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, 674 stretching/compressing factor, 928, 929 substitution method, 1216, 1332 Index 1560 vertex, 476, 635, 810, 893, 1344, 1344, 1387, 1395 vertex form of a quadratic function, 479, 635 vertical asymptote, 582, 586, 592, 635, 951 vertical compression, 336, 380 vertical line, 140, 418, 466 vertical line test, 242, 380 vertical reflection, 328, 380 vertical shift, 318, 380, 411, 670, 704, 760, 908 vertical stretch, 336, 380, 410, 707 vertices, 1344, 1345 volume, 152, 217 volume of a sphere, 499 W whole numbers, 10, 15, 100 X x-axis, 108, 217 x-coordinate, 109, 217 x-intercept, 116, 217, 416 Y y-axis, 108, 217 y-coordinate, 109, 217 y-intercept, 116, 217, 395 Z zero-product property, 173, 217 zeros, 477, 525, 530, 566, 635, 1117 sum and difference formulas for cosine, 990 sum and difference formulas for sine, 993 sum and difference formulas for tangent, 996 sum-to-product formula, 1049 sum-to-product formulas, 1025 summation notation, 1488, 1534 surface area, 609 symmetry test, 1112 synthetic division, 555, 566, 635 system of equations, 1287, 1288, 1290, 1292, 1309 system of linear equations, 444, 1212, 1215, 1216, 1332 system of nonlinear equations, 1246, 1332 system of nonlinear inequalities, 1254, 1332 system of three equations in three variables, 1321 T tangent, 873, 893, 925, 926 tangent function, 926, 927, 928, 944, 978 term, 1446, 1463, 1534 term of a polynomial, 68, 100 term of a polynomial function, 506, 635 terminal point, 1178, 1182, 1202 terminal side, 811, 893 transformation, 317, 410 translation, 1348 transverse axis, 1365, 1437 trigonometric equations, 1158 trigonometric functions, 877 trigon
ometric identities, 1080 trinomial, 68, 100 turning point, 513, 534, 635 U union of two events, 1524, 1534 unit circle, 817, 835, 851, 865, 893, 1032 unit vector, 1188, 1202 upper limit of summation, 1488, 1534 upper triangular form, 1233 V variable, 23, 100 varies directly, 625, 635 varies inversely, 627, 635 vector, 1178, 1202 vector addition, 1183, 1202er cannot be expressed in the form where a and b are integers and b 0. b 18 Number Systems When writing an irrational number, we use three dots (...) after a series of digits to indicate that the number does not terminate. The dots do not indicate a pattern, and no raised bar can be placed over any digits. In an irrational number, we are never certain what the next digit will be when these dots (...) are used. In this section, we will see more examples of irrational numbers, both positive and negative. First, however, we need to review a few terms you learned in earlier mathematics courses. Squares and Square Roots To square a number means to multiply the number by itself. For example: The square of 3 is 9. The square of 4 is 16. 32 3 · 3 9 42 4 · 4 16 Calculators have a special key, x2, that will square a number. ENTER: 5 x2 ENTER DISPLAY: 5 2 2 5 To find a square root of a number means to find a number that, when mul- tiplied by itself, gives the value under the radical sign,. For example: 9 16 3 4 " " A square root of 9 equals 3 because 3 · 3 9. A square root of 16 equals 4 because 4 · 4 16. " Calculators also have a key, This key is often the second function of the ¯ x2 key. For example:, that will display the square root of a number. ENTER: 2nd ¯ 25 ENTER DISPLAY: √ ( 2 5 5 When the square root key is pressed, the calculator displays a square root sign followed by a left parenthesis. It is not necessary to close the parentheses if the entire expression that follows is under the radical sign. However, when other numbers and operations follow that are not part of the expression under the radical sign, the right parenthesis must be entered to indicate the end of the radical
expression. More Irrational Numbers The Irrational Numbers 19 2 When a square measures 1 unit on every side, its diagonal measures units. You can use a ruler to " measure the diagonal and then on a show the placement of number line. " 2 What is the value of 2? Can we find a decimal number that, when multiplied by itself, equals 2? 2 We expect to be somewhere between 1 and 2. " " 1 2 1 1 2 Use a calculator to find the 0 1 2 value. ENTER: 2nd ¯ 2 ENTER DISPLAY Check this answer by multiplying: 1.414213562 1.414213562 1.999999999, too small. 1.414213563 1.414213563 2.000000002, too large. Note that if, instead of rewriting the digits displayed on the screen, we square the answer using 2nd ANS, the graphing calculator will display 2 because in that case it uses the value of calculator, which has more decimal places than are displayed on the screen. that is stored in the memory of the " 2 No matter how many digits can be displayed on a calculator, no terminating decimal, nor any repeating decimal, can be found for 2 because " 2 is an irrational number. " In the same way, an infinite number of square roots are irrational numbers, for example: 3 " 5 " 3.2 " 0.1 " 2 2 " 3 2 " 20 Number Systems The values displayed on a calculator for irrational square roots are called rational approximations. A rational approximation for an irrational number is a rational number that is close to, but not equal to, the value of the irrational number. The symbol ≈ means approximately equal to. Therefore, it is not correct to write 3 5 1.732, but it is correct to write 3 < 1.732. " Another interesting number that you have encountered in earlier courses is p, read as “pi.” Recall that p equals the circumference of a circle divided by its " diameter, or p 5 C d. C d p is an irrational number. There are many rational approximations for p, including: p ≈ 3.14 p < 22 7 If p is doubled, or divided by two, or if a rational number is added to or subtracted from p, the result is again an irrational number. There are infinitely many such irrational numbers, for example: p ≈ 3.1416 2p p 2 p 7 p – 3 App
roximation Scientific calculators have a key that, when pressed, will place in the display a rational approximation for p that is more accurate than the ones given above. p On a graphing calculator, when the key is accessed, the screen shows the symbol p but a rational approximation is used in the calculation. On a graphing calculator: ENTER: 2nd p ENTER DISPLAY The Irrational Numbers 21 With a calculator, however, you must be careful how you interpret and use the information given in the display. At times, the value shown is exact, but, more often, displays that fill the screen are rational approximations. To write a rational approximation to a given number of decimal places, round the number. Procedure To round to a given decimal place: 1. Look at the digit in the place at the immediate right of the decimal place to which you are rounding the number. 2. If the digit being examined is less than 5, drop that digit and all digits to the right. (Example: 3.1415927... rounded to two decimal places is 3.14 because the digit in the third decimal place, 1, is less than 5.) 3. If the digit being examined is greater than or equal to 5, add 1 to the digit in the place to which you are rounding and then drop all digits to the right. (Example: 3.1415927... rounded to four decimal places is 3.1416 because the digit in the fifth decimal place, 9, is greater than 5.) EXAMPLE 1 True or False: 5 1 " Solution Use a calculator. 5 5 " " 10? Explain why. ENTER: 2nd ¯ 5 ) 2nd ¯ 5 ENTER DISPLAY ENTER: 2nd ¯ 10 ENTER DISPLAY Use these rational approximations to conclude that the values are not equal. Answer False. 5 1 5 2 " " " 10 because 5 1 " " 5. 4 while 10, 4. " 22 Number Systems EXAMPLE 2 Find a rational approximation for each irrational number, to the nearest hundredth. a. 0.1 b. 3 " " Solution Use a calculator. a. ENTER: 2nd ¯ 3 ENTER b. ENTER: 2nd ¯.1 ENTER DISPLAY: √ ( 3 DISPLAY Use the rules for rounding. The digit in the thousandths place, 2, is less than 5. Drop this digit and all digits to the right of it. The digit in the thousandths place, 6, is greater than or
equal to 5. Add 1 to the digit in the hundredths place and drop all digits to the right of it. Answer: 3 ≈ 1.73 " Answer: " 0.1 < 0.32 EXAMPLE 3 The circumference C of a circle with a diameter d is found by using the formula C pd. a. Find the exact circumference of a circle whose diameter is 8. b. Find, to the nearest thousandth, a rational approximation of the circumfer- ence of this circle. Solution a. C pd C p · 8 or 8p b. Use a calculator. ENTER: 2nd p 8 ENTER DISPLAY Round the number in the display to three decimal places: 25.133. Answers a. 8p is the exact circumference, an irrational number. b. 25.133 is the rational approximation of the circumference, to the nearest thousandth. The Irrational Numbers 23 EXAMPLE 4 Which of the following four numbers is an irrational number? In each case, the... that follows the last digit indicates that the established pattern of digits repeats. (1) 0.12 (2) 0.12121212... (3) 0.12111111... (4) 0.12112111211112... Solution Each of the first three numbers is a repeating decimal. Choice (1) is a terminating decimal that can be written with a repeating zero. Choice (2) repeats the pair of digits 12 from the first decimal place and choice (3) repeats the digit 1 from the third decimal place. In choice (4), the pattern increases the number of times the digit 1 occurs after each 2. Therefore, (4) is not a repeating decimal and is irrational. Answer (4) 0.12112111211112... is irrational. EXERCISES Writing About Mathematics 1. Erika knows that the sum of two rational numbers is always a rational number. Therefore, she concludes that the sum of two irrational numbers is always an irrational number. Give some examples that will convince Erika that she is wrong. 2. Carlos said that 3.14 is a better approximation for p than 22 7. Do you agree with Carlos? Explain your answer. Developing Skills In 3–22, tell whether each number is rational or irrational. 3. 0.36 4. 0.36363636... 5. 0.36 6. 0.363363336... 7.
8 " 11. 0.989989998... 15. 5.28 8. 10p 12. 0.725 16. 0.14141414... 19. 48 20. 49 " " 9. 0.12131415... 13. 17. 21. 121 " 2 5 " 0.24682 16 10. " 14. p 30 18. –p 22. p – 2 23. Determine which of the following irrational numbers are between 1 and 4. (1) p 2 (2) 5 " 2 (3) " 4 (4) 11 " (5) 2 3 " 24 Number Systems In 24–43 write the rational approximation of each given number: a. as shown on a calculator display, b. rounded to the nearest thousandth (three decimal places) c. rounded to the nearest hundredth (two decimal places). 24. 29. 34. 39. 5 90 " " 12 " 2 " 82 25. 30. 35. 40. 7 " 2 14 " 16 6.5 " " 44. A rational approximation for 22 19 " 2 " 17 " 3 26. 31. 36. 41. 2 55 " is 1.732. 3 " 27. 32. 37. 42. 75 0.2 " " p 3 1,732 " 28. 33. 38. 43. 63 0.3 " " 0.17 " 241 " a. Multiply 1.732 by 1.732. 45. a. Find (3.162)2. b. Which is larger, b. Find (3.163)2. 3 or 1.732? " c. Is 3.162 or 3.163 a better approximation for 10? Explain why. " In 46–50, use the formula C pd to find, in each case, the circumference C of a circle when the diameter d is given. a. Write the exact value of C by using an irrational number. b. Find a rational approximation of C to the nearest hundredth. 46. d 7 47. d = 15 48. d 72 49. d 1 2 50. d 31 3 51. True or False: 52. True or False: 4 1 " 18 1 4 5 " 18 5 " " " 8? Explain why or why not. 36? Explain why or why not. " Hands-On Activity Cut two squares, each of which measures 1 foot on each side. Cut each square along a
diagonal (the line joining opposite corners of the square). Arrange the four pieces of the squares into a larger square. a. What is the area of each of the two squares that you cut out? b. What is the area of the larger square formed by using the pieces of the smaller squares? c. What should be the length of each side of the larger square? Is this length rational or irrational? d. Measure the length of each side of the larger square? Is this measurement rational or irrational? e. Should the answers to parts c and d be the same? Explain your answer. 1-4 THE REAL NUMBERS The Real Numbers 25 Recall that rational numbers can be written as repeating decimals, and that irrational numbers are decimals that do not repeat. Taken together, rational and irrational numbers make up the set of all numbers that can be written as decimals. The set of real numbers is the set that consists of all rational numbers and all irrational numbers. The accompanying diagram shows that the rational numbers are a subset of the real numbers, and the irrational numbers are also a subset of the real numbers. Notice, however, that the rationals and the irrationals take up different spaces in the diagram because they have no numbers in common. Together, these two sets of numbers form the real numbers. The cross-hatched shaded portion in the diagram contains no real numbers. The cross-hatched shading indicates that no other numbers except the rationals and irrationals are real numbers. Real Numbers Irrational Numbers Rational Numbers We have seen that there are an infinite number of rational numbers and an infinite number of irrationals. For every rational number, there is a corresponding point on the number line, and, for every irrational number, there is a corresponding point on the number line. All of these points, taken together, make up the real number line. Since there are no more holes in this line, we say that the real number line is now complete. The completeness property of real numbers may be stated as follows: Every point on the real number line corresponds to a real number, and every real number corresponds to a point on the real number line. Ordering Real Numbers There are two ways in which we can order real numbers: 1. Use a number line. On the standard horizontal real number line, the graph of the greater number is always to the right of the graph of the smaller number. 26 Number Systems 2. Use decimals. Given any two real numbers that are not equal
, we can express them in decimal form (even using rational approximations) to see which is greater. EXAMPLE 1 The number line that was first seen in Section 1-1 is repeated below. –2 13 – 6 –1 2– 0 — –0.43.8 Of the numbers shown here, tell which are: a. counting numbers b. whole numbers c. integers d. rational numbers e. irrational numbers f. real numbers. Solution a. Counting numbers: b. Whole numbers: c. Integers: d. Rational Numbers: e. Irrational numbers f. Real numbers: All: 1, 2, 3, 4 0, 1, 2, 3, 4 2, 1, 0, 1, 2, 3, 4 213 6, 22, 21, 20.43, 0, 1 2, 1, 2, 23 4, 3, 3.8, 4 3, p 2, 2 " " 213 6, 22, 2 " 2, 21, 20.43, 0, 1 2, 1, 3, 2, 23 4, 3, p, 3.8, 4 " EXAMPLE 2 Order these real numbers from least to greatest, using the symbol. 0.3 0.3 " 0.3 Solution STEP 1. Write each real number in decimal form: 0.3 0.3000000... 0.3 ≈ 0.547722575 (a rational approximation, displayed on a calculator) " 0.3 0.3333333... STEP 2. Compare these decimals: 0.3000000... 0.3333333... 0.547722575 STEP 3. Replace each decimal with the number in its original form: 0.3 0.3 0.3 " Answer 0.3 0.3 0.3 " The Real Numbers 27 EXERCISES Writing About Mathematics 1. There are fewer than 6 persons in my family. The board is less than 6 feet long. Each of the given statements can be designated by the inequality x 6. How are the numbers that make the first statement true different from those that make the second statement true? How are they the same? 2. Dell said that it is impossible to decide whether p is larger or smaller than 10 because the calculator gives only rational approximations for these numbers. Do you agree with Dell? Explain. " 3. The decimal form of a real number consists of two digits that repeat for the first one
- hundred decimal places. The digits in the places that follow the one-hundredth decimal place are random, form no pattern, and do not terminate. Is the number rational or irrational? Explain. Developing Skills 4. Twelve numbers have been placed on a number line as shown here. –2 3– – –2.7 –1 0 –0.63 1 0.5 1 3 2 π 2 6 Of these numbers, tell which are: a. counting numbers b. whole numbers c. integers d. rational numbers e. irrational numbers f. real numbers 5. Given the following series of numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 " " " " " " " " " " Of these ten numbers, tell which is (are): a. rational b. irrational c. real 6. Given the following series of numbers: p, 2p, 3p, 4p, 5p Of these five numbers, tell which is (are): a. rational b. irrational c. real In 7–18, determine, for each pair, which is the greater number. 7. 2 or 2.5 11. 0.7 or 0.7 15. 3.14 or p 8 8. 8 or " 12. 5.6 or 5.9 16. 0.5 or 0.5 " 0.2 or 0.22 9. 13. 0.43 or 0.431 2 or 1.414 17. " 10. 0.2 or 0.23 14. 0.21 or 0.2 18. p or.22 7 28 Number Systems In 19–24, order the numbers in each group from least to greatest by using the symbol. 19. 0.202, 0.2, 0.2022 20. 0.4, 0.45, 0.4499 21. 0.67, 0.6, 0.667 22. 2 " 2, 2 " 3, 21.5 23. 0.5, 0.5, 0.3 " 24. p,, 3.15 10 " In 25–34, tell whether each statement is true or false. 25. Every real number is a rational number. 26. Every rational number is a real number. 27. Every irrational number is a real number. 28. Every real number is an irrational number. 29. Every rational number corresponds to a point on the real number line. 30. Every point on the real number
line corresponds to a rational number. 31. Every irrational number corresponds to a point on the real number line. 32. Every point on the real number line corresponds to an irrational number. 33. Some numbers are both rational and irrational. 34. Every repeating decimal corresponds to a point on the real number line. Hands-On Activity a. Using a cloth or paper tape measure, find, as accurately as you can, the distance across and the distance around the top of a can or other object that has a circular top. If you do not have a tape measure, fit a narrow strip of paper around the circular edge and measure the length of the strip with a yardstick. b. Divide the measure of the circumference, the distance around the circular top, by the measure of the diameter, the distance across the circular top at its center. c. Repeat steps a and b for other circular objects and compare the quotients obtained in step b. Compare your results from step b with those of other members of your class. What conclusions can you draw? 1-5 NUMBERS AS MEASUREMENTS In previous sections, we defined the subsets of the real numbers. When we use a counting number to identify the number of students in a class or the number of cars in the parking lot, these numbers are exact. However, to find the length of a block of wood, we must use a ruler, tape measure, or some other measuring instrument. The length that we find is dependent upon the instrument we use to measure and the care with which we make the measurement. Numbers as Measurements 29 For example, in the diagram, a block of wood is placed along the edge of a ruler that is marked in tenths of an inch. We might say that the block of wood is 2.7 inches in length but is this measure exact? Inches 1 2 3 All measurements are approximate. When we say that the length of the block of wood is 2.7 inches, we mean that it is closer to 2.7 inches than it is to 2.6 inches or to 2.8 inches. Therefore, the true measure of the block of wood whose length is given as 2.7 inches is between 2.65 and 2.75 inches. In other words, the true measure is less than 0.05 inches from 2.7 and can be written as 2.7 0.05 inches. The value 0.05 is called the greatest possible error (GPE) of measurement and is half of the place value of
