Apply Properties of 1.1 Real Numbers

Transcription

1 TEKS Apply Properties of 1.1 Real Numbers a.1, a.6 Before Now You performed operations with real numbers. You will study properties of real numbers. Why? So you can order elevations, as in Ex. 58. Key Vocabulary opposite reciprocal KEY CONCEPT Subsets of the Real Numbers For Your Notebook The real numbers consist of the rational numbers and the irrational numbers. Two subsets of the rational numbers are the whole numbers (0, 1, 2, 3,...) and the integers (..., 23, 22, 21, 0, 1, 2, 3,...) Rational Numbers Integers Whole Numbers 16 REAL NUMBERS Irrational Numbers Ï Ï π Rational Numbers can be written as quotients of integers can be written as decimals that terminate or repeat Irrational Numbers cannot be written as quotients of integers cannot be written as decimals that terminate or repeat NUMBER LINE Real numbers can be graphed as points on a line called a real number line, on which numbers increase from left to right. E XAMPLE 1 Graph real numbers on a number line Graph the real numbers 2} 5 and Ï } 3 on a number line. 4 Note that 2} Use a calculator to approximate Ï } 3 to the nearest tenth: 4 Ï } 3 < 1.7. (The symbol < means is approximately equal to.) So, graph 2} 5 between 22 and 21, and graph Ï } 3 between 1 and 2, as shown on 4 the number line below Chapter 1 Equations and Inequalities

2 E XAMPLE 2 TAKS PRACTICE: Multiple Choice The table shows the lowest temperatures ever recorded in six states. Which list shows the temperatures from lowest to highest? State Alaska Colorado Florida Montana New York Rhode Island Lowest temperature 2808F 2618F 228F 2708F 2528F 2238F ELIMINATE CHOICES The problem asks for the temperatures from lowest to highest, not from highest to lowest. So, you can eliminate choice C. A 280, 270, 252, 261, 223, 22 B 280, 270, 22, 261, 252, 223 C 22, 223, 252, 261, 270, 280 D 280, 270, 261, 252, 223, 22 From lowest to highest, the temperatures are 280, 270, 261, 252, 223, and 22. c The correct answer is D. A B C D GUIDED PRACTICE for Examples 1 and 2 1. Graph the numbers 20.2, 7 }, 21, Ï } 2, and 24 on a number line Which list shows the numbers in increasing order? A 20.5, 1.5, 22, 20.75, Ï } 7 B 20.5, 22, 20.75, 1.5, Ï } 7 C 22, 20.75, 20.5, 1.5, Ï } 7 D Ï } 7, 1.5, 20.5, 20.75, 22 PROPERTIES OF REAL NUMBERS You learned in previous courses that when you add or multiply real numbers, there are several properties you can use. KEY CONCEPT For Your Notebook Properties of Addition and Multiplication Let a, b, and c be real numbers. Property Addition Multiplication Closure a 1 b is a real number. ab is a real number. Commutative a 1 b 5 b 1 a ab5 ba Associative (a 1 b) 1 c 5 a 1 (b 1 c) (ab)c 5 a(bc) Identity a a, 0 1 a 5 a a p 1 5 a, 1 p a 5 a Inverse a 1 (2a) 5 0 a p } 1 a 5 1, a? 0 The following property involves both addition and multiplication. Distributive a(b 1 c) 5 ab 1 ac 1.1 Apply Properties of Real Numbers 3

3 E XAMPLE 3 Identify properties of real numbers Identify the property that the statement illustrates. a b. 13 p 1 } a. Commutative property b. Inverse property of of addition multiplication KEY CONCEPT For Your Notebook Defining Subtraction and Division Subtraction is defined as adding the opposite. The opposite, or additive inverse, of any number b is 2b. If b is positive, then 2b is negative. If b is negative, then 2b is positive. a 2 b 5 a 1 (2b) Definition of subtraction Division is defined as multiplying by the reciprocal. The reciprocal, or multiplicative inverse, of any nonzero number b is } 1. b a 4 b 5 a p 1 } b, b? 0 Definition of division E XAMPLE 4 Use properties and definitions of operations Use properties and definitions of operations to show that a 1 (2 2 a) 5 2. Justify each step. a1 (2 2 a) 5 a 1 [2 1 (2a)] Definition of subtraction 5 a 1 [(2a) 1 2] Commutative property of addition 5 [a 1 (2a)] 1 2 Associative property of addition Inverse property of addition 5 2 Identity property of addition GUIDED PRACTICE for Examples 3 and 4 Identify the property that the statement illustrates. 3. (2 p 3) p p (3 p 9) (5 1 25) 5 4(5) 1 4(25) 6. 1 p Use properties and definitions of operations to show that the statement is true. Justify each step. 7. b p (4 4 b) 5 4 when b? x 1 (6 1 4x) 5 7x Chapter 1 Equations and Inequalities

