. 3.1(22.3x 2 8.4) 1 3.5x 5 23.63x 1 (226.04) by evaluating for x 5 1. © Carnegie Learning Remember, evaluating can only prove that two expressions are not equivalent. 282 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 282 28/08/13 12:05 PM ( ( ) ) 1 5 4 __ 3 5 22 __ 1 23x 2 2 ___ 1 by simplifying each side. 1 4x 1 __ 2. 3 __ 2 2 1 __ 5 3 4 2 4 10 3. Graph each expression using graphing technology to determine if these are equivalent expressions. Sketch the graphs. y 36 32 28 24 20 16 12 8 4 © Carnegie Learning –12 –10 –8 –6 –4 –2 –4 –8 –12 –16 –20 –24 –28 –32 –36 2 4 6 8 10 12 x 5.5 Verifying That Expressions Are Equivalent • 283 462345_C2_CH05_pp239-302.indd 283 28/08/13 12:05 PM Problem 3 What About These? Are They Equivalent? For each equation, verify whether the expressions are equivalent using any of these methods. Identify the method and show your work. ( ) ( ) 6 1 12 __ 1 1 5 __ 2 23x 1 1 __ 1 5 26 __ 2 1 6x 2 2 __ 1. 3 __ 3 3 3 5 5 3 © Carnegie Learning 2. 4.2(23.2x 2 8.2) 2 7.6x 5 215.02x 1 3(4.1 2 3x) 284 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 284 28/08/13 12:05 PM ( ) 1 x 2 2 ___ 1 5 10x 2 3( x 1 1 )1 7.2 3. 2223 __ 2 10 y 36 32 28 24 20 16 12 8 4 –6 –4 –2 –4 –8 –12 –16 –20 –24 –28 –32 –36 2 4 6 8 10 12 x © Carnegie Learning –12 –10 –8 Be prepared to share your solutions and methods. 5.5 Verifying That Expressions Are Equivalent • 285 462345_C2_CH05_pp239-302.indd 285 28/08/13 12:05 PM © Carnegie Learning 286 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 286 28/08/13 12:05 PM It is time to justify! Simplifying Algebraic Expressions Using Operations and Their Properties Learning Goal In this lesson, you will: Simplify algebraic expressions using operations and their properties. H ave you ever been asked to give reasons for something that you did? When did this occur? Were your reasons accepted or rejected? Is it always important to © Carnegie Learning have reasons for doing something or believing something? 5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 287 462345_C2_CH05_pp239-302.indd 287 28/08/13 12:05 PM Problem 1 Justifying!! One method for verifying that algebraic expressions are equivalent is to simplify the expressions into two identical expressions. For this equation, the left side is simplified completely to show that the two expressions are equivalent. ( ) 1 x 1 3 1 1__ 1 x 2 __ 22 __ 2 1 2 5 23 __ 5 2 3 3 Step ( Justification ) 1 1 2 __ 22 __ 1__ x 2 1 2 5 5 2 3 2 1 1 ( )( ) ( )( ) 5 4 5 2 2 __ __ x 1 2__ 2__ 1 2 5 5 2 3 2 1 1 1 Distributive Property of Multiplication over Addition 10 2 ___ x 1 1 1 2 5 3 Multiplication 1 13 23 __ x 3 Addition Yes, they are equivalent. © Carnegie Learning Given 288 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 288 28/08/13 12:05 PM 1. Use an operation or a property to justify each step and indicate if the expressions are equivalent. a. 24(23x 2 8) 2 4x 1 8 5 28x 1 40 Step Justification 24(23x 2 8) 2 4x 1 8 5 12x 1 32 1 (24x) 1 8 5 12x 1 (24x) 1 32 1 8 5 28x 1 40 5 b. 22.1(23.2x 2 4) 1 1.2(2x 2 5) 5 9.16x 1 3.4 Step Justification 22.1(23.2x 2 4) 1 1.2(2x 2 5) 5 6.72x 1 8.4 1 2.4x 1 (26) 5 © Carnegie Learning 6.72x 1 2.4x 1 8.4 1 (26) 5 9.16x 1 2.4 5 5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 289 462345_C2_CH05_pp239-302.indd 289 28/08/13 12:05 PM ( ) 13 24x 2 9 23x 1 7 c. ________ 1 ________ 5 23x 1 2___ 6 2 3 Step Justification 1 7 2 9 ________ ________ 24x 23x 1 5 2 3 ( ) 9 7 5 ____ ____ 2 __ 1 23x __ 24x 1 1 2 2 3 3 ( ) 9 7 5 22x 1 2 __ 1 (2x) 1 __ 2 3 ( ) 9 7 5 22x 1 (2x) 1 2 __ 1 __ 2 3 ( ) 27 14 5 23x 1 2 ___ 1 ___ 6 6 ( ) © Carnegie Learning 13 23x 1 2 ___ 5 6 290 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 290 28/08/13 12:05 PM 2. For each equation, simplify the left side completely using the given operation or property that justifies each step and indicate if the expressions are equivalent. 