the last digit. Significant Digits The accuracy of measurement is often indicated in terms of the number of significant digits. Significant digits are those digits used to determine the measure and excludes those zeros that are used as place holders at the beginning of a decimal fraction and at the end of an integer. Rules for Determining Significant Digits RULE 1 All nonzero digits are significant. 135.6 has four significant digits. All digits are significant. RULE 2 All zeros between significant digits are significant. 130.6 has four significant digits. The zero is significant because it is between significant digits. RULE 3 All zeros at the end of a decimal fraction are significant. 135.000 has six significant digits. The three zeros at the end of the decimal fraction are significant. Zeros that precede the first nonzero digit in a decimal fraction are RULE 4 not significant. 0.00424 has three significant digits. The zeros that precede the nonzero digits in the decimal fraction are placeholders and are not significant. 30 Number Systems Zeros at the end of an integer may or may nor be significant. RULE 5 Sometimes a dot is placed over a zero if it is significant. 4,500 has two significant digits. Neither zero is significant. 4,50˙ 0 has three significant digits. The zero in the tens place is significant but the zero in the ones place is not. 4,500˙ has four significant digits. The zero in the ones place is significant. Therefore, the zero in the tens place is also significant because it is between significant digits. In any problem that uses measurement, the rules of greatest possible error and significant digits are used to determine how the answer should be stated. We can apply these rules to problems of perimeter and area. Recall the formulas for perimeter and area that you learned in previous courses. Let P represent the perimeter of a polygon, C the circumference of a circle, and A the area of any geometric figure. Triangle Rectangle Square Circle P a b c P 2l 2w P 4s C pd or C 2pr 2bh A 5 1 A lw A s2 A pr2 Precision The precision of a measurement is the place value of the last significant digit in the number. The greatest possible error of a measurement is one-half the place value of the last significant digit. In the measurement 4,500 feet, the last significant digit is in the hundreds place. Therefore, the greatest possible error is 100 50. We can write
the measurement as 4,500 50 feet. One number is said to be more precise than another if the place value of its last significant digit is smaller. For example, 3.40 is more precise than 3.4 because 3.40 is correct to the nearest hundredth and 3.4 is correct to the nearest tenth. 1 2 When measures are added, the sum can be no more precise than the least precise number of the given values. For example, how should the perimeter of a triangle be stated if the measures of the sides are 34.2 inches, 27.52 inches, and 29 inches? P a b c P 34.2 27.52 29 90.72 Since the least precise measure is 29 which is precise to the nearest integer, the perimeter of the triangle should be given to the nearest integer as 91 inches. Numbers as Measurements 31 Accuracy The accuracy of a measure is the number of significant digits in the measure. One number is said to be more accurate than another if it has a larger number of significant digits. For example, 0.235 is more accurate than 0.035 because 0.235 has three significant digits and 0.035 has two, but 235 and 0.235 have the same degree of accuracy because they both have three significant digits. When measures are multiplied, the product can be no more accurate than the least accurate of the given values. For example, how should the area of a triangle be stated if the base measures 0.52 meters and the height measures 0.426 meters? A A (0.52)(0.426) 0.5(0.52)(0.426) 0.11076 1 2bh 1 2 Since the less accurate measure is 0.52, which has two significant digits, the area should be written with two significant digits as 0.11 square meters. Note that the 1 or 0.5 is not a measurement but an exact value that has been determined by 2 counting or by reasoning and therefore is not used to determine the accuracy of the answer. One last important note: when doing multi-step calculations, make sure to keep at least one more significant digit in intermediate results than is needed in the final answer. For example, if a computation requires three significant digits, then use at least four significant digits in your calculations. Otherwise, you may encounter what is known as round-off error, which is the phenomena that occurs when you discard information contained in the extra digit, skewing your calculations. In this text, you will often be asked
to find the answer to an exercise in which the given numbers are thought of as exact values and the answers are given as exact values. However, in certain problems that model practical applications, when the given data are approximate measurements, you may be asked to use the precision or accuracy of the data to determine how the answer should be stated. EXAMPLE 1 State the precision and accuracy of each of the following measures. a. 5.042 cm b. 12.0 ft c. 93,000,000 mi 32 Number Systems Solution a. 5.042 cm b. 12.0 ft c. 93,000,000 mi EXAMPLE 2 Precision thousandths tenths millions Accuracy 4 significant digits 3 significant digits 2 significant digits Of the measurements 125 feet and 6.4 feet, a. which is the more precise? b. which is the more accurate? Solution The measurement 125 feet is correct to the nearest foot, has an error of 0.5 feet, and has three significant digits. The measurement 6.4 feet is correct to the nearest tenth of a foot, has an error of 0.05 feet, and has two significant digits. Answers a. The measure 6.4 feet is more precise because it has the smaller error. b. The measure 125 feet is more accurate because it has the larger number of significant digits. EXAMPLE 3 The length of a rectangle is 24.3 centimeters and its width is 18.76 centimeters. Using the correct number of significant digits in the answer, express a. the perimeter b. the area. Solution a. Use the formula for the perimeter of a rectangle. P 2l 2w P 2(24.3) 2(18.76) P 86.12 Perimeter is a sum since 2l means l l and 2w means w w. The answer should be no more precise than the least precise measurement. The least precise measurement is 24.3, given to the nearest tenth. The perimeter should be written to the nearest tenth as 86.1 centimeters. b. To find the area of a rectangle, multiply the length by the width. A lw A (24.3)(18.76) A 455.868 Area is a product and the answer should be no more accurate than the least accurate of the given dimensions. Since there are three significant digits in 24.3 and four significant digits in 18.76, there should be three significant digits in the answer. Therefore, the area should be written as 456 square centimeters. Answers
a. 86.1 cm b. 456 sq cm Numbers as Measurements 33 EXERCISES Writing about Mathematics 1. If 12.5 12.50, explain why a measure of 12.50 inches is more accurate and more precise than a measurement of 12.5 inches. 2. A circular track has a radius of 63 meters. Mario rides his bicycle around the track 10 times. Mario multiplied the radius of the track by 2p to find the circumference of the track. He said that he rode his bicycle 4.0 kilometers. Olga said that it would be more correct to say that he rode his bicycle 4 kilometers. Who is correct? Explain your answer. Developing Skills In 3–10, for each of the given measurements, find a. the accuracy b. the precision c. the error. 3. 24 in. 4. 5.05 cm 5. 2,400 ft 6. 454 lb 7. 0.0012 kg 8. 1.04 yd 9. 1.005 m 10. 900 mi In 11–14, for each of the following pairs, select the measure that is a. the more precise b. the more accurate. 11. 57 in. and 4,250 in. 12. 2.50 ft and 2.5 ft 13. 0.0003 g and 32 g 14. 500 cm and 0.055 m Applying Skills In 15–18, express each answer to the correct number of significant digits. 15. Alicia made a square pen for her dog using 72.4 feet of fencing. a. What is the length of each side of the pen? b. What is the area of the pen? 16. Corinthia needed 328 feet of fencing to enclose her rectangular garden. The length of the garden is 105 feet. a. Find the width of the garden. b. Find the area of the garden. 17. Brittany is making a circular tablecloth. The diameter of the tablecloth is 10.5 inches. How much lace will she need to put along the edge of the tablecloth? 18. The label on a can of tomatoes is a rectangle whose length is the circumference of the can and whose width is the height of the can. If a can has a diameter of 7.5 centimeters and a height of 10.5 centimeters, what is the area of the label? 34 Number Systems CHAPTER SUMMARY A set is a collection of distinct objects or elements. The counting numbers or natural numbers are {1, 2,
3, 4,...}. The whole numbers are {0, 1, 2, 3, 4,...}. The integers are {..., 4, 3, 2, 1, 0, 1, 2, 3, 4,...}. These sets of numbers form the basis for a number line, on which the length of a segment from 0 to 1 is called the unit measure of the line. a The rational numbers are all numbers that can be expressed in the form b where a and b are integers and b 0. Every rational number can be expressed as a repeating decimal or as a terminating decimal (which is actually a decimal in which 0 is repeated). The irrational numbers are decimal numbers that do not terminate and do not repeat. On calculators and in the solution of many problems, rational approximations are used to show values that are close to, but not equal to, irrational numbers. The real numbers consist of all rational numbers and all irrational numbers taken together. On a real number line, every point represents a real number and every real number is represented by a point. The precision of a measurement is determined by the place value of the last significant digit. The accuracy of a measurement is determined by the number of significant digits in the measurement. VOCABULARY 1-1 Mathematics • Real number • Number • Numeral • Counting numbers • Natural numbers • Successor • Whole numbers • Set • Finite set • Digit • Infinite set • Empty set • Null set • Numerical expression • Simplify • Negative numbers • Opposites • Integers • Subset • Number line • Graph • Standard number line • Unit measure • Absolute value • Inequality 1-2 Rational numbers • Everywhere dense • Common fraction • Decimal fraction • Terminating decimal • Repeating decimal • Periodic decimal 1-3 Irrational numbers • Square • Square root • Radical sign • Rational approximation • Pi (p) • Round 1-4 Real numbers • Real number line • Completeness property of real numbers 1-5 Greatest possible error (GPE) • Significant digits • Precision • Accuracy REVIEW EXERCISES In 1–5, use a calculator to evaluate each expression and round the result to the nearest hundredth. Review Exercises 35 1. 29.73 14.6 " 6. Order the numbers 5, 3, and 1 using the symbol. 2. 38 9 3. 12.232 4. 216 5. p 12 In 7–10, state
whether each sentence is true of false. 7. 7 8 8. –7 2 9. 4 8 10. 9 9 In 11–16, write each rational number in the form, where a and b are integers and b 0. a b 11. 0.9 12. 0.45 13. 81 2 14. 14 15. 0.3 16. 63 17. Find a rational number between 19.9 and 20. In 18–22, tell whether each number is rational or irrational. 18. 0.64 19. 22. 0.040040004... 6 " 20. 64 " 21. p In 23–27, write a rational approximation of each given number: a. as shown on a calculator display b. rounded to the nearest hundredth. 23. 11 " 24. 0.7 " 25. 905 " 1,599 26. " 27. p In 28–32, determine which is the greater number in each pair. 20 28. 5 or " 31. 0.41 or 0.4 32. 0.12 or 0.121 29. 12 8 or 12 8 30. 3.2 or p In 33–37, tell whether each statement is true or false. 33. Every integer is a real number. 34. Every rational number is an integer. 35. Every whole number is a counting number. 36. Every irrational number is a real number. 37. Between 0 and 1, there is an infinite number of rational numbers. 38. Draw a number line, showing the graphs of these numbers: 0, 1, 4, 3, 1.5, and p. 36 Number Systems In 39 and 40, use the given number line where the letters are equally spaced 39. Find the real number that corresponds to each point indicated by a letter shown on the number line when C 0 and E 1. 40. Between what two consecutive points on this number line is the graph of: a. 1.8 b. 0.6 c. 2 d. p e. 6 " " 41. The distance across a circular fountain (the diameter of the fountain) is 445 centimeters. The distance in centimeters around the fountain (the circumference of the fountain) can be found by multiplying 445 by p. a. Find the circumference of the fountain in centimeters. Round your answer to the nearest ten centimeters. b. When the circumference is rounded to the nearest ten centimeters, are the zeros significant? Exploration Using only the digits 5 and 6, and
without using a radical sign: a. Write an irrational number. b. Write three irrational numbers that are between 5 and 6 in increasing order. c. Write three irrational numbers that are between 0.55 and 0.56 in increasing order. d. Write three irrational numbers that are between 0.556 and 0.556 in increas- ing order. OPERATIONS AND PROPERTIES Jesse is fascinated by number relationships and often tries to find special mathematical properties of the five-digit number displayed on the odometer of his car. Today Jesse noticed that the number on the odometer was a palindrome and an even number divisible by 11, with 2 as three of the digits.What was the five-digit reading? (Note: A palindrome is a number, word, or phrase that is the same read left to right as read right to left, such as 57375 or Hannah.) In this chapter you will review basic operations of arithmetic and their properties. You will also study operations on sets. CHAPTER 2 CHAPTER TABLE OF CONTENTS 2-1 Order of Operations 2-2 Properties of Operations 2-3 Addition of Signed Numbers 2-4 Subtraction of Signed Numbers 2-5 Multiplication of Signed Numbers 2-6 Division of Signed Numbers 2-7 Operations With Sets 2-8 Graphing Number Pairs Chapter Summary Vocabulary Review Exercises Cumulative Review 37 38 Operations and Properties 2-1 ORDER OF OPERATIONS The Four Basic Operations in Arithmetic Bicycles have two wheels. Bipeds walk on two feet. Biceps are muscles that have two points of origin. Bilingual people can speak two languages. What do these bi-words have in common with the following examples? 6.3 0.9 7.2 113 7 5 92 7 7 2 21 21.4 3 64.2 9 4 2 5 41 2 The prefix bi- means “two.” In each example above, an operation or rule was followed to replace two rational numbers with a single rational number. These familiar operations of addition, subtraction, multiplication, and division are called binary operations. Each of these operations can be performed with any pair of rational numbers, except that division by zero is meaningless and is not allowed. In every binary operation, two elements from a set are replaced by exactly one element from the same set. There are some important concepts to remember when working with binary operations: 1. A set must be identified, such as the set of whole numbers or
the set of rational numbers. When no set is identified, use the set of all real numbers. 2. The rule for the binary operation must be clear, such as the rules you know for addition, subtraction, multiplication, and division. 3. The order of the elements is important. Later in this chapter, we will use the notation (a, b) to indicate an ordered pair in which a is the first element and b is the second element. For now, be aware that answers may be different depending on which element is first and which is second. Consider subtraction. If 8 is the first element and 5 is the second element, then: 8 5 3. But if 5 is the first element and 8 is the second element, then: 5 8 3 4. Every problem using a binary operation must have an answer, and there must be only one answer. We say that each answer is unique, meaning there is one and only one answer. DEFINITION A binary operation in a set assigns to every ordered pair of elements from the set a unique answer from the set. Note that, even when we find the sum of three or more numbers, we still add only two numbers at a time, indicating the binary operation: 4 9 7 (4 9) 7 13 7 20 Order of Operations 39 Factors When two or more numbers are multiplied to give a certain product, each number is called a factor of the product. For example: • Since 1 16 16, then 1 and 16 are factors of 16. • Since 2 8 16, then 2 and 8 are factors of 16. • Since 4 4 16, then 4 is a factor of 16. • The numbers 1, 2, 4, 8, and 16 are all factors of 16. Prime Numbers A prime number is a whole number greater than 1 that has no whole number factors other than itself and 1. The first seven prime numbers are 2, 3, 5, 7, 11, 13, 17. Whole numbers greater than 1 that are not prime are called composite numbers. Composite numbers have three or more whole number factors. Some examples of composite numbers are 4, 6, 8, 9, 10. Bases, Exponents, Powers When the same number appears as a factor many times, we can rewrite the expression using exponents. For example, the exponent 2 indicates that the factor appears twice. In the following examples, the repeated factor is called a base. 4 4 16 can be written as 42 16. 42 is read as “4 squared,
” or “4 raised to the second power,” or “the second power of 4.” Exponent Base 42 = 16 Power The exponent 3 indicates that a factor is used three times. 4 4 4 64 can be written as 43 64. 43 is read as “4 cubed,” or “4 raised to the third power,” or “the third power of 4.” Exponent Base 43 = 64 Power 40 Operations and Properties The examples shown above lead to the following definitions: DEFINITION A base is a number that is used as a factor in the product. An exponent is a number that tells how many times the base is to be used as a factor. The exponent is written, in a smaller size, to the upper right of the base. A power is a number that is a product in which all of its factors are equal. A number raised to the first power is equal to the number itself, as in 61 6. Also, when no exponent is shown, the exponent is 1, as in 9 91. EXAMPLE 1 Compute the value of 45. Solution 4 4 4 4 4 1,024 Calculator Solution Use the exponent key, ^, on a calculator. ENTER: 4 ^ 5 ENTER DISPLAY: 4 ^ 5 1 0 2 4 Answer 1,024 EXAMPLE 2 Find 2 3 A Solution a. The exact value is B 3 a. as an exact value b. as a rational approximation. a fraction 27 3 2 3 A B b. Use a calculator. ENTER: 2 ( 3 ) ^ 3 ENTER DISPLAY Note: The exact value is a rational number that can also be written as the repeating decimal 0.296. Answers a. 8 27 b. 0.2962962963 Order of Operations 41 Computations With More Than One Operation When a numerical expression involves two or more different operations, we need to agree on the order in which they are performed. Consider this example: 11 3 2 Suppose that one person multiplied first. Suppose another person subtracted first. 11 3 2 11 6 5 11 3 2 8 2 16 In order that there will be one and only one correct answer to problems like this, mathematicians have agreed to follow this order of operations: Who is correct? 1. Simplify powers (terms with exponents). 2. Multiply and divide, from left to right. 3. Add and subtract, from left to right. Therefore, we multiply before we subtract, and 11 3