4 UNIT ANALYSIS When you use operations in real-life problems, you should use unit analysis to check that the units in your calculations make sense. E XAMPLE 5 Use unit analysis with operations a. You work 4 hours and earn $36. What is your earning rate? b. You travel for 2.5 hours at 50 miles per hour. How far do you go? c. You drive 45 miles per hour. What is your speed in feet per second? a. 36 dollars } 4 hours 5 9 dollars per hour b. ( miles hours)1} 1 hour miles c. 45 miles 1 hour 1 minute 5280 feet 1} 1 hour 21} 60 minutes 21} 60 seconds 21} 1 mile feet per second at classzone.com E XAMPLE 6 Use unit analysis with conversions DRIVING DISTANCE The distance from Montpelier, Vermont, to Montreal, Canada, is about 132 miles. The distance from Montreal to Quebec City is about 253 kilometers. a. Convert the distance from Montpelier to Montreal to kilometers. b. Convert the distance from Montreal to Quebec City to miles. a. 132 miles p b. 253 kilometers p 1.61 kilometers } 1 mile ø 213 kilometers 1 mile } 1.61 kilometers ø 157 miles GUIDED PRACTICE for Examples 5 and 6 Solve the problem. Use unit analysis to check your work. 9. You work 6 hours and earn $69. What is your earning rate? 10. How long does it take to travel 180 miles at 40 miles per hour? REVIEW MEASURES For help with converting units, see the Table of Measures on p You drive 60 kilometers per hour. What is your speed in miles per hour? Perform the indicated conversion yards to feet gallons to pints years to seconds 1.1 Apply Properties of Real Numbers 5

5 1.1 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 21, 31, and 59 5 TAKS PRACTICE AND REASONING Exs. 9, 10, 23, 24, 60, 61, 63, and VOCABULARY Copy and complete: The? of any nonzero number b is 1 } b. 2. WRITING Express the associative property of addition in words. EXAMPLE 1 on p. 2 for Exs. 3 8 GRAPHING NUMBERS Graph the numbers on a number line. 3. 2} 3, 5, 9 }, 22, , } 5, 2, 2} 9, , Ï } 3, 2} 2, 2} 5, , 2Ï } 5, 2.7, 22, 7 } , 3 } 2, 0, Ï } 10, , 5, 9 } 2, 2Ï } 8, 23 EXAMPLE 2 on p. 3 for Exs ORDERING NUMBERS In Exercises 9 and 10, use the table of elevations below. State Alabama California Kentucky Louisiana Tennessee Highest elevation 2407 ft 14,494 ft 4145 ft 535 ft 6643 ft Lowest elevation 0 ft 2282 ft 257 ft 28 ft 178 ft 9. MULTIPLE TAKS REASONING CHOICE Which list shows the highest elevations in order from least to greatest? A 2407; 14,494; 4145; 535; 6643 B 535; 2407; 4145; 6643; 14,494 C 14,494; 2407; 4145; 535; 6643 D 14,494; 6643; 4145; 2407; 535 Louisiana bayou 10. TAKS REASONING LE CHOICE Which list shows the lowest elevations in order from greatest to least? A 0, 28, 178, 257, 2282 B 2282, 28, 0, 178, 257 C 2282, 257, 178, 28, 0 D 257, 178, 0, 28, 2282 EXAMPLE 3 on p. 4 for Exs EXAMPLE 4 on p. 4 for Exs IDENTIFYING PROPERTIES Identify the property that the statement illustrates. 11. (4 1 9) (9 1 3) p p p (25) (2 1 8) 5 7(2) 1 7(8) 16. (6 p 5) p p (5 p 7) USING PROPERTIES Use properties and definitions of operations to show that the statement is true. Justify each step p (a 4 3) 5 2a p (3 4 b) b 19. (c 2 3) c 20. (a 1 b) 2 c 5 a 1 (b 2 c) 21. 7a 1 (4 1 5a) 5 12a (12b 1 15) 2 3b b 23. OPEN-ENDED TAKS REASONING MATH Find values of a and b such that a is a whole number, b is a rational number but not an integer, and a 4 b OPEN-ENDED TAKS REASONING MATH Write three equations using integers to illustrate the distributive property. 6 Chapter 1 Equations and Inequalities