26 ___ 6x 2 7 5 2 x 1 __ 7 a. 24x 2 _______ 5 5 5 Step Justification Given Distributive Property of Division over Subtraction Division Addition of Like Terms Yes, they are equivalent. b. 24x 2 3(3x 2 6) 1 8(2.5x 1 3.5) 5 7x 1 46 Step Justification Given Distributive Property of Multiplication over Addition © Carnegie Learning Addition Commutative Property of Addition Addition of Like Terms Yes, they are equivalent. 5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 291 462345_C2_CH05_pp239-302.indd 291 28/08/13 12:05 PM Problem 2 Simplify and Justify! For each equation, simplify the left side completely to determine if the two expressions are equivalent. Use an operation or a property to justify each step and indicate if the expressions are equivalent. 1. 24x 1 3(27x 1 3) 2 2(23x 1 4) 5 219x 1 1 Step Justification I think it's easier to simplify first and then come up with the reasons when I'm done. What do you think? 2. 25(x 1 4.3) 2 5(x 1 4.3) 5 210x 2 43 Justification © Carnegie Learning Step 292 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 292 28/08/13 12:05 PM For each equation, simplify the left side and the right side completely to determine if the two expressions are equivalent. Use an operation or a property to justify each step and indicate if the expressions are equivalent. 3. 23(2x 1 17) 1 7(2x 1 30) 28x 5 26x 1 3(2x 1 50) 1 9 Left side Step Justification Right side Justification © Carnegie Learning Step 5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 293 462345_C2_CH05_pp239-302.indd 293 28/08/13 12:05 PM 1 (4x 1 6) 1 1 __ 1 (26x 2 3) 2 4x 5 25 __ 1 x 2 13 1 2 __ 1 (26x 1 1) 2 5 __ 1 x 2 14 __ 1 4. 23 __ 2 3 2 2 2 2 Left side Step Justification Right side Justification © Carnegie Learning Step Be prepared to share your solutions and methods. 294 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 294 28/08/13 12:05 PM Chapter 5 Summary Key Terms Properties Distributive Property of Multiplication variable (5.2) over Addition (5.3) algebraic expression (5.2) Distributive Property of Multiplication evaluate an algebraic expression (5.2) over Subtraction (5.3) factor (5.4) Distributive Property of Division over common factor (5.4) Addition (5.3) greatest common factor (GCF) (5.4) Distributive Property of Division over coefficient (5.4) Subtraction (5.3) like terms (5.4) combining like terms (5.4) Modeling Integer Addition and Integer Subtraction on a Number Line A number line can be used to model integer addition. When adding a positive integer, move to the right on the number line. When adding a negative integer, move to the left on the number line. Subtraction means to move in the opposite direction on the number line. When subtracting a positive integer, move to the left on the number line. When subtracting a negative integer, move to the right on the number line. Examples 28 1 3 5 25 –8 3 © Carnegie Learning –15 –10 –5 0 5 10 15 0 5 10 15 210 2 (26) 5 24 (–10) –(–6) –15 –10 –5 Chapter 5 Summary • 295 462345_C2_CH05_pp239-302.indd 295 28/08/13 12:05 PM Adding and Subtracting Integers When adding two integers with the same sign, add the integers and keep the sign. When adding integers with opposite signs, subtract the integers and keep the sign of the integer with the greater absolute value. Because subtraction is the inverse of addition, it is the same as adding the opposite number. Examples 29 1 (212) 5 2(9 1 12) 5 221 7 1 (215) 5 28 27 2 19 5 27 1 (219) 5 226 12 2 21 5 12 1 (221) 5 29 Adding and Subtracting Rational Numbers When adding and subtracting positive and negative rational numbers, follow the same rules as when adding and subtracting integers. When adding rational numbers with the same sign, add the numbers and keep the sign. When the rational sign of the number with the greater absolute value. Because subtraction is the inverse of addition, it is the same as adding the opposite number. Examples 28.54 1 (23.4) 5 2(8.54 1 3.4) 5 211.94 3 ) 5 2(10 __ 3 2 5 __ 1 1 (210 __ 2 ) 5 25 __ 1 5 __ 2 4 4 4 4 5 ) 5 27 __ 5 5 3 __ 3 1 2 (210 __ 2 1 10 __ 27 __ 4 8 8 8 8 28.5 2 3.4 5 28.5 1 (23.4)5 211.9 Having trouble remembering something new? Your brain learns by association so create a mnemonic, a song, or a story about the information and it will be easier to remember! © Carnegie Learning numbers have different signs, subtract the numbers and keep the 296 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 296 28/08/13 12:05 PM Multiplying Integers When multiplying integers, multiplication can be thought of as repeated addition. Example Consider the expression 3 3 (23). As repeated addition, it means (23) 1 (23) 1 (23) 5 29. The expression 3 3 (23) can be thought of as three groups of (23). = (-3) -15 -10 -9 (-3) -5 (-3) 0 5 10 15 Determining the Sign of a Product or Quotient The sign of a product or quotient of two integers depends on the signs of the two integers © Carnegie Learning being multiplied or divided. The product or quotient will be positive when both integers have the same sign. The product or quotient will be negative when one integer is positive and the other is negative. Examples 3 3 2 5 6 64253 (23) 3 (22) 5 6 (26) 4 (22) 5 3 3 3 (22) 5 26 6 4 (22) 5 23 (23) 3 2 5 26 (26) 4 2 5 23 Chapter 5 Summary • 297 462345_C2_CH05_pp239-302.indd 297 28/08/13 12:05 PM Picturing the Rational Numbers The Venn diagram represents relationships between natural numbers, whole numbers, integers, and rational numbers. From the Venn diagram, you can see that natural numbers are a subset of whole numbers. Natural numbers and whole numbers are subsets of integers. Natural numbers, whole numbers, and integers are subsets of rational numbers. The set of rational numbers is all numbers that can be written as __ a , where a and b are b integers and b fi 0. Rational numbers include mixed numbers, fractions, and their equivalent decimals. Rational Integers Whole Natural Multiplying and Dividing Rational Numbers The rules used to determine the sign of a product or quotient of two integers also apply when multiplying and dividing rational numbers. Example The product or quotient of each are shown following the rules for determining the sign for each. 4 1 5 ___ 1 1 3 5 __ 13 3 ___ 16 5 ___ 13 3 __ 4 5 ___ 52 5 17 __ 3 __ 4 3 4 3 1 3 3 3 1 9 2 1 5 9 2 18 27 ___ 27 3 4 1 __ 3 7 5 2 ___ 26 __ 4 15 5 2 ___ 3 ___ 8 5 2 __ 3 __ 5 2 ___ 5 23 __ 5 4 4 1 5 5 4 8 8 15 258.75 4 26.25 5 9.40 © Carnegie Learning 12.1 3 25.6 5 267.76 298 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 298 28/08/13 12:05 PM Writing Algebraic Expressions When a mathematical process is repeated over and over, a mathematical phrase, called an algebraic expression, can be used to represent the situation. An algebraic expression is a mathematical phrase involving at least one variable and sometimes numbers and operation symbols. Example The algebraic expression 2.49p represents the cost of p pounds of apples. One pound of apples costs 2.49(1), or $2.49. Two pounds of apples costs 2.49(2), or $4.98. Evaluating Algebraic Expressions To evaluate an algebraic expression, replace each variable in the expression with a number or numerical expression and then perform all possible mathematical operations. Example The expression 2x 2 7 has been evaluated for these values of x: 9, 2, 23, and 4.5. 2(9) 2 7 5 18 2 7 5 11 2(2) 2 7 5 4 2 7 2(23) 2 7 5 26 2 7 5 23 2(4.5) 2 7 5 9 2 7 5 213 52 Using the Distributive Property of Multiplication over Addition to Simplify Numerical Expressions The Distributive Property of Multiplication over Addition states that if a, b, and c are any real numbers, then a • (b 1 c) 5 a • b 1 a • c. © Carnegie Learning Example A model is drawn and an expression written to show how the Distributive Property of Multiplication over Addition can be used to solve a multiplication problem. 