2 11 6 5 is correct. A different problem involving powers is solved in this way: 1. Simplify powers: 5 23 + 3 5 8 3 2. Multiply and divide: 3. Add and subtract: 40 3 43 Expressions with Grouping Symbols In mathematics, parentheses ( ) act as grouping symbols, giving different meanings to expressions. For example, (4 6) 7 means “add 7 to the product of 4 and 6,” while 4 (6 7) means “multiply the sum of 6 and 7 by 4.” When simplifying any numerical expression, always perform the operations within parentheses first. (4 6) 7 24 7 31 4 (6 7) 4 13 52 Besides parentheses, other symbols are used to indicate grouping, such as brackets [ ]. The expressions 2(5 9) and 2[5 9] have the same meaning: 2 is multiplied by the sum of 5 and 9. A bar, or fraction line, also acts as a symbol of grouping, telling us to perform the operations in the numerator and/or denominator first. 20 2 8 3 5 12 11 2 42 Operations and Properties However, when entering expressions such as these into a calculator, the line of the fraction is usually entered as a division and a numerator or denominator that involves an operation must be enclosed in parentheses. ENTER: ( 20 8 ) 3 ENTER DISPLAY: ( 2 0 8 ) / 3 4 ENTER: 6 ( 3 1 ) ENTER DISPLAY When there are two or more grouping symbols in an expression, we perform the operations on the numbers in the innermost symbol first. For example: 5 2[6 (3 1)3] = 5 2[6 23] = 5 2[6 8] = 5 2[14] = 5 28 = 33 Procedure To simplify a numerical expression, follow the correct order of operations: 1. Simplify any numerical expressions within parentheses or within other grouping symbols, starting with the innermost. 2. Simplify any powers. 3. Do all multiplications and divisions in order from left to right. 4. Do all additions and subtractions in order from left to right. EXAMPLE 3 Simplify the numerical expression 80 4(7 5). Solution Remember that, in the given expression, 4(7 5) means 4 times the value in Order of Operations 43 the parentheses. How to Proceed (1) Write the expression: (2) Simplify the value within the parentheses: (3) Multip
ly: (4) Subtract: 80 4(7 5) 80 4(2) 80 8 72 Calculator Solution ENTER: 80 4 ( 7 5 ) ENTER DISPLAY Answer 72 EXERCISES Writing About Mathematics 1. Explain why 2 is the only even prime. 2. Delia knows that every number except 2 that ends in a multiple of 2 is composite. Therefore, she concludes that every number except 3 that ends in a multiple of 3 is composite. Is Delia correct? Explain how you know. Developing Skills In 3–10, state the meaning of each expression in part a and in part b, and simplify the expression in each part. 3. a. 20 (6 1) 5. a. 12 (3 0.5) 7. a. (12 8 ) 4 9. a. 7 52 b. 20 6 1 b. 12 3 0.5 b. 12 8 4 b. (7 5)2 4. a. 18 (4 3) 6. a. 15 (2 1) 8. a. 48 (8 4 ) 10. a. 4 32 11. Noella said that since the line of a fraction indicates division, 10 15 5 3. Do you agree with Noella? Explain why or why not. 10 3 15 5 3 3 b. 18 4 3 b. 15 2 1 b. 48 8 4 b. (4 3)2 is the same as 44 Operations and Properties In 12–15: a. Find, in each case, the value of the three given powers. b. Name, in each case, the expression that has the greatest value. 12. 52, 53, 54 13. (0.5)2, (0.5)3, (0.5)4 14. (0.5)2, (0.6)2, (0.7)2 15. (1.1)2, (1.2)2, (1.3)2 In 16–23: a. List all of the whole numbers that are factors of each of the given numbers. b. Is the number prime, composite, or neither? 16. 82 20. 1 17. 101 21. 808 18. 71 22. 67 19. 15 23. 397 Applying Skills In 24–28, write a numerical expression for each of the following and find its value to answer the question. 24. What is the cost of two chocolate chip and three peanut butter cookies if each cookie costs
28 cents? 25. What is the cost of two chocolate chip cookies that cost 30 cents each and three peanut but- ter cookies that cost 25 cents each? 26. How many miles did Ms. McCarthy travel if she drove 30 miles per hour for hour and 55 3 4 miles per hour for 11 2 hours? 27. What is the cost of two pens at $0.38 each and three notebooks at $0.69 each? 28. What is the cost of five pens at $0.29 each and three notebooks at $0.75 each if ordered from a mail order company that adds $1.75 in postage and handling charges? In 29–30, use a calculator to find each answer. 29. The value of $1 invested at 6% for 20 years is equal to (1.06)20. Find, to the nearest cent, the value of this investment after 20 years. 30. The value of $1 invested at 8% for n years is equal to (1.08)n. How many years will be required for $1 invested at 8% to double in value? (Hint: Guess at values of n to find the value for which (1.08)n is closest to 2.00.) 31. In each box insert an operational symbol +, –,,, and then insert parentheses if needed to make each of the following statements true. a. 3 □ 2 □ 1 4 d. 1 □ 3 □ 1 4 e-2 PROPERTIES OF OPERATIONS Properties of Operations 45 When numbers behave in a certain way for an operation, we describe this behavior as a property. You are familiar with these operations from your study of arithmetic. As we examine the properties of operations, no proofs are given, but the examples will help you to see that these properties make sense and to identify the sets of numbers for which they are true. The Property of Closure A set is said to be closed under a binary operation when every pair of elements from the set, under the given operation, yields an element from that set. 1. Add any two numbers. The sum of two whole numbers is a whole number. 23 11 34 7.8 4.8 12.6 The sum of two rational numbers is a rational number. The sum of p and its opposite, p, two irrational nump (p) 0 bers, is 0, a rational number. Even though the sum of two irrational numbers is usually an irrational number, the set of
irrational numbers is not closed under addition. There are some pairs of irrational numbers whose sum is not an irrational number. However, p, p, 0, and each of the other numbers used in these examples are real numbers and the sum of two real numbers is a real number. The sets of whole numbers, rational numbers, and real numbers are each closed under addition. 2. Multiply any two numbers. (2)(4 The product of two whole numbers is a whole number. The product of two rational numbers is a rational number. 2, two irrational numThe product of " bers, is 2, a rational number. and " 2 Though the product of two irrational numbers is usually an irrational number, there are some pairs of irrational numbers whose product is not an irrational number. The set of irrational numbers is not closed under multiplication. 2, 2, and each of the other numbers used in these examples are real However, numbers and the product of two real numbers is a real number. " The sets of whole numbers, rational numbers, and real numbers are each closed under multiplication. 46 Operations and Properties 3. Subtract any two numbers. 7 12 5 12.7 8.2 4.5 3 2 " 3 5 0 " The difference of two whole numbers is not a whole number, but these whole numbers are also integers and the difference between two integers is an integer. The difference of two rational numbers is a rational number. 3 The difference of, two irrational num" bers, is 0, a rational number. 3 " and Even though the difference of two irrational numbers is usually an irrational number, there are some pairs of irrational numbers whose difference is not an irrational number. The set of irrational numbers is not closed under 3, 0, and each of the other numbers used in these examsubtraction. However, ples are real numbers and the difference of two real numbers is a real number. " The sets of integers, rational numbers, and real numbers are each closed under subtraction. 4. Divide any two numbers by a nonzero number. (Remember that division by 0 is not allowed.) 9 2 4. The quotient of two whole numbers or two integers is not always a whole number or an integer. The quotient of two rational numbers is a rational number. 5 4 " 5 5 1 " 5 The quotient of " bers, is 1, a rational number. 5 " and, two irrational num- Though the quotient of two irrational numbers is usually an irrational number,
there are some pairs of irrational numbers whose quotient is not an irrational number. The set of irrational numbers is not closed under division. 5, 1, and each of the other numbers used in these examples are real However, numbers and the quotient of two nonzero real numbers is a nonzero real number. " The sets of nonzero rational numbers, and nonzero real numbers are each closed under division. Later in this book, we will study operations with signed numbers and operations with irrational numbers in greater detail. For now, we will simply make these observations: The set of whole numbers is closed under the operations of addition and multiplication. Properties of Operations 47 The set of integers is closed under the operations of addition, subtraction, and multiplication. The set of rational numbers is closed under the operations of addition, subtraction, and multiplication, and the set of nonzero rational numbers is closed under division. The set of real numbers is closed under the operations of addition, subtraction, and multiplication, and the set of nonzero real numbers is closed under division. Commutative Property of Addition When we add rational numbers, we assume that we can change the order in which two numbers are added without changing the sum. For example, 4 5 5 4 and the commutative property of addition. These examples illustrate In general, we assume that for every number a and every number b: a b b a Commutative Property of Multiplication In the same way, when we multiply rational numbers, we assume that we can change the order of the factors without changing the product. 2 3 1 1 For example, 5 4 4 5, and the commutative property of multiplication. 4 5 1 4 3 1 2. These examples illustrate In general, we assume that for every number a and every number b: a b b a Subtraction and division are not commutative, as shown by the following counterexample. 12 7 7 12 5 5 12 3 3 12 4 3 12 Associative Property of Addition Addition is a binary operation; that is, we add two numbers at a time. If we wish to add three numbers, we find the sum of two and add that sum to the third. For example: 48 Operations and Properties 2 5 8 (2 5) 8 or 2 5 8 2 (5 8) 7 8 15 2 13 15 The way in which we group the numbers to be added does not change the sum. Therefore, we see that (2 5) 8 2 (5
8). This example illustrates the associative property of addition. In general, we assume that for every number a, every number b, and every number c: (a b) c a (b c) Associative Property of Multiplication In a similar way, to find a product that involves three factors, we first multiply any two factors and then multiply this result by the third factor. We assume that we do not change the product when we change the grouping. For example: 5 4 2 (5 4) 2 or 5 4 2 5 (4 2) 20 2 40 5 8 40 Therefore, (5 4) 2 5 (4 2). This example illustrates the associative property of multiplication. In general, we assume that for every number a, every number b, and every number c: a (b c) (a b) c Subtraction and division are not associative, as shown in the following coun- terexamples. (15 4) 3 15 (4 3) 11 3 15 1 8 14 (8 4) 2 8 (4 2) 2 2 8 2 1 4 The Distributive Property We know 4(3 2) 4(5) 20, and also 4(3) 4(2) 12 8 20. Therefore, we see that 4(3 2) 4(3) 4(2). This result can be illustrated geometrically. Recall that the area of a rectan- gle is equal to the product of its length and its width. Properties of Operations 49 3 2 4 4(3+2) = 4 4(3) + 4 4(2) (3+2) 3 2 This example illustrates the distributive property of multiplication over addition, also called the distributive property. This means that the product of one number times the sum of a second and a third number equals the product of the first and second numbers plus the product of the first and third numbers. In general, we assume that for every number a, every number b, and every number c: a(b c) ab ac and (a b)c ac bc The distributive property is also true for multiplication over subtraction: a(b c) ab ac and (a b)c ac bc The distributive property can be useful for mental computations. Observe how we can use the distributive property to find each of the following products as a sum: 1. 6 23 6(20 3) 6 20 6 3 120 18 138 9 3 31 3
2. 3. 6.5 8 (6 0.5)8 6 8 0.5 8 48 4 52 1 9 3 9 27 3 30 3 311 3 9 B A Working backward, we can also use the distributive property to change the form of an expression from a sum or a difference to a product: 1. 5(12) 5(8) 5(12 8) 5(20) 100 2. 7(14) 7(4) 7(14 4) 7(10) 70 Addition Property of Zero and the Additive Identity Element The equalities 5 + 0 5 and 0 2.8 2.8 are true. They illustrate that the sum of a rational number and zero is the number itself. These examples lead us to observe that: 50 Operations and Properties 1. The addition property of zero states that for every number a: a 0 a and 0 a a 2. The identity element of addition, or the additive identity, is 0. Thus, for any number a: If a x a, or if x a a, it follows that x 0. Additive Inverses (Opposites) When we first studied integers, we learned about opposites. For example, the opposite of 4 is 4, and the opposite of 10 is 10. Every rational number a has an opposite, –a, such that their sum is 0, the identity element in addition. The opposite of a number is called the additive inverse of the number. In general, for every rational number a and its opposite a: a (a) 0 On a calculator, the (-) key, is used to enter the opposite of a number. The following example shows that the opposite of 4.5 is 4.5. ENTER: (-) (-) 4.5 ENTER DISPLAY: - - 4. 5 4. 5 Multiplication Property of One and the Multiplicative Identity Element The sentences 5 1 5 and 1 4.6 4.6 are true. They illustrate that the product of a number and one is the number itself. These examples lead us to observe that: 1. The multiplication property of one states that for every number a: a 1 a and 1 a a 2. The identity element of multiplication, or the multiplicative identity, is 1. Properties of Operations 51 Multiplicative Inverses (Reciprocals) When the product of two numbers is 1 (the identity element for multiplication), then each of these numbers is called the multiplicative inverse or reciprocal of the
other. Consider these examples: 4? 1 4 5 1 1 The reciprocal of 4 is. 4 The reciprocal of 1 4 is 4. 21 The multiplicative inverse of B A The multiplicative inverse of 21 2 or 5 2 2 is. 5 2 5 5 is or 2 21 2. Since there is no number that, when multiplied by 0, gives 1, the number 0 has no reciprocal, or no multiplicative inverse. In general, for every nonzero number a, there is a unique number 1 a such that: a? 1 a 5 1 On the calculator, a special key, displays the reciprocal. For example, if each of the numbers shown above is entered and the reciprocal key is pressed, the reciprocal appears in decimal form. x1 ENTER: 4 x1 ENTER ENTER: ( 5 2 ) x1 ENTER DISPLAY: 4 – 1 DISPLAY Note: Parentheses must be used when calculating the reciprocal of a fraction. For many other numbers, however, the decimal form of the reciprocal is not shown in its entirety in the display. For example, we know that the reciprocal of 1 6 is, but what appears is a rational approximation of. 6 1 6 ENTER: 6 x1 ENTER DISPLAY The display shows the rational approximation of rounded to the last decimal place displayed by the calculator. A calculator stores more decimal places in its operating system than it has in its display. The decimal displayed times the original number will equal 1. 1 6 52 Operations and Properties To display the reciprocal of 6 from the example on the previous page as a common fraction, we use 6 x1 MATH ENTER ENTER. Multiplication Property of Zero The sentences 7 0 0 and 0 0 are true. They illustrate that the product of a rational number and zero is zero. This property is called the multiplication property of zero: 3 4 In general, for every number a: a 0 0 and 0 a 0 EXAMPLE 1 Write, in simplest form, the opposite (additive inverse) and the reciprocal (multiplicative inverse) of each of the following: a. 7 b. d. 0.2 e. p c. 23 8 11 5 Solution Number Opposite Reciprocal a. 7 b. 23 8 –7 3 8 1 7 28 3 5 222 3 c. 5 5 6 11 5 211 5 5 2 6 5 d. 0.2 e. p 0.2 p 5 6 5 1 p EXAMPLE 2 Express 6t + t as a product and give the reason for each step. Solution
Step Reason (1) 6t t 6t 1t (6 1)t 7t (2) (3) Multiplication property of 1. Distributive property. Addition. Answer 7t Properties of Operations 53 EXERCISES Writing About Mathematics 1. If x and y represent real numbers and xy x: a. What is the value of y if the equation is true for all x? Explain your answer. b. What is the value of x if the equation is true for all y? Explain your answer. 2. Cookies and brownies cost $0.75 each. In order to find the cost of 2 cookies and 3 brownies Lindsey added 2 3 and multiplied the sum by $0.75. Zachary multiplied $0.75 by 2 and then $0.75 by 3 and added the products. Explain why Lindsey and Zachary both arrived at the correct cost of the cookies and brownies. Developing Skills 3. Give the value of each expression. a. 9 0 1 (p) g. p b. 9 0 h. 1 4.5 A B c. 9 1 0 1 i. 7 " 2 3 3 0 1.63 3 0 d. j. e. 0 1 3 k. 2 3 5 " f. 1 0 3 l. 2 3 225 " In 4–13: a. Replace each question mark with the number that makes the sentence true. b. Name the property illustrated in each sentence that is formed when the replacement is made. 4. 8 6 6? 6. (3 9) 15 3 (9?) 8. (0.5 0.2) 0.7 0.5 (? 0.7) 10. (3 7) 5 (? 3) 5 12. 7(4?) 7(4) 5. 17 5? 17 7. 6(5 8) 6(5)?(8) 9. 4 0? 11. (?)(8 2) (8 2)(9) 13.?x x In 14–25: a. Name the additive inverse (opposite) of each number. b. Name the multiplicative inverse (reciprocal) of each number. 14. 17 21 3 20. 15. 1 21. 2p 16. –10 3 7 22. 17. 2.5 23. 1.780 18. –1.8 2 1 11 24. 19. 25. 1 9 3 5 71 In 26–31, state whether each sentence is
a correct application of the distributive property. If you believe that it is not, state your reason. 26. 6(5 8) 6(5) 6(8) 28. 5 (8 6) (5 8) (5 6) 30. 14a 4a (14 4)a 10 5 10 3 1 2 1 1 2 1 1 1 27. 5 5 B 29. 3(x 5) 3x 3 5 31. 18(2.5) 18(2) 18(0.5) A 54 Operations and Properties In 32–35: a. Tell whether each sentence is true or false. b. Tell whether the commutative property holds for the given operation. 32. 357 19 19 357 34. 25 7 7 25 33. 2 1 1 2 35. 18(3.6) 3.6(18) In 36–39: a. Tell whether each sentence is true of false. b. Tell whether the associative property holds for the given operation 36. (73 68) 92 73 (68 92) 38. (19 8) 5 19 (8 5) 37. (24 6) 2 24 (6 2) 39. 9 (0.3 0.7) (9 0.3) 0.7 40. Insert parentheses to make each statement true. a. 3 2 1 3 3 d. 3 3 3 3 3 1 b. 4 3 2 2 3 e. 3 3 3 3 3 0 c. 8 8 8 8 8 8 f. 0 12 3 16 8 0 Applying Skills 41. Steve Heinz wants to give a 15% tip to the taxi driver. The fare was $12. He knows that 10% of $12 is $1.20 and that 5% would be half of $1.20. Explain how this information can help Steve calculate the tip. What mathematical property is he using to determine the tip? 42. Juana rides the bus to and from work each day. Each time she rides the bus the fare is $1.75. She works five days a week. To find what she will spend on bus fare each week, Juana wants to find the product 2(1.75)(5). Juana rewrote the product as 2(5)(1.75). a. What property of multiplication did Juana use when she changed 2(1.75)5 to 2(5)(1.75)? b. What is her weekly bus fare? 2-3 ADDITION