6 EXAMPLE 5 on p. 5 for Exs OPERATIONS AND UNIT ANALYSIS Solve the problem. Use unit analysis to check your work. 25. You work 10 hours and earn $85. What is your earning rate? 26. You travel 60 kilometers in 1.5 hours. What is your average speed? 27. You work for 5 hours at $7.25 per hour. How much do you earn? 28. You buy 6 gallons of juice at $1.25 per gallon. What is your total cost? 29. You drive for 3 hours at 65 miles per hour. How far do you go? 30. You ride in a train for 175 miles at an average speed of 50 miles per hour. How many hours does the trip take? EXAMPLE 6 on p. 5 for Exs CONVERSION OF MEASUREMENTS Perform the indicated conversion feet to yards meters to millimeters kilograms to grams hours to minutes quarts to gallons tons to pounds ounces to tons seconds to hours at classzone.com ERROR ANALYSIS Describe and correct the error in the conversion dollars p 1 dollar } 0.82 euro ø 30.5 euros pints p 1 cup } 2 pints cups CONVERSION OF RATES Convert the rate into the given units mi/h to feet per second ft/sec to miles per hour km/h to miles per hour mi/h to kilometers per hour gal/h to ounces per second oz/sec to gallons per hour 47. ROCKET SLED On a track at an Air Force base in New Mexico, a rocket sled travels 3 miles in 6 seconds. What is the average speed in miles per hour? 48. ELEVATOR SPEED The elevator in the Washington Monument takes 60 seconds to rise 500 feet. What is the average speed in miles per hour? REASONING Tell whether the statement is always, sometimes, or never true for real numbers a, b, and c. Explain your answer. 49. (a 1 b) 1 c 5 a 1 (b 1 c) 50. (a p b) p c 5 a p (b p c) 51. (a 2 b) 2 c 5 a 2 (b 2 c) 52. (a 4 b) 4 c 5 a 4 (b 4 c) 53. a(b 2 c) 5 ab 2 ac 54. a(b 4 c) 5 ab 4 ac 55. REASONING Show that a } b 4 c } d 5 a } c 4 b } d for nonzero real numbers a, b, c, and d. Justify each step in your reasoning. 56. CHALLENGE Let a } b and c } d be two distinct rational numbers. Find the rational number that lies exactly halfway between a } b and c } d on a number line. 1.1 Apply Properties of Real Numbers 7

7 PROBLEM SOLVING EXAMPLE 2 on p. 3 for Exs MINIATURE GOLF The table shows the scores of people playing 9 holes of miniature golf. Lance Darcy Javier Sandra a. Find the sum of the scores for each player. b. List the players from best (lowest) to worst (highest) total score. 58. VOLCANOES The following list shows the elevations (in feet) of several volcano summits above or below sea level. 641, 3976, 610, 259, 1718, 1733, 2137 Order the elevations from lowest to highest. 59. MULTI-STEP PROBLEM The chart shows the average daytime surface temperatures on the planets in our solar system. a. Sort by Temperature List the planets in order from least to greatest daytime surface temperature. b. Sort by Distance List the planets in order from least to greatest distance from the sun. c. Find Patterns What pattern do you notice between surface temperature and distance from the sun? d. Analyze Which planet does not follow the general pattern you found in part (c)? Mercury 7258F Mars 2248F Uranus 23208F Daytime Surface Temperatures Venus 8658F Jupiter 21608F Neptune 23308F Earth 688F Saturn 22208F Pluto 23708F EXAMPLES 5 and 6 on p. 5 for Exs TAKS REASONING RESPONSE The average weight of the blue whale (the largest mammal) is 120 tons, and the average weight of the bumblebee bat (the smallest mammal) is 0.07 ounce. a. Convert Convert the weight of the blue whale from tons to pounds. Convert the weight of the bumblebee bat from ounces to pounds. b. Compare About how many times as heavy as the bat is the blue whale? c. Find a Method Besides converting the weights to pounds, what is another method for comparing the weights of the mammals? 8 5 WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING

8 61. SHORT TAKS REASONING RESPONSE The table shows the maximum speeds of various animals in miles per hour or feet per second. Animal Speed (mi/h) Speed (ft/s) Cheetah 70? Three-toed sloth? 0.22 Squirrel 12? Grizzly bear? 44 Three-toed sloth a. Copy and complete the table. b. Compare the speeds of the fastest and slowest animals in the table. 62. CHALLENGE A newspaper gives the exchange rates of some currencies with the U.S. dollar, as shown below. Copy and complete the statements. 1 USD in USD Australian dollar Canadian dollar Hong Kong dollar New Zealand dollar Singapore dollar This row indicates that $1 U.S. ø $1.31 Australian and $1 Australian ø $.76 U.S. a. 1 Singapore dollar ø? Canadian dollar(s) b. 1 Hong Kong dollar ø? New Zealand dollar(s) MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 976; TAKS Workbook 63. TAKS PRACTICE Susan purchased a television on sale for $315. The original price of the television was $370. Which expression can be used to determine the percent of the original price that Susan saved on the purchase of this television? TAKS Obj. 10 A 315 } B 370 } C D } } REVIEW TAKS Preparation p. 324; TAKS Workbook 64. TAKS PRACTICE In the figure, what is the length of } QR in inches? TAKS Obj. 6 P 102 in. 80 in. 48 in. P S R F 86 in. G 90 in. H 122 in. J 154 in. EXTRA PRACTICE for Lesson 1.1, p ONLINE 1.1 Apply QUIZ Properties at classzone.com of Real Numbers 9