6(820) 800 20 6 4800 120 6(800 1 20) 5 6(800) 1 6(20) 5 4800 1 120 5 4920 Chapter 5 Summary • 299 462345_C2_CH05_pp239-302.indd 299 28/08/13 12:05 PM Using the Distributive Properties to Simplify and Evaluate Algebraic Expressions Including the Distributive Property of Multiplication over Addition, there are a total of four different forms of the Distributive Property. Another Distributive Property is the Distributive Property of Multiplication over Subtraction, which states that if a, b, and c are any real numbers, then a • (b 2 c) 5 a • b 2 a • c. The Distributive Property of Division over Addition states that if a, b, and c are real a 1 __ b 5 __ c b numbers and c fi 0, then ______ a 1 c c . The Distributive Property of Division over Subtraction states that if a, b, and c are real a 2 __ a 2 b 5 __ c b numbers and c fi 0, then ______ c c . Example The Distributive Properties have been used to simplify the algebraic expression. The simplified expression is then evaluated for x 5 2. 4(6x 2 7) 1 10 ______________ 1 10 ______________ 5 24x 2 28 3 3 18 5 _________ 24x 2 3 8x 2 6 5 8(2) 2 6 18 24x 2 ___ 5 ____ 3 3 5 8x 2 6 5 16 2 6 5 10 Using the Distributive Properties to Factor Expressions The Distributive Properties can be used in reverse to rewrite an expression as a product of factors. When factoring expressions, it is important to factor out the greatest common factor from all the terms. The greatest common factor (GCF) is the largest factor that two Example The expression has been rewritten by factoring out the greatest common factor. 24x3 1 3x2 2 9x 5 3x(8x2) 1 3x(x) 2 3x(3) 5 3x(8x2 1 x 2 3) © Carnegie Learning or more numbers or terms have in common. 300 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 300 28/08/13 12:05 PM Combining Like Terms to Simplify Expressions Like terms are terms whose variable portions are the same. When you add like terms together, you are combining like terms. You can combine like terms to simplify algebraic expressions to make them easier to evaluate. Example Like terms have been combined to simplify the algebraic expression. The simplified expression is then evaluated for x 5 5. 7x 2 2(3x 2 4) 5 7x 2 6x 1 8 5x18 x185518 5 13 Determining If Expressions Are Equivalent by Evaluating Evaluate the expression on each side of the equal sign for the same value of x. If the results are the same, then the expressions are equivalent. Example The expressions are equivalent. (x 1 12) 1 (4x 2 9) 5 5x 1 3 for x 5 1 (1 1 12) 1 (4 ? 1 2 9) 0 5 ? 1 1 3 13 1 (25) 0 5 1 3 8 5 8 Determining If Expressions Are Equivalent by Simplifying The Distributive Properties and factoring can be used to determine if the expressions on © Carnegie Learning each side of the equal sign are equivalent. Example The expressions are equivalent. ( ) 1 2 5 2 __ (45x 1 6) 5 23x 2 __ 3 5 215x 2 2 5 215x 2 2 Chapter 5 Summary • 301 462345_C2_CH05_pp239-302.indd 301 28/08/13 12:05 PM Determining If Two Expressions Are Equivalent by Graphing Expressions can be graphed to determine if the expressions are equivalent. If the graph of each expression is the same, then the expressions are equal. Example 3(x 1 3) 2 x 5 3x 1 3 is not true because the graph of each expression is not the same. y 14 3(x + 16 3)x 18 12 3x+3 10 8 6 4 2 2 4 6 8 10 12 14 16 18 x Simplifying Algebraic Expressions Using Operations and Their Properties Simplify each side completely to determine if the two expressions are equivalent. Use an operation or a property to justify each step and indicate if the expressions are equivalent. Example Step Justification 23(22x 1(29)) 2 3x 2 7 5 Given 6x 1 27 1 (23x) 1 (27) 5 Distributive Property of Multiplication over Addition 6x 1 (23x) 1 27 1 (27) 5 Commutative Property of Addition 3x 1 20 5 Addition of Like Terms Yes, they are equivalent. © Carnegie Learning 23(22x 2 9) 2 3x 2 7 5 3x 1 20 302 • Chapter 5 Numerical and Algebraic Expressions and Equations 462345_C2_CH05_pp239-302.indd 302 28/08/13 12:05 PM