OF SIGNED NUMBERS Adding Numbers That Have the Same Signs The number line can be used to find the sum of two numbers. Start at 0. To add a positive number, move to the right. To add a negative number, move to the left. EXAMPLE 1 Add 3 and 2. Addition of Signed Numbers 55 +3 +2 –1 0 +1 +2 +3 +4 +5 +6 Solution Start at 0 and move 3 units to the right to 3; then move 2 more units to the right, arriving at +5. Calculator Solution ENTER: 3 2 ENTER DISPLAY: 3 + 2 5 Answer 5 The sum of two positive integers is the same as the sum of two whole numbers. The sum +5 is a number whose absolute value is the sum of the absolute values of 3 and 2 and whose sign is the same as the sign of 3 and 2. EXAMPLE 2 Add 3 and 2. Solution Start at 0 and move 3 units to the left to 3: then move 2 more units to the left, arriving at 5. Calculator Solution ENTER: (-) 3 (-) 2 ENTER DISPLAY: - 3 + - 2 - 5 Answer 5 –2 –3 –6 –5 –4 –3 –2 –1 0 +1 The sum 5 is a number whose absolute value is the sum of the absolute values of 3 and 2 and whose sign is the same as the sign of 3 and 2. Examples 1 and 2 illustrate that the sum of two numbers with the same sign is a number whose absolute value is the sum of the absolute values of the numbers and whose sign is the sign of the numbers. Procedure To add two numbers that have the same sign: 1. Find the sum of the absolute values. 2. Give the sum the common sign. 56 Operations and Properties Adding Numbers That Have Opposite Signs EXAMPLE 3 Add: 3 and 2. Solution Start at 0 and move 3 units to the right to 3; then move 2 units to the left, arriving at 1. –2 +3 Calculator Solution ENTER: 3 (-) 2 ENTER –1 0 +1 +2 +3 +4 DISPLAY: 3 + - 2 1 Answer 1 This sum can also be found by using properties. In the first step, substitution is used, replacing (3) with the sum (1) (2). (3) ( 2) [(1) (2)] (2) Substitution (1) [(2) ( 2)] Associative property 1 0 1
Addition property of zero Addition property of opposites The sum 1 is a number whose absolute value is the difference of the absolute values of 3 and 2 and whose sign is the same as the sign of 3, the number with the greater absolute value. EXAMPLE 4 Add: 3 and 2. Solution Start at 0 and move 3 units to the left to 3; then move 2 units to the right, arriving at 1. +2 –3 Calculator Solution ENTER: (-) 3 2 ENTER –4 –3 –2 –1 0 +1 DISPLAY: - 3 + 2 - 1 Answer 1 Addition of Signed Numbers 57 This sum can also be found by using properties. In the first step, substitution is used, replacing (3) with the sum (1) (2). (3) (2) [(1) (2)] (2) Substitution (1) [(2) (2)] Associative property 1 0 1 Addition property of zero Addition property of opposites The sum 1 is a number whose absolute value is the difference of the absolute values of 3 and 2 and whose sign is the same as the sign of 3, the number with the greater absolute value. Examples 3 and 4 illustrate that the sum of a positive number and a negative number is a number whose absolute value is the difference of the absolute values of the numbers and whose sign is the sign of the number having the larger absolute value. Procedure To add two numbers that have different signs: 1. Find the difference of the absolute values of the numbers. 2. Give this difference the sign of the number that has the greater absolute value. 3. The sum is 0 if both numbers have the same absolute value. EXAMPLE 5 Find the sum of 233 4 and 11 4. Solution How to Proceed (1) Since the numbers have different signs, find the difference of their absolute values: (2) Give the difference the sign of the number with the greater absolute value: 33 4 2 11 4 5 22 4 222 4 5 221 2 Answer 58 Operations and Properties Calculator Solution 33 The number 4 33 233, is the sum of 3 and site of 4 4, 23 4. is the sum of the whole number 3 and the fraction. The oppo- 3 4 Enclose the absolute value of the sum of 3 and 3 4 in parentheses. ENTER: (-) ( 3 3 4 ) 1 1 4 ENTER DISPLAY Answer 2.5 or 221 2 The method used in Example 5 to enter the negative mixed number illustrate the following property
: 233 4 Property of the Opposite of a Sum For all real numbers a and b: (a b) (a) (b) When adding more than two signed numbers, the commutative and associative properties allow us to arrange the numbers in any order and to group them in any way. It may be helpful to add positive numbers first, add negative numbers next, and then add the two results. EXERCISES Writing About Mathematics 1. The sum of two numbers is positive. One of the numbers is a positive number that is larger than the sum. Is the other number positive or negative? Explain your answer. 2. The sum of two numbers is positive. One of the numbers is negative. Is the other number positive or negative? Explain your answer. 3. The sum of two numbers is negative. One of the numbers is a negative number that is smaller than the sum. Is the other number positive or negative? Explain your answer. 4. The sum of two numbers is negative. One of the numbers is positive. Is the other number positive or negative? Explain your answer. 5. Is it possible for the sum of two numbers to be smaller than either of the numbers? If so, give an example. Developing Skills In 6–10, find each sum or difference. 7. 10 5 6. 6 4 8. 4.5 4.5 9. 6 4 10. 6 4 Subtraction of Signed Numbers 59 In 11–27, add the numbers. Use a calculator to check your answer. 13. 87 (87) 12. 23 (35) 11. 17 (28) 15. 2331 3 192 3 19. (47) (35) (47) 16. 253 4 22. 2143 4 2173 4 B 25. 6.25 (0.75) A 81 2 17. +7 20. 343 8 (73) 262 7 A B 23. 13 (13) 26. (12.4) 13.0 14. 2.06 1.37 18. (3.72) (5.28) 21. 2731 2 86 24. 42 43 B A 27. 12.4 13.0 Applying Skills 28. In 1 hour, the temperature rose 4° Celsius and in the next hour it dropped 6° Celsius. What was the net change in temperature during the two-hour period? 29. An elevator started on the first floor and rose 30 floors. Then it came down 12 floors. At which floor was it
at that time? 30. A football team gained 7 yards on the first play, lost 2 yards on the second, and lost 8 yards on the third. What was the net result of the three plays? 31. Fay has $250 in a bank. During the month, she made a deposit of $60 and a withdrawal of $80. How much money did Fay have in the bank at the end of the month? 32. During a four-day period, the dollar value of a share of stock rose $1.50 on the first day, dropped $0.85 on the second day, rose $0.12 on the third day, and dropped $1.75 on the fourth day. What was the net change in the stock during this period? 2-4 SUBTRACTION OF SIGNED NUMBERS In arithmetic, to subtract 3 from 7, we find the number that, when added to 3, gives 7. We know that 7 3 4 because 3 4 7. Subtraction is the inverse operation of addition. DEFINITION In general, for every number c and every number b, the expression c b is the number a such that b a c. We use this definition in order to subtract signed numbers. To subtract 2 from 3, written as (3) (2), we must find a number that, when added to 2, will give 3. We write: (2) (?) 3 60 Operations and Properties We can use a number line to find the answer to this open sentence. From a point 2 units to the left of 0, move to the point that represents 3, that is, 3 units to the right of 0. We move 5 units to the right, a motion that represents 5. +5 –2 –3 –2 –1 0 +1 +2 +3 +4 Therefore, (3) (2) 5 because (2) (5) 3. Notice that (3) (2) can also be represented as the directed distance from 2 to 3 on the number line. Subtraction can be written vertically as follows: (3) or (2) 5 Subtract: (3) (2) 5 minuend subtrahend difference When you first learned to subtract numbers, you learned to check your answer. The answer (the difference) plus the number being subtracted (the subtrahend) must be equal to the number from which you are subtracting (the minuend). Check each of the following examples using:
subtrahend difference minuend: Subtract Now, consider another way in which addition and subtraction are related. In each of the following examples, compare the result obtained when a signed number is subtracted with the result obtained when the opposite of that signed number is added. Subtract Add 9 9 6 6 3 3 Subtract Add 7 7 2 2 5 5 Subtract Add 5 5 2 2 7 7 Subtract Add 3 3 1 1 4 4 Observe that, in each example, adding the opposite (the additive inverse) of a signed number gives the same result as subtracting that signed number. It therefore seems reasonable to define subtraction as follows: DEFINITION If a is any signed number and b is any signed number, then: a b a (b) Procedure To subtract one signed number from another, add the opposite (additive inverse) of the subtrahend to the minuend. Subtraction of Signed Numbers 61 Uses of the Symbol We have used the symbol in three ways: 1. To indicate that a number is negative: 2 2. To indicate the opposite of a number: (4) Negative 2 Opposite of negative 4 3. To indicate subtraction: Opposite of a a 4 (3) Difference between 4 and 3 Note that an arithmetic expression such as 3 7 can mean either the dif- ference between 3 and 7 or the sum of 3 and 7. 3 7 3 (7) 3 (7) When writing an arithmetic expression, we use the same sign for both a neg, is ative number and subtraction. On a calculator, the key for subtraction, not the same key as the key for a negative number, will result in an error message. (-). Using the wrong key EXAMPLE 1 Perform the indicated subtractions. a. (30) (12) b. (19) (7) c. (4) (0) d. 0 8 Answers a. 18 b. 12 c. 4 d. 8 EXAMPLE 2 From the sum of 2 and 8, subtract 5. Solution (2 8) (5) 6 (5) 11 Calculator Solution ENTER: (-) 2 8 (-) 5 ENTER DISPLAY Answer 11 62 Operations and Properties EXAMPLE 3 Subtract the sum of 7 and 2 from 4. Solution 4 (7 2) 4 (5) 4 (5) 1 Calculator Solution ENTER: (-) 4 ( (-) 7 2 ) ENTER DISPLAY Answer 1
EXAMPLE 4 How much greater than 3 is 9? Solution 9 (3) 9 3 12 Answer 9 is 12 greater than 3. +12 –3 0 +9 Note that parentheses need not be entered in the calculator in Example 2 since the additions and subtractions are to be done in the order in which they occur in the expression. Parentheses are needed, however, in Example 3 since the sum of 7 and 2 is to be found first, before subtracting this sum from 4. EXERCISES Writing About Mathematics 1. Does 8 12 mean the difference between 8 and 12 or the sum of 8 and 12? Explain your answer. 2. How is addition used to check subtraction? Developing Skills In 3–10, perform each indicated subtraction. Check your answers using a calculator. 3. 23 (35) 4. 87 (87) 5. 5.4 (8.6) 6. 8.8 (3.7) 7. 2.06 (1.37) 8. 2331 3 1192 3 A B 9. 253 4 281 2 A B 10. 7 62 7 A B In 11–18, Find the value of each given expression. Subtraction of Signed Numbers 63 13. (12) (57 12) 16. 2143 4 2 2173 4 A B A B 18. 15. 8 2 2343 12. (3.72) (5.28) 2731 86 2 32 (32) 11. (18) (14) 14. (47) (35 47) 273 2 723 17. 8 19. How much is 18 decreased by 7? 20. How much greater than 15 is 12? 21. How much greater than 4 is 1? 22. What number is 6 less than 6? 23. Subtract 8 from the sum of 6 and 12. 24. Subtract 7 from the sum of 18 and 10. 25. State whether each of the following sentences is true or false: a. (5) (3) (3) (5) b. (7) (4) (4) (7) 26. If x and y represent real numbers: a. Does x y y x for all replacements of x and y? Justify your answer. b. Does x y y x for any replacements of x and y? For which values of x and y? c. What is the relation between x y and y x for all replacements of x and y? d. Is the
operation of subtraction commutative? In other words, for all signed numbers x and y, does x y y x? 27. State whether each of the following sentences is true or false: a. (15 9) 6 15 (9 6) b. [(10) (4)] (8) (10) [(4) (8)] 28. Is the operation of subtraction associative? In other words, for all signed numbers x, y, and z, does (x y) z x (y z)? Justify your answer. Applying Skills 29. Express as a signed number the increase or decrease when the Celsius temperature changes from: a. 5° to 8° b. 10° to 18° c. 6° to 18° d. 12° to 4° 30. Find the change in altitude when you go from a place that is 15 meters below sea level to a place that is 95 meters above sea level. 31. In a game, Sid was 35 points “in the hole.” How many points must he make in order to have a score of 150 points? 32. The record high Fahrenheit temperature in New City is 105°; the record low is 9°. Find the difference between these temperatures. 33. At one point, the Pacific Ocean is 0.50 kilometers in depth; at another point it is 0.25 kilo- meters in depth. Find the difference between these depths. 64 Operations and Properties 2-5 MULTIPLICATION OF SIGNED NUMBERS Four Possible Cases in the Multiplication of Signed Numbers We will use a common experience to illustrate the various cases that can arise in the multiplication of signed numbers. 1. Represent a gain in weight by a positive number and a loss of weight by a negative number. 2. Represent a number of months in the future by a positive number and a number of months in the past by a negative number. Multiplying a Positive Number by a Positive Number CASE 1 If a boy gains 2 pounds each month, 4 months from now he will be 8 pounds heavier than he is now. Using signed numbers, we may write: (2)(4) 8 The product of the two positive numbers is a positive number. Multiplying a Negative Number by a Positive Number CASE 2 If a boy loses 2 pounds each month, 4 months from now he will be 8 pounds lighter than he is now. Using signed numbers, we may write: (2)(4) 8 The product of the negative
number and the positive number is a negative number. Multiplying a Positive Number by a Negative Number CASE 3 If a girl gained 2 pounds each month, 4 months ago she was 8 pounds lighter than she is now. Using signed numbers, we may write: (2)(4) 8 The product of the positive number and the negative number is a negative number. Multiplying a Negative Number by a Negative Number CASE 4 If a girl lost 2 pounds each month, 4 months ago she was 8 pounds heavier than she is now. Using signed numbers, we may write: (2)(4) 8 The product of the two negative numbers is a positive number. In all four cases, the absolute value of the product, 8, is equal to the product of the absolute values of the factors, 4 and 2. Multiplication of Signed Numbers 65 Using the Properties of Real Numbers to Multiply Signed Numbers You know that the sum of any real number and its inverse is 0, a (a) 0, and that for all real numbers a, b, and c, a(b c) ab ac. These two facts will enable us to demonstrate the rules for multiplying signed numbers. 4[7 (7)] 4(7) 4(7) 4(0) 28 4(7) 0 28 4(7) In order for this to be a true statement, 4(7) must equal 28, the opposite of 28. Since addition is a commutative operation, 4(7) 7(4). Then 7[4 (4)] 7(4) (7)(4) 7(0) 28 (7)(4) 0 28 (7)(4) In order for this to be a true statement, 7(4) must be 28, the additive inverse of 28. Rules for Multiplying Signed Numbers RULE 1 The product of two positive numbers or of two negative numbers is a positive number whose absolute value is the product of the absolute values of the numbers. The product of a positive number and a negative number is a negative RULE 2 number whose absolute value is the product of the absolute values of the numbers. In general, if a and b are both positive or are both negative, then: ab a · b If one of the numbers, a or b, is positive and the other is negative, then: ab (a · b)) Procedure To multiply two signed numbers: 1. Find the product of the absolute values. 2. Write a plus sign before this product when
the two numbers have the same sign. 3. Write a minus sign before this product when the two numbers have different signs. 66 Operations and Properties EXAMPLE 1 Find the product of each of the given pairs of numbers. a. (12)(4) c. (18)(3) Answers 48 54 e. (3.4)(3) 10.2 b. (13)(5) d. (15)(6) 271 8 f. A B (23) Answers 5 65 5 90 5 1213 8 Use the distributive property of multiplication over addition to find the product 8(72). 8(72) 8[(70) (2)] 8(70) 8(2) (560) (16) 576 Answer EXAMPLE 2 Solution EXAMPLE 3 Find the value of (2)3. Solution Answer 8 (2)3 (2)(2)(2) 4(2) 8 Note: The product of an odd number (3) of negative factors is negative. EXAMPLE 4 Find the value of (3)4. Solution (3)4 (3)(3)(3)(3) [(3)(3)][(3)(3)] (9)(9) 81 Answer 81 Note: The product of an even number (4) of negative factors is positive. In this example, the value of (3)4 was found to be 81. This is not equal to 34, which is the opposite of 34 or 1(34). To find the value of 34, first find the value of 34, which is 81, and then write the opposite of this power, 81. Thus, (3)4 81 and 34 81. Multiplication of Signed Numbers 67 On a calculator, evaluate (3)4. On a calculator, evaluate 34. ENTER: ( (-) 3 ) ^ 4 ENTER ENTER: (-) 3 ^ 4 ENTER DISPLAY: ( - 3 ) ^ 4 DISPLAY EXERCISES Writing About Mathematics 1. Javier said that (5)(4)(2) 40 because the product of numbers with the same sign is positive. Explain to Javier why he is wrong. 2. If a(b) means the opposite of ab, explain how knowing that 3(4) 12 can be used to show that 3(4) 12. Developing Skills In 3–12, find the product of each pair of numbers. Check your answers using a calculator. 3. 17(6) 4. +27(6) 5. 9
( 27) 6. 23(15) 7. +4(4) 8. +5.4(0.6) 9. 2.6( 0.05) 10. 231 3 112 3 A B 11. 253 4 21 2 A B 12. 262 3(7) In 13–18, find the value of each expression. 13. (18)(4)(5) 16. (12)(7 3) 14. (3.72)(0.5)(0.2) 273 1 723 8 28 17. A B 15. (4)(35 7) 18. 4 23 4 1 1 4 In 19–28, find the value of each power. 19. (3)2 24. 24 20. 32 25. (0.5)2 21. (5)3 26. (0.5)2 22. (2)4 27. 0.52 23. (2)4 28. (3)3 In 29–34, name the property that is illustrated in each statement. 29. 2(1 3) 2(1) 2(3) 31. (1 2) 3 1 (2 3) 33. (5)(7) (7)(5) 30. (2) (1) (1) (2) 32. 2 0 2 34. 3 (2 3) (3 2) (3) 68 Operations and Properties 2-6 DIVISION OF SIGNED NUMBERS Using the Inverse Operation in Dividing Signed Numbers Division may be defined as the inverse operation of multiplication, just as subtraction is defined as the inverse operation of addition. To divide 6 by 2 means to find a number that, when multiplied by 2, gives 6. That number is 3 because 3(2) 6. Thus, 3, or 6 2 3. The number 6 is the dividend, 2 is the divisor, and 3 is the quotient. 6 2 It is impossible to divide a signed number by 0; that is, division by 0 is undefined. For example, to solve (9) 0, we would have to find a number that, when multiplied by 0, would give 9. There is no such number since the product of any signed number and 0 is 0. In general, for all signed numbers a and b (b 0), a b or means to find a b the unique number c such that cb a In dividing nonzero signed numbers, there are four possible cases. Consider the
following examples: CASE 1 CASE 2 CASE 3 CASE 4 16 13 26 23 26 13 16 23? implies (?)(3) 6. Since (2)(3) 6, 2 16 13? implies (?)(3) 6. Since (2)(3) 6, 2 26 23? implies (?)(3) 6. Since (2)(3) 6, 2 26 13? implies (?)(3) 6. Since (2)(3) 6, 2 16 23 In the preceding examples, observe that: 1. When the dividend and divisor are both positive or both negative, the quo- tient is positive. 2. When the dividend and divisor have opposite signs, the quotient is nega- tive. 3. In all cases, the absolute value of the quotient is the absolute value of the dividend divided by the absolute value of the divisor. Rules for Dividing Signed Numbers The quotient of two positive numbers or of two negative numbers is a RULE 1 positive number whose absolute value is the absolute value of the dividend divided by the absolute value of the divisor. Division of Signed Numbers 69 The quotient of a positive number and a negative number is a nega- RULE 2 tive number whose absolute value is the absolute value of the dividend divided by the absolute value of the divisor. In general, if a and b are both positive or are both negative, then: a b a b or a b \|a\| \|b\| If one of the numbers, a or b, is positive and the other is negative, then: a b (a b) or a b \|a\| \|b\| A B Procedure To divide two signed numbers: 1. Find the quotient of the absolute values. 2. Write a plus sign before this quotient when the two numbers have the same sign. 3. Write a minus sign before this quotient when the two numbers have different signs. Rule for Dividing Zero by a Nonzero Number If the expression 0 (5) or?, then (?)(5) 0. Since 0 is the only number that can replace? and result in a true statement, 0 (5) or 0. This illustrates that 0 divided by any nonzero number is 0. In general, if a is a nonzero number (a 0), then 0 25 0 25 EXAMPLE 1 0 a 0. 0 a Perform each indicated division, if possible. 160 a. 115 a.
4 b. b. 110 290 21 9 227 c. 23 c. 9 d. (45) 9 d. 5 e. 0 (9) f. 3 0 e. 0 f. Undefined Answers Using the Reciprocal in Dividing Signed Numbers In Section 2-2, we learned that for every nonzero number a, there is a unique number, called the reciprocal or multiplicative inverse, such that a? 1 a 5 1 1 a. 70 Operations and Properties Using the reciprocal of a number, we can define division in terms of multi- plication as follows: For all numbers a and b (b 0): a b a · (b 0) 1 b a b Procedure To divide a signed number by a nonzero signed number, multiply the dividend by the reciprocal of the divisor. Notice that we exclude division by 0. The set of nonzero real numbers is closed with respect to division because every nonzero real number has a unique reciprocal, and multiplication by this reciprocal is always possible. EXAMPLE 2 Perform each indicated division by using the reciprocal of the divisor. a. 130 12 230 190 b. c. (54) 6 21 3 B 23 5 e. A f. 0 (9) (13) 4 227 4 d. A B Answers 11 2 B 1 1 90 B 1 9 6 5 115 5 21 3 5 181 B 23 1 B 5 25 A B 5 0 (130) (254) (230) A A A 227 3 25 3 13 0 A A 21 9 B EXERCISES Writing About Mathematics 1. If x and y represent nonzero numbers, what is the relationship between x y and y x? 2. If x y, are there any values of x and y for which x y y x? Developing Skills In 3–10, name the reciprocal (the multiplicative inverse) of each given number. 3. 6 7. 1 2 4. 5 2 1 10 8. 5. 1 9. 23 4 6. 1 10. x if x 0 In 11–26, find the indicated quotients or write “undefined” if no quotient exists. Operations with Sets 71 14. 0 213 10.01 20.001 13. 210 110 20.25 22.5 18. 17. 20. (75) (0) 22. (1.5) (0.03) 12. 248 116 13.6 20.12 11. 263 29 215 245 16. 15. 19
. (100) (2.5) 21. (0.5) (0.25) 23. 25. (112) 4 17 8 4 A B A 21 3 A 221 32 B B 24. 26. A 23 4 B 211 4 A 4 (16) 221 2 4 B A B In 27–28, state whether each sentence is true or false: 27. a. [(16) (4)] (2) (16) [(4) (2)] b. [(36) (6)] (2) (36) [(6) (2)] c. Division is associative. 28. a. (12 6) 2 12 2 6 2 b. [(25) (10)] (5) (25) (5) (10) (5) c. 2 (3 5) 2 3 2 5 d. Division is distributive over addition and subtraction. 2-7 OPERATIONS WITH SETS Recall that a set is simply a collection of distinct objects or elements, such as a set of numbers in arithmetic or a set of points in geometry. And, just as there are operations in arithmetic and in geometry, there are operations with sets. Before we look at these operations, we need to understand one more type of set. The universal set, or the universe, is the set of all elements under consider- ation in a given situation, usually denoted by the letter U. For example: 1. Some universal sets, such as all the numbers we have studied, are infinite. Here, U = {real numbers}. 2. In other situations, such as the scores on a classroom test, the universal set can be finite. Here, using whole-number grades, U = {0, 1, 2, 3,..., 100}. Three operations with sets are called intersection, union, and complement. Intersection of Sets The intersection of two sets, A and B, denoted by ments that belong to both sets, A and B. For example: A d B, is the set of all ele- 72 Operations and Properties 1. When A = {1, 2, 3, 4, 5}, and B = {2, 4, 6, 8, 10}, then A d B is {2, 4}. 2. In the diagram, two lines called 4 AB 4 CD and inter- sect. Each line is an infinite set of points although only three points are marked on each line. The intersection is a set that has one
element, point E, the point that is on line AB ( ) and on line CD 4 AB ( ). We write the intersection of the lines in the 4 d CD example shown as 4 AB 4 CD A D E C B 3. Intersection is a binary operation, are subsets of the universal set and example: 5 E. F d G 5 H, where sets F, G, and H d is the operation symbol. For U set of natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,...} F multiples of 2 {2, 4, 6, 8, 10, 12,...} G multiples of 3 {3, 6, 9, 12, 15, 18,...} multiples of 6 {6, 12, 18, 24, 30,...} F d G 4. Two sets are disjoint sets if their intersection is the empty set ( or {}); that is, if they do not have a common element. For example, when K = {1, 3, 5, 7, 9, 11, 13} and L = {2, 4, 6, 8},. Therefore, K and L are disjoint sets. K d L 5 Union of Sets The union of two sets, A and B, denoted by belong to set A or to set B, or to both set A and set B. For example: A < B, is the set of all elements that 1. If A = {1, 2, 3, 4} and B = {2, 4, 6}, then {1, 2, 3, 4, 6}. Note that an element is not repeated in the union of two sets even if it is an element of each set. A < B 2. In the diagram, both region R (gray shading) and region S (light color shading) represent sets of points. The shaded parts of both R < S regions represents, and the dark color shading where the regions overlap represents R d S. R S 3. If A {1, 2} and B = {1, 2, 3, 4, 5}, then the union of A and B is {1, 2, 3, 4, 5}. A < B B. Once again we have We can write an example of a binary operation, where the elements are taken from a universal set and where the operation here is
union. {1, 2, 3, 4, 5}, or A < B Operations with Sets 73 4. The union of the set of all rational numbers and the set of all irrational numbers is the set of real numbers. {real numbers} {rational numbers} {irrational numbers} Complement of a Set The complement of a set A, denoted by, is the set of all elements that belong to the universe U but do not belong to set A. Therefore, before we can determine the complement of A, we must know U. For example: A 1. If A {3, 4, 5} and U = {1, 2, 3, 4, 5}, then {1, 2} because 1 and 2 A belong to the universal set U but do not belong to set A. 2. If the universe is {whole numbers} and A = {even whole numbers} then {odd whole numbers} because the odd whole numbers belong to the A universal set but do not belong to set A. A Although it seems at first that only one set is being considered in writing the, actually there are two sets. This fact suggests a binary complement of A as operation, in which the universe U and the set A are the pair of elements, complement is the operation, and the unique result is. The complement of any universe is the empty set. Note that the complement can also be written as U \ A to emphasize that it is a binary operation. A EXAMPLE 1 If U {1, 2, 3, 4, 5, 6, 7}, A {6, 7}, and B {3, 5, 7}, determine A d B. Solution U = {1, 2, 3, 4, 5, 6, 7} Since A {6, 7}, then {1, 2, 3, 4, 5}. Since B = {3, 5, 7}, then {1, 2, 4, 6}. A B Since 1, 2, and 4 are elements in both, we can write: B A and {1, 2, 4} Answer EXAMPLE 2 A d B Using sets U, A, and B given for Example 1, find the complement of the set A < B, that is, determine A < B. Solution Since A < B contains all of the elements that are common to A and B, A < B 5 3, 5, 6, 7 6 5. Therefore, A < B 5 1, 2, 4 6 5
Answer 74 Operations and Properties EXERCISES Writing About Mathematics 1. A line is a set of points. Can the intersection of two lines be the empty set? Explain. 2. Is the union of the set of prime numbers and the set of composite numbers equal to the set of counting numbers? Explain. Developing Skills In 3–10, A {1, 2, 3}, B {3, 4, 5, 6}, and C = {1, 3, 4, 6}. In each case, perform the given operation and list the element(s) of the resulting set. 3. 7. A d B B d C 4. A < B 8. B < C 5. 9. A d C B < 6. 10. A < C B d 11. Using the sets A, B, and C given for Exercises 3–10, list the element(s) of the smallest possi- ble universal set of which A, B, and C are all subsets. In 12–19, the universe U = {1, 2, 3, 4, 5}, A {1, 5}, B = {2, 5}, and C {2}. In each case, perform the given operation and list the element(s) of the resulting set. 12. A A d B 13. B 14. C A d B 15. A < B A < B 17. A < B 16. 20. If U {2, 4, 6, 8} and {6}, what are the elements of A? 21. If U {2, 4, 6, 8}, A {2}, and {2, 4}, what are the elements of B 22. If U {2, 4, 6, 8}, A {2}, and {2, 4}, what is the set A d B? 18. 19. A B A < B? 23. Suppose that the set A has two elements and the set B has three elements. a. What is the greatest number of elements that A < B can have? b. What is the least number of elements that A < B can have? c. What is the greatest number of elements that A d B can have? d. What is the least number of elements that A d B can have? 24. Let the universe U {2, 4, 6, 8, 10, 12}, A {2, 8, 12} and B {4, 10}. a. A b. A (
the complement of A ) c. B d. B B (the complement of ) 25. Let the universe U {1, 2, 3, 4, 5, 6, 7, 8}. a. Find the elements of A d B, A d B, and A < B when A and B are equal to: (1) A = {1, 2, 3, 4}; B {5, 6, 7, 8} (3) A {1, 3, 5, 7}; B {2, 4, 6, 8} (2) A {2, 4}; B {6, 8} (4) A {2}; B {4} A d B. b. When A and B are disjoint sets, describe, in words, the set c. If A and B are disjoint sets, what is the set A < B? Explain. 2-8 GRAPHING NUMBER PAIRS Even though we know that the surface of the earth is approximately the surface of a sphere, we often model the earth by using maps that are plane surfaces. To locate a place on a map, we choose two reference lines, the equator and the prime meridian. The location of a city is given in term of east or west longitude (distance from the prime meridian) and north or south latitude (distance from the equator). For example, the city of Lagos in Nigeria is located at 3° east longitude and 6° north latitude, and the city of Dakar in Senegal is located 17° west longitude and 15° north latitude. Points on a Plane Graphing Number Pairs 75 20° W 10° W 0° 10° E 20° E 30° E 40° E Dakar Equator Lagos 50° E 30°N 20°N 10°N 0° 10°S 20°S 30°s The method used to locate cities on a map can be used to locate any point on a plane. The reference lines are a horizontal number line called the x-axis and a vertical number line called the y-axis. These two number lines, which have the same scale and are drawn perpendicular to each other, are called the coordinate axes. The plane determined by the axes is called the coordinate plane. In a coordinate plane, the intersection of the two axes is called the origin and is indicated as point O. This point of intersection is assigned the value 0 on both the x- and y-axes.
Moving to the right and moving up are regarded as movements in the positive direction. In the coordinate plane, points to the right of O on the x-axis and on lines parallel to the x-axis and points above O on the y-axis and on lines parallel to the y-axis are assigned positive values. Moving to the left and moving down are regarded as movements in the negative direction. In the coordinate plane, points to the left of O on the x-axis and on lines parallel to the xaxis and points below O on the y-axis and on lines parallel to the y-axis are assigned negative values. y 4 3 2 1 O –4 –3 –2 –1 1 2 3 4 x –1 –2 –3 –4 76 Operations and Properties Quadrant II y 3 2 1 –3 –2 –1 –1 –2 –3 Quadrant III The x-axis and the y-axis separate the plane into four regions called quadrants. These quadrants are numbered I, II, III, and IV in a counterclockwise order, beginning at the upper right, as shown in the accompanying diagram. The points on the axes are not in any quadrant. Quadrant I O 1 2 3 x Quadrant IV Coordinates of a Point Every point on the plane can be described by two numbers, called the coordinates of the point, usually written as an ordered pair. The first number in the pair is called the x-coordinate or the abscissa. The second number is the y-coordinate or the ordinate. In general, the coordinates of a point are represented as (x, y). In the graph at the right, point A, which is the graph of the ordered pair (2, 3), lies in quadrant I. Here, A lies a distance of 2 units to the right of the origin (in a positive direction along the x-axis) and then up a distance of 3 units (in a positive direction parallel to the y-axis). Point B, the graph of (4, 1) in quadrant II, lies a distance of 4 units to the left of the origin (in a negative direction along the x-axis) and then up 1 unit (in a positive direction parallel to the y-axis). B(–4,+1) s i x a y O A(+2,+3) x-axis C(–2,–3) D(+3,–5) In quadrant III
, every ordered pair (x, y) consists of two negative numbers. For example, C, the graph of (2, 3), lies 2 units to the left of the origin (in the negative direction along the x-axis) and then down 3 units (in the negative direction parallel to the y-axis). Point D, the graph of (3, 5) in quadrant IV, lies 3 units to the right of the origin (in a positive direction along the x-axis) and then down 5 units (in a negative direction parallel to the y-axis). Point O, the origin, has the coordinates (0, 0). Locating a Point on the Coordinate Plane An ordered pair of signed numbers uniquely determines the location of a point. Graphing Number Pairs 77 Procedure To find the location of a point on the coordinate plane: 1. Starting from the origin O, move along the x-axis the number of units given by the x-coordinate. Move to the right if the number is positive or to the left if the number is negative. If the x-coordinate is 0, there is no movement along the x-axis. 2. Then, from the point on the x-axis, move parallel to the y-axis the number of units given by the y-coordinate. Move up if the number is positive or down if the number is negative. If the y-coordinate is 0, there is no movement in the y direction. To locate the point A(3, 4), from O, move 3 units to the left along the x-axis, then 4 units down, parallel to the y-axis. To locate the point B(4, 0), from O, move 4 units to the right along the xaxis. There is no movement parallel to the y-axis. To locate the point C(0, 5), there is no movement along the x-axis. From O, move 5 units down along the y-axis. y 2 1 O –1 –2 –3 –4 –5 –4 –3 –2 –1 A(–3, –4) B(4, 0) 1 2 3 4 x5 C(0, –5) Finding the Coordinates of a Point on a Plane The location of a point on the coordinate plane uniquely determines the coordinates of the point. Procedure To find the coordinates of a point: 1. From the point, move along a vertical line to the x-axis.
The number assigned to that point on the x-axis is the x-coordinate of the point. 2. From the point, move along a horizontal line to the y-axis.The number assigned to that point on the y-axis is the y-coordinate of the point. 78 Operations and Properties To find the coordinates of point R, from point R, move in the vertical direction to 5 on the x-axis and in the horizontal direction to 6 on the yaxis. The coordinates of R are (5, 6) To find the coordinates of point S, from point S, move in the vertical direction to 2 on the x-axis and in the horizontal direction to 4 on the y-axis. The coordinates of S are (2, 4) Point T, lies at 5 on the x-axis. Therefore, the x-coordinate is 5 and the y-coordinate is 0. The coordinates of T are (5, 0). Note that if a point lies on the yaxis, the x-coordinate is 0. If a point lies on the x-axis, the y-coordinate is 0. Graphing Polygons A polygon is a closed figure whose sides are line segments. A quadrilateral is a polygon with four sides. The endpoints of the sides are called vertices. A quadrilateral can be represented in the coordinate plane by locating its vertices and then drawing the sides, connecting the vertices in order. The graph at the right shows the rectangle ABCD. The vertices are A(3, 2), B(3, 2), C(3, 2) and D(3, 2). From the graph, note the following5 –4 –3 –2 –1 O –1 –2 –3 –4 –5 –6 1 2 3 4 x5 R y 1 O 1 –1 –1 B(–3, 2) C(–3, –2) A(3, 2) x D(3, –2) 1. Points A and B have the same y-coordinate and are on a line parallel to the x-axis. 2. Points C and D have the same y-coordinate and are on a line parallel to the x-axis. 3. Lines parallel to the x-axis are parallel to each other. 4. Lines parallel to the x-axis are perpendicular to the y-axis. 5. Points B and C have the same x-coord
inate and are on a line parallel to the y-axis. Graphing Number Pairs 79 6. Points A and D have the same x-coordinate and are on a line parallel to the y-axis. 7. Lines parallel to the y-axis are parallel to each other. 8. Lines parallel to the y-axis are perpendicular to the x-axis. Now, we know that ABCD is a rectangle, because it is a parallelogram with right angles. From the graph, we can find the dimensions of this rectangle. To find the length of the rectangle, we can count the number of units from A to B or from C to D. AB CD 6. Because points on the same horizontal line have the same y-coordinate, we can also find AB and CD by subtracting their xcoordinates. AB CD 3 (3) 3 3 6 To find the width of the rectangle, we can count the number of units from B to C or from D to A. BC DA 4. Because points on the same vertical line have the same x-coordinate, we can find BC and DA by subtracting their ycoordinates. BC DA 2 (2) 2 2 4 EXAMPLE 1 Graph the following points: A(4, 1), B(1, 5), C(2, 1). Then draw ABC and find its area. Solution The graph at the right shows ABC. To find the area of the triangle, we need to know the lengths of the base and of the altitude drawn to that base. The base of ABC is AC. AC 4 (2) 4 2 6 y B(1, 5) C(–2,–1) D(1, 1) O 1 –1 –1 A(4, 1) x The line segment drawn from B perpendicular to is the altitude AC BD. BD 5 1 4 1 2(AC)(BD) 1 2(6)(4) Area 12 Answer The area of ABC is 12 square units. 80 Operations and Properties EXERCISES Writing About Mathematics 1. Mark is drawing triangle ABC on the coordinate plane. He locates points A(2, 4) and C(5, 4). He wants to make AC BC, C a right angle, and point B lie in the first quadrant. What must be the coordinates of point B? Explain how you found your answer. 2. Phyllis graphed the points D(3, 0), E(0, 5), F(
2, 0), and G(0, 4) on the coordinate plane and joined the points in order. Explain how Phyllis can find the area of this polygon, then find the area. Developing Skills 3. Write as ordered number pairs the coordinates of points A, B, C, D, E, F, G, H, and O in the graph. B y F O 2 1 G –2 C –1 –1 –2 H A E x 21 D In 4–15, draw a pair of coordinate axes on graph paper and locate the point associated with each ordered number pair. Label each point with its coordinates. 4. (5, 7) 8. (1, 6) 12. (3, 0) 5. (3, 2) 9. (8, 5) 13. (0, 4) 6. (2, 6) 10. (4, 4) 14. (0, 6) 7. (4, 5) 11. (5, 0) 15. (0, 0) In 16–20, name the quadrant in which the graph of each point described appears. 16. (5, 7) 17. (3, 2) 18. (7, 4) 19. (1, 3) 20. (2, 3) 21. Graph several points on the x-axis. What is the value of the y-coordinate for every point in the set of points on the x-axis? 22. Graph several points on the y-axis. What is the value of the x-coordinate for every point in the set of points on the y-axis? 23. What are the coordinates of the origin in the coordinate plane? Applying Skills In 24–33: a. Graph the points and connect them with straight lines in order, forming a polygon. b. Identify the polygon. c. Find the area of the polygon. 24. A(1, 1), B(8, 1), C(1, 5) 26. C(8, 1), A(9, 3), L(4, 3), F(3, 1) 25. P(0, 0), Q(5, 0), R(5, 4), S(0, 4) 27. H(4, 0), O(0, 0), M(0, 4), E(4, 4) Graphing Number Pairs 81 28. H(5, 3),
E(5, 3), N(2, 0) 30. B(3, 2), A(2, 2), R(2, 2), N(3, 2) 32. R(4, 2), A(0, 2), M(0, 7) 29. F(5, 1), A(5, 5), R(0, 5), M(2, 1) 31. P(3, 0), O(0, 0), N(2, 2), D(1, 2) 33. M(1, 1), I(3, 1), L(3, 3), K(1, 3) 34. Graph points A(1, 1), B(5, 1), and C(5, 4). What must be the coordinates of point D if ABCD is a rectangle? 35. Graph points P(1, 4) and Q(2, 4). What are the coordinates of R and S if PQRS is a square? (Two answers are possible.) 36. a. Graph points S(3, 0), T(0, 4), A(3, 0), and R(0, 4), and draw the quadrilateral STAR. b. Find the area of STAR by adding the areas of the triangles into which the axes divide it. 37. a. Graph points P(2, 0), L(1, 1), A(1, 1), N(2, 0), E(1, 1), and T(1, 1). Draw PLANET, a six-sided polygon called a hexagon. b. Find the area of PLANET. (Hint: Use the x-axis to separate the hexagon into two parts.). CHAPTER SUMMARY A binary operation in a set assigns to every ordered pair of elements from the set a unique answer from that set. The general form of a binary operation is a b c, where a, b, and c are elements of the set and is the operation symbol. Binary operations exist in arithmetic, in geometry, and in sets. Operations in arithmetic include addition, subtraction, multiplication, divi- sion, and raising to a power. Powers are the result of repeated multiplication of the same factor, as in 53 125. Here, the base 5 with an exponent of 3 equals 5 5 5 or 125, the power. Numerical expressions are simplified by following a clear order of opera- tions: (1) simplify within
parentheses or other grouping symbols; (2) simplify powers; (3) multiply and divide from left to right; (4) add and subtract from left to right. Many properties are used in operations with real numbers, including: • closure under addition, subtraction, and multiplication; • commutative properties for addition and multiplication, a b b a; • associative properties for addition and multiplication, (a b) c a (b c); 82 Operations and Properties • distributive property of multiplication over addition or subtraction, a (b c) a b a c and (a b) c a c b c; • additive identity (0); • additive inverses (opposites), called a for every element a; • multiplicative identity (1); • multiplicative inverses (reciprocals), 1 a for every nonzero element a; the nonzero real numbers are closed under division. Operations with sets include the intersection of sets, the union of sets, and the complement of a set. Basic operations with signed numbers: • To add two numbers that have the same sign, find the sum of the absolute values and give this sum the common sign. • To add two numbers that have different signs, find the difference of the absolute values of the numbers. Give this difference the sign of the number that has the greater absolute value. The sum is 0 if both numbers have the same absolute value. • To subtract one signed number from another, add the opposite (additive inverse) of the subtrahend to the minuend. • To multiply two signed numbers, find the product of the absolute values. Write a plus sign before this product when the two numbers have the same sign. Write a minus sign before this product when the two numbers have different signs. • To divide two signed numbers, find the quotient of the absolute values. Write a plus sign before this quotient when the two numbers have the same sign. Write a minus sign before this quotient when the two numbers have different signs. When 0 is divided by any number, the quotient is 0. Division by 0 is not defined. The location of a point on a plane is given by an ordered pair of numbers that indicate the distance of the point from two reference lines, a horizontal line and a vertical line, called coordinate axes. The horizontal line is called the xaxis, the vertical line is the y-axis, and their intersection is the origin. The pair of numbers that are used to locate a point on
the plane are called the coordinates of the point. The first number in the pair is called the x-coordinate or abscissa, and the second number is the y-coordinate or ordinate. The coordinates of a point are represented as (x, y). VOCABULARY 2-1 Binary operation • Factor • Prime • Composite • Base • Exponent • Power • Order of operations • Parentheses • Brackets Review Exercises 83 2-2 Property • Closure • Commutative property • Associative property • Distributive property of multiplication over addition and subtraction • Addition property of zero • Additive identity • Additive inverse (opposite) • Multiplication property of one • Multiplicative inverse (reciprocal) • Multiplication property of zero 2-3 Property of the opposite of a sum 2-7 Universal set • Intersection • Disjoint sets • Empty set • Union • Complement 2-8 x-axis • y-axis • Coordinate axes • Coordinate plane • Origin • Quadrants • Coordinates • Ordered pair • x-coordinate • Abscissa • y-coordinate • Ordinate • Graph of the ordered pair 2-9 Polygon • Quadrilateral • Vertices REVIEW EXERCISES In 1–8, simplify each numerical expression. 1. 20 3 4 4. (8 16) (4 2) 7. (6 8)2 2. (20 3) ( 4) 5. (0.16)2 8. 7(9 7)3 3. –8 16 4 2 6. 62 82 In 9–14: a. Replace each question mark with a number that makes the sentence true. b. Name the property illustrated in each sentence that is formed when the replacement is made. 9. 8 (2 9) 8 (9?) 11. 3(?) 3 13. 5(7 4) 5(7)?(4) 10. 8 (2 9) (8?) 9 12. 3(?) 0 14. 5(7 4) (7 4)? In 15–24, the universe U {1, 2, 3, 4, 5, 6}, set A {1, 2, 4, 5}, and set B {2, 4, 6}. In each case, perform the given operation and list the element(s) of the resulting set. 15. A d B 20. A < A 16. 21. A < B A d
B 17. A 22. A < B 18. 23. A d B A d B 19. A d A 24. A < B In 25–32, find each sum or difference. 25. 6 6 29. 23 0 26. 6 6 30. 54 52 27. 3.2 4.5 31. 100 25 28. 4 5 32. 0 7 84 Operations and Properties In 33–40, to each property named in Column I, match the correct application of the property found in Column II. Column I Column II 33. Associative property of multiplication 34. Associative property of addition 35. Commutative property of addition 36. Commutative property of multiplication 37. Identity element of multiplication 38. Identity element of addition 39. Distributive property 40. Multiplication property of zero a. 3 4 4 3 b. 3 1 3 c. 0 4 0 d. 3 0 3 e. 3 4 4 3 f. 3(4 5) 3(4) 3(5) g. 3(4 5) (3 4)5 h. (3 4) 5 3 (4 5) 41. a. Find the number on the odometer of Jesse’s car described in the chapter opener on page 37. b. What would the number be if it were an odd number? 42. a. Rectangle ABCD is drawn with two sides parallel to the x-axis. The coordinates of vertex A are (2, 4) and the coordinates of C are (3, 5). Find the coordinates of vertices B and D. b. What is the area of rectangle ABCD? 43. Maurice answered all of the 60 questions on a multiple-choice test. The where S is the score on test was scored by using the formula S the test, R is the number right, and W is the number wrong. R 2 W 4 a. What is the lowest possible score? b. How many answers did Maurice get right if his score was 5? c. Is it possible for a person who answers all of the questions to get a score of 4? Explain why not or find the numbers of right and wrong answers needed to get this score. d. Is it possible for a person who does not answer all of the questions to get a score of 4? Explain why not or find the numbers of right and wrong answers needed to get this score. 44. Solve the following problem using signed numbers. Doug, the team’s replacement quarterback, started out
on his team’s 30-yard line. On the first play, one of his linemen was offsides for a loss of 5 yards. On the next play, Doug gave the ball to the runningback who made a gain of 8 yards. He then made a 17-yard pass. Then Doug was tackled, for a loss of 3 yards. Where was Doug on the field after he was tackled? Cumulative Review 85 Exploration In this activity, you will derive a rule to determine if a number is divisible by 3. We will start our exploration with the number 23,568. For steps 1–4, fill in the blanks with a digit that will make the equality true. STEP 1. Write the number as a sum of powers of ten. 23,568 20,000 3,000 500 60 8 □ 10,000 □ 1,000 □ 100 □ 10 □ STEP 2. Rewrite the powers of ten as a multiple of 9, plus 1. □ 1,000 □ □ (9,999 1) □ (999 1) □ (99 1) □ (9 1) □ □ 10,000 □ 100 □ 10 STEP 3. Use the distributive property to expand the product terms. Do not multiply out products involving the multiples of 9. □ (9,999 1) □ (999 1) □ (99 1) □ (9 1) □ (□ 9,999 □) (□ 999 □) (□ 99 □) (□ 9 □) □ STEP 4. Group the products involving the multiples of 9 first and then the remaining digit terms. (□ 9,999 □) (□ 999 □) (□ 99 □) (□ 9 □) □ (□ 9,999 □ 999 □ 99 □ 9) (□ □ □ □ □) STEP 5. Compare the expression involving the digit terms with the original number. What do they have in common? STEP 6. The expression involving the multiples of 9 is divisible by 3. Why? STEP 7. If the expression involving the digit terms is divisible by 3, will the entire expression be divisible by 3? Explain. STEP 8. Based on steps 1–7 write the rule to determine if a number is divisible by 3. CUMUL
ATIVE REVIEW CHAPTERS 1–2 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. Which number is not an integer? (1) 7 (2) 2 (3) 0.2 (4) 9 " 86 Operations and Properties 2. Which inequality is false? (1) 4 3 (2) 3 3 3. What is the opposite of 4? (1) 21 4 (2) 1 4 (3) 4 3 (4) 3 3 4. Which of the following numbers has the greatest value? (1) 5 7 (2) 0.7 (3) p 5 5. Under which operation is the set of integers not closed? (3) 4 (4) 4 (4) 25 36 (1) Addition (2) Subtraction (3) Multiplication (4) Division 6. Which of the following numbers has exactly three factors? (1) 1 (2) 2 (3) 9 (4) 15 7. The graph of the ordered pair (3, 5) is in which quadrant? (1) I (2) II (3) III (4) IV 8. Put the following numbers in order, starting with the smallest: 21, 3 21 3 21 3 (1) (2) 21 5 22, 7 22 7 21 5 21 5 22 7 (3) (4) 21 5 21 5 21 3 22 7 22 7 21 3 9. When rounded to the nearest hundredth, 3 is approximately equal to (1) 1.732 (2) 1.730 10. For which value of t is t t2? 1 t (2) 2 (1) 1 Part II " (3) 1.73 (3) 0 (4) 1.74 (4) 1 2 Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. Mrs. Ling spends more than $4.90 and less than $5.00 for meat for tonight’s dinner. Write the set of all possible amounts that she could have paid for the meat. Is this a finite or an infinite set? Explain. 12. In a basketball league, 100 students play on 8 teams. Each team has at
least 12 players. What is the largest possible number of players on any one team? Cumulative Review 87 Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. A teacher wrote the sequence 2, 4, 6,... and asked the class what the next number could be. Three students each gave a different answer. The teacher said that each of the answers was correct. a. Josie said 8. Explain the rule that she used. b. Emil said 10. Explain the rule that he used. c. Ross said 12. Explain the rule that he used. 14. Evaluate the following expression without using a calculator. Show each step in your computation. 4(7 3) 8 (2 6)2 Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. The vertices of triangle ABC are A(2, 3), B(5, 3), and C(0, 4). Draw triangle ABC on the coordinate plane and find its area. 16. A survey to which 250 persons responded found that 140 persons said that they watch the news on TV at 6 o’clock, 120 persons said that they watch the news on TV at 11 o’clock and 40 persons said that they do not watch the news on TV at any time. a. How many persons from this group watch the news both at 6 and at 11? b. How many persons from this group watch the news at 6 but not at 11? c. How many persons from this group watch the news at 11 but not at 6? CHAPTER 3 CHAPTER TABLE OF CONTENTS 3-1 Using Letters to Represent Numbers 3-2 Translating Verbal Phrases Into Symbols 3-3 Algebraic Terms and Vocabulary 3-4 Writing Algebraic Expressions in Words 3-5 Evaluating Algebraic Expressions 3-6 Open Sentences and Solution Sets 3-7 Writing Formulas Chapter Summary Vocabulary Review Exercises Cumulative Review 88 ALGEBRAIC EXPRESSIONS AND OPEN S
ENTENCES An express delivery company will deliver a letter or package locally, within two hours.The company has the following schedule of rates. In addition to the basic charge of $25, the cost is $3 per mile or part of a mile for the first 10 miles or less and $4.50 per mile or part of a mile for each additional mile over 10. Costs such as those described, that vary according to a schedule, are often shown by a formula or set of formulas. Formulas can be used to solve many different problems. In this chapter, you will learn to write algebraic expressions and formulas, to use algebraic expressions and formulas to solve problems, and to determine the solution set of an open sentence. 3-1 USING LETTERS TO REPRESENT NUMBERS Using Letters to Represent Numbers 89 Eggs are usually sold by the dozen, that is, 12 in a carton. Therefore, we know that: In 1 carton, there are 12 1 or 12 eggs. In 2 cartons, there are 12 2 or 24 eggs. In 3 cartons, there are 12 3 or 36 eggs. In n cartons, there are 12 n or 12n eggs. Here, n is called a variable or a placeholder that can represent different numbers from the set of whole numbers, {1, 2, 3,...}. The set of numbers that can replace a variable is called the domain or the replacement set of that variable. Recall that a numerical expression contains only numbers. An algebraic expression, such as 12n, however, is an expression or phrase that contains one or more variables. In this section, you will see how verbal phrases are translated into algebraic expressions, using letters as variables and using symbols to represent operations. Verbal Phrases Involving Addition The algebraic expression a b may be used to represent several different verbal phrases, such as: a plus b the sum of a and b a added to b b is added to a a is increased by b b more than a The word exceeds means “is more than.” Thus, the number that exceeds 5 by 2 can be written as “2 more than 5” or 5 2. Compare the numerical and algebraic expressions shown below. A numerical expression: An algebraic expression: The number that exceeds 5 by 2 is 5 2, or 7. The number that exceeds a by b is a b. Verbal Phrases Involving Subtraction The algebraic expression
a b may be used to represent several different verbal phrases, such as: a minus b b subtracted from a a decreased by b a diminished by b b less than a a reduced by b the difference between a and b Verbal Phrases Involving Multiplication The algebraic expressions a b, a b, (a)(b) and ab may be used to represent several different verbal phrases, such as: a times b the product of a and b b multiplied by a 90 Algebraic Expressions and Open Sentences The preferred form to indicate multiplication in algebra is ab. Here, a product is indicated by using no symbol between the variables being multiplied. The multiplication symbol is avoided in algebra because it can be confused with the letter or variable x. The raised dot, which is sometimes mistaken for a decimal point, is also avoided. Parentheses are used to write numerical expressions: (3)(5)(2) or 3(5)(2). Note that all but the first number must be in parentheses. In algebraic expressions, parentheses may be used but they are not needed: 3(b)(h) 3bh. Verbal Phrases Involving Division The algebraic expressions a b and may be used to represent several different verbal phrases, such as: a b a divided by b the quotient of a and b a 4 The symbols a 4 and mean one-fourth of a as well as a divided by 4. Phrases and Commas In some verbal phrases, using a comma can prevent misreading. For example, in “the product of x and y, decreased by 2,” the comma after y makes it clear that the x and y are to be multiplied before subtracting 2 and can be written as (xy) 2 or xy 2. Without the comma, the phrase, “the product of x and y decreased by 2,” would be written x(y 2). EXAMPLE 1 Use mathematical symbols to translate the following verbal phrases into algebraic language: a. w more than 3 b. w less than 3 c. r decreased by 2 Answers 3 w 3 w r 2 d. the product of 5r and s e. twice x, decreased by 10 f. 25, diminished by 4 times n g. the sum of t and u, divided by 6 h. 100 decreased by twice (x 5) 5rs 2x 10 25 4n t1u 6 100 2(x 5) Translating Verbal Phrases into
Symbols 91 EXERCISES Writing About Mathematics 1. Explain why the sum of a and 4 can be written as a 4 or as 4 a. 2. Explain why 3 less than a can be written as a 3 but not as 3 a. Developing Skills In 3–20, use mathematical symbols to translate the verbal phrases into algebraic language. 3. y plus 8 6. x times 7 4. 4 minus r 5. 7 times x 7. x divided by 10 8. 10 divided by x 9. c decreased by 6 10. one-tenth of w 11. the product of x and y 12. 5 less than d 13. 8 divided by y 14. y multiplied by 10 15. t more than w 16. one-third of z 17. twice the difference of p and q 18. a number that exceeds m by 4 19. 5 times x, increased by 2 20. 10 decreased by twice a In 21–30, using the letter n to represent “number,” write each verbal phrase in algebraic language. 21. a number increased by 2 22. 20 more than a number 23. 8 increased by a number 24. a number decreased by 6 25. 2 less than a number 26. 3 times a number 27. three-fourths of a number 28. 4 times a number, increased by 3 29. 3 less than twice a number 30. 10 times a number, decreased by 2 In 31–34, use the given variable(s) to write an algebraic expression for each verbal phrase. 31. the number of baseball cards, if b cards are added to a collection of 100 cards 32. Hector’s height, if he was h inches tall before he grew 2 inches 33. the total cost of n envelopes that cost $0.39 each 34. the cost of one pen, if 12 pens cost d dollars 3-2 TRANSLATING VERBAL PHRASES INTO SYMBOLS A knowledge of arithmetic is important in algebra. Since the variables represent numbers that are familiar to you, it will be helpful to solve each problem by first using a simpler related problem; that is, relate similar arithmetic problems to the given algebraic one. 92 Algebraic Expressions and Open Sentences Procedure To write an algebraic expression involving variables: 1. Think of a similar problem in arithmetic. 2. Write an expression for the arithmetic problem, using numbers. 3. Write a similar expression for the problem, using letters or variables
. EXAMPLE 1 Represent each phrase by an algebraic expression. a. a distance that is 20 meters shorter than x meters b. a bill for n baseball caps, each costing d dollars c. a weight that is 40 pounds heavier than p pounds d. an amount of money that is twice d dollars Solution a. How to Proceed (1) Think of a similar problem in arithmetic: (2) Write an expression for this arithmetic problem: Think of a distance that is 20 meters shorter than 50 meters. 50 20 (3) Write a similar expression using x 20 Answer the letter x in place of 50. b. How to Proceed (1) Think of a similar problem in arithmetic: Think of a bill for 6 caps, each costing 5 dollars. (2) Write an expression for this arithmetic problem. Multiply the number of caps by the cost of one cap: 6(5) (3) Write a similar expression nd Answer using n and d: Note: After some practice, you will be able to do steps (1) and (2) mentally. c. (p 40) pounds or (40 p) pounds d. 2d dollars Answers a. (x 20) meters c. (p 40) or (40 p) pounds b. nd dollars d. 2d dollars Translating Verbal Phrases into Symbols 93 EXAMPLE 2 Brianna paid 17 dollars for batteries and film for her camera. If the batteries cost x dollars, express the cost of the film in terms of x. Solution If Brianna had spent 5 dollars for the batteries, the amount that was left is found by subtracting the 5 dollars from the 17 dollars, (17 5) dollars. This would have been the cost of the film. If Brianna spent x dollars for the batteries, then the difference, (17 x) dollars would have been the cost of the film. Answer (17 x) dollars Note: In general, if we know the sum of two quantities, then we can let x represent one of these quantities and (the sum x) represent the other. EXERCISES Writing About Mathematics 1. a. Represent the number of pounds of grapes you can buy with d dollars if each pound costs b dollars. b. Does the algebraic expression in part a always represent a whole number? Explain your answer by showing examples using numbers. 2. a. If x apples cost c cents, represent the cost of one apple. b. If x apples cost c cents,
represent the cost of n apples. c. Do the algebraic expressions in parts a and b always represent whole numbers? Explain your answer. Developing Skills In 3–18, represent each answer in algebraic language, using the variable mentioned in the problem. 3. The number of kilometers traveled by a bus is represented by x. If a train traveled 200 kilo- meters farther than the bus, represent the number of kilometers traveled by the train. 4. Mr. Gold invested $1,000 in stocks. If he lost d dollars when he sold the stocks, represent the amount he received for them. 5. The cost of a mountain bike is 5 times the cost of a skateboard. If the skateboard costs x dollars, represent the cost of the mountain bike. 6. The length of a rectangle is represented by l. If the width of the rectangle is one-half of its length, represent its width. 7. After 12 centimeters had been cut from a piece of lumber, c centimeters were left. Represent the length of the original piece of lumber. 94 Algebraic Expressions and Open Sentences 8. Paul and Martha saved 100 dollars. If the amount saved by Paul is represented by x, repre- sent the amount saved by Martha. 9. A ballpoint pen sells for 39 cents. Represent the cost of x pens. 10. Represent the cost of t feet of lumber that sells for g cents a foot. 11. If Hilda weighed 45 kilograms, represent her weight after she had lost x kilograms. 12. Ronald, who weighs c pounds, is d pounds overweight. Represent the number of pounds Ronald should weigh. 13. A woman spent $250 for jeans and a ski jacket. If she spent y dollars for the ski jacket, rep- resent the amount she spent for the jeans. 14. A man bought an article for c dollars and sold it at a profit of $25. Represent the amount for which he sold it. 15. The width of a rectangle is represented by w meters. Represent the length of the rectangle if it exceeds the width by 8 meters. 16. The width of a rectangle is x centimeters. Represent the length of the rectangle if it exceeds twice the width by 3 centimeters. 17. If a plane travels 550 kilometers per hour, represent the distance it will travel in h hours. 18. If a car traveled for 5 hours at an average rate of r kilometers per hour, represent the dis- tance it traveled. 19. a. Represent the total number of
days in w weeks and 5 days. b. Represent the total number of days in w weeks and d days. Applying Skills 20. An auditorium with m rows can seat a total of c people. If each row in the auditorium has the same number of seats, represent the number of seats in one row. 21. Represent the total number of calories in x peanuts and y potato chips if each peanut con- tains 6 calories and each potato chip contains 14 calories. 22. The charges for a telephone call are $0.45 for the first 3 minutes and $0.09 for each additional minute or part of a minute. Represent the cost of a telephone call that lasts m minutes when m is greater than 3. 23. A printing shop charges a 75-cent minimum for the first 8 photocopies of a flyer. Additional copies cost 6 cents each. Represent the cost of c copies if c is greater than 8. 24. A utility company measures gas consumption by the hundred cubic feet, CCF. The company has a three-step rate schedule for gas customers. First, there is a minimum charge of $5.00 per month for up to 3 CCF of gas used. Then, for the next 6 CCF, the charge is $0.75 per CCF. Finally, after 9 CCF, the charge is $0.55 per CCF. Represent the cost of g CCF of gas if g is greater than 9. 3-3 ALGEBRAIC TERMS AND VOCABULARY Terms Algebraic Terms and Vocabulary 95 A term is a number, a variable, or any product or quotient of numbers and vari5 c ables. For example: 5, x, 4y, 8ab,, and are terms. k2 5 An algebraic expression that is written as a sum or a difference has more than one term. For example, 4a 2b 5c has three terms. These terms, 4a, 2b, and 5c, are separated by and signs. Factors of a Term If a term contains two or more numbers or variables, then each number, each variable, and each product of numbers and variables is called a factor of the term, or factor of the product. For example, the factors of 3xy are 1, 3, x, y, 3x, 3y, xy, and 3xy. When we factor whole numbers, we write only factors that are integers. Any factor of an algebra
ic term is called the coefficient of the remaining factor, or product of factors, of that term. For example, consider the algebraic term 3xy: 3 is the coefficient of xy 3y is the coefficient of x 3x is the coefficient of y xy is the coefficient of 3 When an algebraic term consists of a number and one or more variables, the number is called the numerical coefficient of the term. For example: In 8y, the numerical coefficient is 8. In 4abc, the numerical coefficient is 4. When the word coefficient is used alone, it usually means a numerical coefficient. Also, since x names the same term as 1x, the coefficient of x is understood to be 1. This is true of all terms that contain only variables. For example: 7 is the coefficient of b in the 1 is the coefficient of b in the term b. term 7b. 2.25 is the coefficient of gt in 1 is the coefficient of gt in the term gt. the term 2.25gt. Bases, Exponents, and Powers You learned in Chapter 2 that a power is the product of equal factors. A power has a base and an exponent. The base is one of the equal factors of the power. 96 Algebraic Expressions and Open Sentences The exponent is the number of times the base is used as a factor. (If a term is written without an exponent, the exponent is understood to be 1.) 42 4(4): x3 = x(x)(x): 35m: 5d2 = 5(d)(d): base 4 base x base m base d exponent 2 exponent 3 exponent 1 exponent 2 power 42 16 power x3 power m1 or m power d2 An exponent refers only to the number or variable that is directly to its left, as seen in the last example, where 2 refers only to the base d. To show the product 5d as a base (or to show any sum, difference, product, or quotient as a base), we must enclose the base in parentheses. (5d)2 (5d)(5d): base 5d exponent 2 power (5d)2 exponent 2 power (a 4)2 (a + 4)2 (a + 4)( a + 4): base (a + 4) Note that –d4 is not the same as (d)4. –d 4 –1(d)(d)(d)(d) is always a negative number.
(d)4 1(d)(d)(d)(d) is always a positive number since the exponent is even. EXAMPLE 1 For each term, name the coefficient, base, and exponent. a. 4x5 b. w8 c. 2pr coefficient 4 coefficient –1 coefficient 2p Answers base x base w base r exponent 5 exponent 8 exponent 1 Note: Remember that coefficient means numerical coefficient, and that 2p is a real number. EXERCISES Writing About Mathematics 1. Does squaring distribute over multiplication, that is, does (ab)2 = (a2)(b2)? Write (ab)2 as (ab)(ab) and use the associative and commutative properties of multiplication to justify your answer. 2. Does squaring distribute over addition, that is, does (a b)2 a2 b2? Substitute values for a and b to justify your answer. Algebraic Terms and Vocabulary 97 Developing Skills In 3–6, name the factors (other than 1) of each product. 3. xy 4. 3a 5. 7mn 6. 1st In 7–14, name, in each case, the numerical coefficient of x. 7. 8x 11. 1.4x 8. (5 2)x 12. 2 7x 9. 1 2x 13. 3.4x In 15–22, name, in each case, the base and exponent of the power. 15. m2 19. 106 16. s3 20. (5y)4 17. t 21. (x y)5 10. x 14. x 18. (a)4 22. 12c3 In 23–29, write each expression, using exponents. 23. b b b b b 26. 7 r r r s s 24. p r r 27. (6a)(6a)(6a) 29. the fourth power of (m + 2n) 25. a a a a b b 28. (a b)(a b)(a b) In 30–33, write each term as a product without using exponents. 30. r6 31. 5x4 32. 4a4b2 33. (3y)5 In 34–41, name, for each given term, the coefficient, base, and exponent. 34. 3k 38. 2y " Applying Skills 35. k3 39. 0.0004t12 36. pr2
3 2a4 40. 37. (ax)5 41. (b)3 42. If x represents the cost of a can of soda, what could 5x represent? 43. If r represents the speed of a car in miles per hour, what could 3r represent? 44. If n represents the number of CDs that Alice has, what could n 5 represent? 45. If d represents the number of days until the end of the year, what could 46. If s represents the length of a side of a square, what could 4s represent? d 7 represent? 47. If r represents the measure of the radius of a circle, what could 2r represent? 48. If w represents the number of weeks in a school year, what could 49. If d represents the cost of one dozen bottles of water, what could w 4 d 12 represent? represent? 50. If q represents the point value of one field goal, what could 7q represent? 98 Algebraic Expressions and Open Sentences 3-4 WRITING ALGEBRAIC EXPRESSIONS IN WORDS In Section 1 of this chapter, we listed the words that can be represented by each of the four basic operations. We can use these same lists to write algebraic expressions in words and to write problems that can be represented by a given algebraic expression. For an algebraic expression such as 2n 3, n could be any real number. That is, associated with any real number n, there is exactly one real number that is the value of 2n 3. However, if n and 2n 3 represent the number of cans of tuna that two customers buy, then n must be a whole number greater than or equal to 2 in order for both n and 2n 3 to be whole numbers. For this situation, the domain or replacement set would be the set of whole numbers. EXAMPLE 1 If n represents the number of points that Hradish scored in a basketball game and 2n 3 represents the number of points that his friend Brad scored, describe in words the number of points that Brad scored. What is a possible domain for the variable n? Solution The number of points scored is always a whole number. In order that 2n 3 be a whole number, n must be at least 2. Answer The number of points that Brad scored is 3 less than twice the number that Hradish scored. A possible domain for n is the set of whole numbers greater than or equal to 2. EXAMPLE
2 Molly earned d dollars in July and the number of dollars that Molly earned in August. 1 2d 1 10 dollars in August. Describe in words Answer In August, Molly earned 10 more than half the number of dollars that she earned in July. EXAMPLE 3 Describe a situation in which x and 12 x can be used to represent variable quantities. List the domain or replacement set for the answer. Solution If x eggs are used from a full dozen of eggs, there will be 12 x eggs left. Answer The domain or replacement set is the set of whole numbers less than or equal to 12. Writing Algebraic Expressions in Words 99 Another Solution The distance from my home to school is 12 miles. On my way to school, after I have traveled x miles, I have 12 x miles left to travel. Answer The domain or replacement set is the set of non-negative real numbers that are less than or equal to 12. Many other answers are possible. EXERCISES Writing About Mathematics 1. a. If 4 n represents the number of books Ken read in September and 4 n represents the number of books he read in October, how many books did he read in these two months? b. What is the domain of the variable n? 2. Pedro said that the replacement set for the amount that we pay for any item is the set of rational numbers of the form 0.01x where x is a whole number. Do you agree with Pedro? Explain why or why not. Developing Skills In 3–14: a. Write in words each of the given algebraic expressions. b. Describe a possible domain for each variable. 3. By one route, the distance that Ian walks to school is d miles. By a different route, the dis- tance is d 0.2 miles. 4. Juan pays n cents for a can of soda at the grocery store. When he buys soda from a machine, he pays n 15 cents. 5. Yesterday Alexander spent a minutes on leisurely reading and 3a 10 minutes doing homework. 6. The width of a rectangle is w meters and the length is 2w 8 meters. 7. During a school day, Abby spends h hours in class, hours at lunch and hours on sports. d 8. Jen spends d hours at work and 12 9. Alicia’s score for 18 holes of golf was g and her son’s score was 10 g. 10. Tom paid d cents for a notebook and 5d 30 cents for
a pen. h 6 hours driving to and from work. h 3 11. Seema’s essay for English class had w words and Dominic’s had 12. Virginia read r books last month and Anna read 3r 5 books. 3 4 w 1 80 words. 13. Mario and Pete are playing a card game where it is possible to have a negative score. Pete’s score is s and Mario’s score is s 220. 100 Algebraic Expressions and Open Sentences 14. In the past month, Agatha has increased the time that she walks each day from m minutes to 3m 10 minutes. 3-5 EVALUATING ALGEBRAIC EXPRESSIONS Benjamin has 1 more tape than 3 times the number of tapes that Julia has. If Julia has n tapes, then Benjamin has 3n 1 tapes. The algebraic expression 3n 1 represents an unspecified number. Only when the variable n is replaced by a specific number does 3n 1 become a specific number. For example: If n 10, then 3n 1 3(10) 1 30 1 31. If n 15, then 3n 1 3(15) 1 45 1 46. Since in this example, n represents the number of tapes that Julia has, only whole numbers are reasonable replacements for n. Therefore, the replacement set is the set of whole numbers or some subset of the set of whole numbers. When we substitute specific values for the variables in an algebraic expression and then determine the value of the resulting expression, we are evaluating the algebraic expression. When we determine the number that an algebraic expression represents for specific values of its variables, we are evaluating the algebraic expression. Procedure To evaluate an algebraic expression, replace the variables by the given values, and then follow the rules for the order of operations. 1. Replace the variables by the given values. 2. Evaluate the expression within the grouping symbols such as parentheses, always simplifying the expressions in the innermost groupings first. 3. Simplify all powers and roots. 4. Multiply and divide, from left to right. 5. Add and subtract, from left to right. EXAMPLE 1 Evaluate 50 3x when x 7. Solution How to Proceed (1) Write the expression: 50 3x Evaluating Algebraic Expressions 101 (2) Replace the variable by its given value: (3) Multiply: (4) Subtract: 50 3(7) 50
21 29 Answer 29 EXAMPLE 2 Evaluate 2x2 5x 4 when: a. x = 7 b. x = 1.2 Solution How to Proceed a. (1) Write the expression: (2) Replace the variable by the value 7: (3) Evaluate the power: (4) Multiply: (5) Add: b. (1) Write the expression: (2) Replace the variable by the value 1.2: (3) Evaluate the power: (4) Multiply: (5) Add and subtract: 2x2 5x 4 2(7)2 5(7) 4 2(49) 5(7) 4 98 35 4 137 2x2 5x 4 2(1.2)2 5(1.2) 4 2(1.44) 5(1.2) 4 2.88 6 4 0.88 Answers a. 137 b. 0.88 EXAMPLE 3 Evaluate 2a 5 1 (n 2 1)d when a 4, n 10, and d 3. Solution How to Proceed (1) Write the expression: (2) Replace the variables with their given values: (3) Simplify the expressions grouped by parentheses or fraction bar: (4) Multiply and divide: (5) Add: 2a 5 1 (n 2 1)d 2(24) 5 1 (10 – 1)(3) 28 5 1(9)(3) 213 213 252 5 5 1 27 5 1 265 5 Answer 102 Algebraic Expressions and Open Sentences Calculator Solution The values given for the variables can be stored in the calculator. ENTER: (-) 4 STO ALPHA A ENTER 10 STO ALPHA N ENTER 3 STO ALPHA D ENTER DISPLAY Now enter the algebraic expression to be evaluated. ENTER: 2 ALPHA A 5 ( ALPHA N 1 ) ALPHA D ENTER DISPLAY Answer 252 5 5 25.4 EXAMPLE 4 Evaluate (2x)3 2x3 when x 0.40. Solution How to Proceed (1) Write the expression: (2) Replace the variable by its given value: (3) Simplify the expression within brackets: (4) Evaluate the powers: (5) Multiply: (6) Subtract: Answer 0.384 (2x)3 2x3 [2(0.40)]3 2(0.40)3 [0.80]
3 2(0.40)3 0.512 2(0.064) 0.512 0.128 0.384 Evaluating Algebraic Expressions 103 EXERCISES Writing About Mathematics 1. Explain why, in an algebraic expression such as 12ab, 12 is called a constant and a and b are called variables? 2. Explain why, in step 2 of Example 1, parentheses were needed when x was replaced by its value. Developing Skills To understand this topic, you should first evaluate the expressions in Exercises 3 to 27 without a calculator. Then, store the values of the variables in the calculator and enter the given algebraic expressions to check your work. In 3–27, find the numerical value of each expression. Use a 8, b 6, d 3, x 4, and y 0.5. 15. 3. 5a 7. b 2 11. 7xy3 3 4x3 19. (ay)3 23. 3y (x d) 26. (2a 5d)2 Applying Skills 4. 1 2x 8. ax2 12. ab dx 16. (3y)2 20. x(y 2) 9. 13. 5. 0.3y 3bd 9 5a 1 1 2 5b 1 4x2y 17. 21. 4(2x 3y) 6. a 3 10. 5x 2y 14. 0.2d 0.3b 18. a2 3d2 22. 1 2x(y 1 0.1)2 24. 2(x y) 5 27. (2a)2 (5d)2 25. (x d)5 28. At one car rental agency, the cost of a car for one day can be determined by using the alge- braic expression $32.00 $0.10m where m represents the number of miles driven. Determine the cost of rental for each of the following: a. Mike Baier drove the car he rented for 35 miles. b. Dana Morse drove the car he rented for 435 miles. c. Jim Szalach drove the car he rented for 102 miles. 29. The local pottery co-op charges $40.00 a year for membership and $0.75 per pound for firing pottery pieces made by the members. The algebraic expression 40 0.75p represents the yearly cost to a member who brings p pounds of pottery to be fired. Determine
the yearly cost for each of the following: a. Tiffany is an amateur potter who fired 35 pounds of work this year. b. Nia sells her pottery in a local craft shop and fired 485 pounds of work this year. 30. If a stone is thrown down into a deep gully with an initial velocity of 30 feet per second, the distance it has fallen, in feet, after t seconds can be found by using the algebraic expression 16t2 30t. Find the distance the stone has fallen: a. after 1 second. b. after 2 seconds. c. after 3 seconds. 104 Algebraic Expressions and Open Sentences 31. The Parkside Bread Company sells cookies and scones as well as bread. Bread (b) costs $4.50 a loaf, cookies (c) cost $1.10 each, and scones (s) cost $1.50 each. The cost of a bakery order can be represented by 4.50b 1.10c 1.50s. Determine the cost of each of the following orders: a. six cookies and two scones b. three loaves of bread and one cookie c. one loaf of bread, a dozen cookies, and a half-dozen scones 32. A Green Thumb volunteer can plant shrubbery at a rate of 6 shrubs per hour and a Friendly Garden volunteer can plant shrubbery at a rate of 8 shrubs per hour. The total number of shrubs that g Green Thumb volunteers and f Friendly Garden volunteers can plant in h hours is given by the algebraic expression 6gh 8fh. Determine the number of shrubs planted: a. in 3 hours by 2 Green Thumb and 1 Friendly Garden volunteers. b. in 2 hours by 4 Green Thumb and 4 Friendly Garden volunteers. 3-6 OPEN SENTENCES AND SOLUTION SETS In this chapter, you learned how to translate words into algebraic expressions. The value of an algebraic expression depends on the value of the variables. When the values of the variables change, the value of the algebraic expression changes. For example, x 6 is an algebraic expression. The value of x 6 depends on the value of x. If one value is assigned to an algebraic expression, an algebraic sentence is formed. These sentences may be formulas, equations, or inequalities. For example, when the value 9 is assigned to the algebraic expression x 6, we can write the sentence “
Six more than x is 9.” This sentence can be written in symbols as x 6 9. Every sentence that contains a variable is called an open sentence. x 6 9 An open sentence is neither true nor false. x 5 8 3y 12 2n 0 The sentence will be true or false only when the variables are replaced by numbers from a domain or a replacement set, such as {0, 1, 2, 3}. The numbers from the domain that make the sentence true are the elements of the solution set of the open sentence. A solution set, as seen below, can contain one or more numbers or, at times, no numbers at all, from the replacement set. EXAMPLE 1 Using the domain {0, 1, 2, 3}, find the solution set of each open sentence: a. x 6 9 b. 2n 0 Solution a. Procedure: Replace x in the open sentence with numbers from the domain {0, 1, 2, 3}. x + 6 9 Let x 0. Then 0 6 9 is false. Let x 1. Then l 6 9 is false. Let x 2. Then 2 6 9 is false. Let x 3. Then 3 6 9 is true. Here, only when x 3 does the open sentence become a true sentence. Open Sentences and Solution Sets 105 b. Procedure: Replace n in the open sentence with numbers from the domain {0, 1, 2, 3}. 2n 0 Let n 0. Then 2(0) 0 or 0 0 is false. Let n 1. Then 2(l) 0 or 2 0 is true. Let n 2. Then 2(2) 0 or 4 0 is true. Let n 3. Then 2(3) 0 or 6 0 is true. Here, three elements of the domain make the open sentence true. Answer: a. Solution set {3}. Answer: b. Solution set {1, 2, 3} EXAMPLE 2 Find the solution set for the open sentence 3y 12 using: a. the domain {3, 5, 7} b. the domain {whole numbers} Solution a. Procedure: Replace y with 3, 5, and 7. If y 3, then 3(3) 12 is false. If y 5, then 3(5) 12 is false. If y 7, then 3(7) 12 is false. When y is replaced by each of the numbers from the domain, no true statement is found. The solution set for this domain
is the empty set or the null set, written in symbols as { } or as. Answer: a. The solution set is { } or. b. Procedure: Of course, you cannot replace y with every whole number, but you can use multiplication facts learned previously. You know that 3(4) 12. Let y 4. Then 3(4) 12 is true. No other whole number would make the open sentence 3y 12 a true sentence. Answer: b. The solution set is {4}. 106 Algebraic Expressions and Open Sentences EXERCISES Writing About Mathematics 1. For the open sentence x 7 12, write a domain for which the solution set is the empty set. 2. For the open sentence x 7 12, write a domain for which the solution set has only one element. 3. For the open sentence x 7 12, write a domain for which the solution set is an infinite set. Developing Skills In 4–11, tell whether each is an open sentence, a true sentence, or an algebraic expression. 4. 2 3 5 0 8. n 7 5. x 10 14 9. 3 2 10 6. y 4 10. 3r 2 7. 3 7 2(5) 11. 2x 7 15 In 12–15, name the variable in each open sentence. 12. x 5 9 13. 4y 20 14. r 6 2 15. 7 3 a In 16–23, using the domain {5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5}, find the solution set for each open sentence. 16. n 3 7 20. 2n 1 8 Applying Skills 17. x x 0 2n 1 1 3 5 4 21. 18. 5 n 2 22. 3x 2, x 19. n 3 9 23. 2x 4 24. Pencils sell for $0.19 each. Torry wants to buy at least one but not more than 10 pencils and has $1.50 in his pocket. a. Use the number of pencils that Torry wants to buy to write a domain for this problem. b. The number of pencils that Torry might buy, x, can be found using the open sentence 0.19x 1.50. Find the solution set of this open sentence using the domain from part a. c. How many pencils can Torry buy? 25. The local grocery store has frozen orange juice on sale for
$0.99 a can but limits the number of cans that a customer may buy at the sale price to no more than 5. a. The domain for this problem is the number of cans of juice that a customer may buy at the sale price. Write the domain. b. If Mrs. Dajhon does not want to spend more than $10, the number of cans that she might buy at the sale price, y, is given by the equation 0.99y 10. Find the solution set of this equation using the domain from part a. c. How many cans can Mrs. Dajhon buy if she does not want to spend more than $10? Writing Formulas 107 26. Admission to a recreation park is $17.50. This includes all rides except for a ride called The Bronco that costs $1.50 for each ride. Ian has $25 to spend. a. Find the domain for this problem, the number of times a person might ride The Bronco. b. The number of times Ian might ride The Bronco, z, can be found using the open sen- tence 17.50 1.50z 25. Find the solution set of this open sentence using the domain from part a. c. How many times can Ian ride The Bronco? 3-7 WRITING FORMULAS A formula uses mathematical language to express the relationship between two or more variables. Some formulas are found by the strategy of looking for patterns. For example, how many square units are shown in the rectangle on the left? This rectangle, measuring 4 units in length and 3 units in width, contains a total of 12 square units of area. Many such examples led to the conclusion that the area of a rectangle is equal to the product of its length and width. This relationship is expressed by the formula A lw where A, l, and w are variables that represent, respectively, the area, the length, and the width of a rectangle. A formula is an open sentence that states that two algebraic expressions are equal. In formulas, the word is is translated into the symbol. EXAMPLE 1 Write a formula for each of the following relationships. a. The perimeter P of a square is equal to 4 times the length of one side. b. The total cost C of an article is equal to its price p plus an 8% tax on the price. c. The sum S of the measures of the interior angles of an n-sided polygon is 180 times 2
less than the number of sides. Solution a. Let s represent the length of each side of a square. b. 8% (or 8 percent) means 8 hundredths, written as 0.08 or 8 100. P 4s Answer 100p c. “2 less than the number of sides” means (n 2). C p 0.08p or C 5 p 1 8 Answer S 180(n 2) Answer 108 Algebraic Expressions and Open Sentences EXAMPLE 2 Write a formula that expresses the number of months m that are in y years. Solution Look for a pattern. In 1 year, there are 12 months. In 2 years, there are 12(2) or 24 months. In 3 years, there are 12(3) or 36 months. In y years, there are 12(y) or 12y months. This equals m, the number of months. Answer m 12y EXAMPLE 3 The Short Stop Diner pays employees $6.00 an hour for working 40 hours a week or less. For working overtime, an employee is paid $9.00 for each hour over 40 hours. Write a formula for the wages, W, of an employee who works h hours in a week. Solution Two formulas are needed, one for h 40 and the other for h 40. If h 40, the wage is 6.00 times the number of hours, h. W 6.00h If h 40, the employee has worked 40 hours at $6.00 an hour and the remaining hours, h 40, at $9.00 an hour. W 6.00(40) 9.00(h 40). Answer W 6h if h 40 and W 6(40) 9(h 40) if h 40. (Note that the formula for h 40 may also be given as W 240 9(h 40).) EXERCISES Writing About Mathematics 1. Fran said that a recipe is a type of formula. Do you agree or disagree with Fran? Explain your answer. 2. a. Is an algebraic expression a formula? Explain why or why not. b. Is a formula an open sentence? Explain why or why not. Writing Formulas 109 Developing Skills In 3–17, write a formula that expresses each relationship. 3. The total length l of 10 pieces of lumber, each m meters in length, is 10 times the length of each piece of lumber. 4. An article’s selling price S equals its
cost c plus the margin of profit m. 5. The perimeter P of a rectangle is equal to the sum of twice its length l and twice its width w. 6. The average m of three numbers, a, b, and c is their sum divided by 3. 7. The area A of a triangle is equal to one-half the length of the base b multiplied by the length of the altitude h. 8. The area A of a square is equal to the square of the length of a side s. 9. The volume V of a cube is equal to the cube of the length of an edge e. 10. The surface area S of a cube is equal to 6 times the square of the length of an edge e. 11. The surface area S of a sphere is equal to the product of 4p and the square of the radius r. 12. The average rate of speed r is equal to the distance that is traveled d divided by the time spent on the trip t. 13. The Fahrenheit temperature F is 32° more than nine-fifths of the Celsius tempera- ture C. 14. The Celsius temperature C is equal to five-ninths of the difference between the Fahrenheit temperature F and 32°. 15. The dividend D equals the product of the divisor d and the quotient q plus the remainder r. 16. A sales tax T that must be paid when an article is purchased is equal to 8% of the price of the article v. 17. A salesman’s weekly earnings F is equal to his weekly salary s increased by 2% of his total volume of sales v. Applying Skills 18. A ferry takes cars, drivers, and passengers across a body of water. The total ferry charge C in dollars is $20.00 for the car and driver, plus d dollars for each passenger. a. Write a formula for C in terms of d and the number of passengers, n. b. Find the cost of the ferry for a car if d $15 and there are 5 persons in the car. c. Find the cost of the ferry for a car with only the driver. 110 Algebraic Expressions and Open Sentences 19. The cost C in cents of an internet telephone call lasting m minutes is x cents for the first 3 minutes and y cents for each additional minute. a. Write two formulas for C, one for the cost of calls lasting 3 minutes or less (m 3), and another for the cost of calls lasting more
than 3 minutes (m 3). b. Find the cost of a 2.5 minute telephone call if x $0.25 and y $0.05. c. Find the cost of a 10 minute telephone call if x $0.25 and y $0.05. 20. The cost D in dollars of sending a fax of p pages is a dollars for sending the first page and b dollars for each additional page. a. Write two formulas for D, one for the cost of faxing 1 page (p 1), and another for the cost of faxing more than 1 page (p 1). b. Find the cost of faxing 1 page if a $1.00 and b $0.60. c. Find the cost of faxing 5 pages if a $1.00 and b $0.60. 21. A gasoline dealer is allowed a profit of 12 cents a gallon for each gallon sold. If more than 25,000 gallons are sold in a month, an additional profit of 3 cents for every gallon over that number is given. a. Write two formulas for the gasoline dealer’s profit, P, one for when the number of gallons sold, n, is not more than 25,000 (n 25,000), and another for when more than 25,000 gallons are sold (n 25,000). b. Find P when 21,000 gallons of gasoline are sold in one month. c. Find P when 30,000 gallons of gasoline are sold in one month. 22. Gabriel earns a bonus of $25 for each sale that he makes if the number of sales, s, in a month is 20 or less. He earns an extra $40 for each additional sale if he makes more than 20 sales in a month. a. Write a formula for Gabriel’s bonus, B, when s 20. b. Write a formula for B when s 20. c. In August, Gabriel made 18 sales. Find his bonus for August. d. In September, Gabriel made 25 sales. Find his bonus for September. 23. Mrs. Lucy is selling cookies at a local bake sale. If she sells exactly 3 dozen cookies, the cost of ingredients will equal her earnings. If she sells more than 3 dozen cookies, Mrs. Lucy will make a profit of 25 cents for each cookie sold. a. Write a formula for Mrs. Lucy’s earnings, E, when the number of cookies sold, c, is equal to 36. b. Write a
formula for E when c 36. c. Find the number of cookies Mrs. Lucy sold if she makes a profit of $2.00. CHAPTER SUMMARY Chapter Summary 111 An algebraic expression, such as x 6, is an expression or a phrase that contains one or more variables, such as x. The variable is a placeholder for numbers. To evaluate an expression, replace each variable with a number and follow the order of operations. A term is a number, a variable, or any product or quotient of numbers and variables. In the term 6by, 6 is the numerical coefficient. In the term n3, the base is n, the exponent is 3, and the power is n3. The power n3 means that base n is used as a factor 3 times. An open sentence, which can be an equation or an inequality, contains a variable. When the variable is replaced by numbers from a domain, the numbers that make the open sentence true are the elements of the solution set of the sentence. A formula is a sentence that shows the relationship between two or more variables. VOCABULARY 3-1 Variable • Placeholder • Domain • Replacement set • Algebraic expression 3-3 Term • Factor • Coefficient • Numerical coefficient 3-5 Evaluating an algebraic expression 3-6 Open sentence • Solution set 3-7 Formula REVIEW EXERCISES 1. Explain the difference between an algebraic expression and an open sentence. 2. Explain the difference between 2a2 and (2a)2. In 3–6, use mathematical symbols to translate the verbal phrases into algebraic language. 3. x divided by b 4. 4 less than r 5. q decreased by d 6. 3 more than twice g In 7–14, find the value of each expression. 7. 6ac d when a 10, c 8, and d 5 8. 4b2 when b 2.5 112 Algebraic Expressions and Open Sentences 9. 3b c when b 7 and c 14 10. km 9 when k 15 and m 0.6 11. when a 5, b 3, and c 12 bc a 12. 2a2 2a when a 5 1 4 a 5 1 13. (2a)2 2a when 4 14. a(b c) when a 2.5, b 1.1, and c 8.9 15. Write an algebraic expression for the total number of cents in n nickels
and q quarters. 16. In the term 2xy3 what is the coefficient? 17. In the term 2xy3 what is the exponent of y? 18. In the term 2xy3, what is the base that is used 3 times as a factor? 19. If distance is the product of rate and time, write a formula for distance, d, in terms of rate, r, and time, t. 20. What is the smallest member of the solution set of 19.4 y 29 if the domain is {46.25, 47.9, 48, 48.5, 49, 50, 51.3 }? 21. What is the smallest member of the solution set of 19.4 y 29 if the domain is the set of whole numbers? 22. What is the smallest member of the solution set of 19.4 y 29 if the domain is the set of real numbers? 23. In a baseball game, the winning team scored n runs and the losing team scored 2n 5 runs. a. Describe in words the number of runs that the losing team scored. b. What could have been the score of the game? Is there more than one answer? c. What are the possible values for n? 24. A mail order book club offers books for $8.98 each plus $3.50 for shipping and handling on each order. The cost of Bethany’s order, which totaled less than $20, can be expressed as 8.98b 3.50 20 where b represents the number of books Bethany ordered. a. What could be the domain for this problem? b. What is the solution set for this open sentence? 25. Write two algebraic expressions to represent the cost of sending an express delivery based on the rates given in the chapter opener on page 88, the first if the delivery distance is 10 miles or less, and the second if the delivery distance is more than 10 miles. 26. A list of numbers that follows a pattern begins with the numbers 2, 5, 8, 11,.... Chapter Summary 113 a. Find the next number in the list. b. Write a rule or explain how the next number is determined. c. What is the 25th number in the list? 27. Each of the numbers given below is different from the others, that is, it belongs to a set of numbers to which the others do not belong. Explain why each is different. 3 6 9 35 28. Two oranges
cost as much as five bananas. One orange costs the same as a banana and an apple. How many apples cost the same as three bananas? Exploration STEP 1. Write a three-digit multiple of 11 by multiplying any whole number from 10 to 90 by 11. Add the digits in the hundreds and the ones places. If the sum is greater than or equal to 11, subtract 11. Compare this result to the digit in the tens place. Repeat the procedure for other three-digit multiples of 11. STEP 2. Write a three-digit number that is not a multiple of 11 by adding any counting number less than 11 to a multiple of 11 used in step 1. Add the digits in the hundreds and the ones places. If the sum is greater than or equal to 11, subtract 11. Compare this result to the digit in the tens place. Repeat the procedure with another number. STEP 3. Based on steps 1 and 2, can you suggest a way of determining whether or not a three-digit number is divisible by 11? STEP 4. Write a four-digit multiple of 11 by multiplying any whole number from 91 to 909 by 11. Add the digits in the hundreds and ones places. Add the digits in the thousands and tens places. If one sum is greater than or equal to 11, subtract 11. Compare these results. Repeat the procedure for another four-digit multiple of 11. STEP 5. Write a four-digit number that is not a multiple of 11 by adding any counting number less than 11 to a multiple of 11 used in step 4. Add the digits in the hundreds place and ones place. Add the digits in the thousands place and tens place. If one sum is greater than or equal to 11, subtract 11. Compare these results. Repeat the procedure starting with another number. STEP 6. Based on steps 4 and 5, can you suggest a way of determining whether or not a four-digit number is divisible by 11? STEP 7. Write a rule for determining whether or not any whole number is divis- ible by 11. 114 Algebraic Expressions and Open Sentences CUMULATIVE REVIEW CHAPTERS 1–3 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. Which of the following is the set of negative integers greater than –3? (1) {4, 5, 6, 7,...} (2) {3, 4, 5, 6
,...} (3) {3, 2, 1} (4) {2, 1} 5 3 (3) 1.666666667 2. The exact value of the rational number can be written as (1) 1.6 (2) 1.6 (4) 1.666666666 3. Rounded to the nearest hundredth, (1) 2.23 (2) 2.236 5 is approximately equal to (3) 2.24 (4) 2.240 " 4. Which of the following numbers is rational? (1) p (2) (3) 1.42 2 " (4) 0.4 " 5. Which of the following inequalities is a true statement? (1) 0.026 0.25 0.2 (3) 0.2 0.25 0.026 (2) 0.2 0.026 0.25 (4) 0.026 0.2 0.25 6. The length of a rectangle is given as 30.02 yards. This measure has how many significant digits? (2) 2 (1) 1 7. Which of the following is not a prime? (1) 7 (2) 23 (3) 3 (3) 37 (4) 4 (4) 51 8. The arithmetic expression 8 5(0.2)2 10 is equal to (1) 0.9 (2) 0.78 (3) 7.98 (4) 8.02 9. Which of the following is a correct application of the distributive prop- erty? (1) 4(8 0.2) 4(8) 4(0.2) (2) 6(3 1) 6(1 3) 10. The additive inverse of 7 is (1) –7 (2) 0 Part II (3) 8(5 4) 8(5) 4 (4) 8(5)(4) 8(5) 8(4) (3) 1 7 (4) 7 Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. Cumulative Review 115 11. Of the 80 students questioned about what they had read in the past month, 35 had read nonfiction, 55 had read fiction, and 22 had read neither fiction nor nonfiction.
How many students had read both fiction and nonfiction? 12. What is the largest number that is the product of three different two-digit primes? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. sides of equal measure) is 13. The formula for the area of an equilateral triangle (a triangle with three 3 4 s2 A 5 " triangle if the measure of one side is 12.6 centimeters. Express your answer to the number of significant digits determined by the given data.. Find the area of an equilateral 14. A teacher wrote the sequence 1, 2, 4,... and asked what the next number could be. Three students each gave a different answer and the teacher said that all three answers were correct. a. Adam said 7. Explain what rule Adam used. b. Bette said 8. Explain what rule Bette used. c. Carlos said 5. Explain what rule Carlos used. Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. c 5 1 15. If a 7, b –5, and 3, evaluate the expression a2b c 1 3(b 2 2). Do not use a calculator. Show each step in your calculation. 16. Michelle bought material to make a vest and skirt. She used half of the material to make the skirt and two-thirds of what remained to make the vest. She had yards of material left. 11 4 a. How many yards of material did she buy? b. How many yards of material did she use for the vest? c. How many yards of material did she use for the skirt? CHAPTER 4 CHAPTER TABLE OF CONTENTS 4-1 Solving Equations Using More Than One Operation 4-2 Simplifying Each Side of an Equation 4-3 Solving Equations That Have the Variable in Both Sides 4-4 Using Formulas to Solve Problems 4-5 Solving for a Variable in Terms of Another Variable 4-6 Transforming Formulas 4-7 Properties of Inequalities 4-8 Finding and G
raphing the Solution Set of an Inequality 4-9 Using Inequalities to Solve Problems Chapter Summary Vocabulary Review Exercises Cumulative Review 116 FIRST DEGREE EQUATIONS AND INEQUALITIES IN ONE VARIABLE An equation is an important problem-solving tool. A successful business person must make many decisions about business practices. Some of these decisions involve known facts, but others require the use of information obtained from equations based on expected trends. For example, an equation can be used to represent the following situation. Helga sews hand-made quilts for sale at a local craft shop. She knows that the materials for the last quilt that she made cost $76 and that it required 44 hours of work to complete the quilt. If Helga received $450 for the quilt, how much did she earn for each hour of work, taking into account the cost of the materials? Most of the problem-solving equations for business are complex. Before you can cope with complex equations, you must learn the basic principles involved in solving any equation. 4-1 SOLVING EQUATIONS USING MORE THAN ONE OPERATION Solving Equations Using More Than One Operation 117 Some Terms and Definitions An equation is a sentence that states that two algebraic expressions are equal. For example, x 3 9 is an equation in which x 3 is called the left side, or left member, and 9 is the right side, or right member. An equation may be a true sentence such as 5 2 7, a false sentence such as 6 3 4, or an open sentence such as x 3 9. The number that can replace the variable in an open sentence to make the sentence true is called a root, or a solution, of the equation. For example, 6 is a root of x + 3 9. As discussed in Chapter 3, the replacement set or domain is the set of possible values that can be used in place of the variable in an open sentence. If no replacement set is given, the replacement set is the set of real numbers. The set consisting of all elements of the replacement set that are solutions of the open sentence is called the solution set of the open sentence. For example, if the replacement set is the set of real numbers, the solution set of x 3 9 is {6}. If no element of the replacement set makes the open sentence true, the solution set is the empty or null set, or {}. If every element of the domain satisfies an equation, the
equation is called an identity. Thus, 5 x x (5) is an identity when the domain is the set of real numbers because every element of the domain makes the sentence true. Two equations that have the same solution set are equivalent equations. To solve an equation is to find its solution set. This is usually done by writing simpler equivalent equations. If not every element of the domain makes the sentence true, the equation is called a conditional equation, or simply an equation. Therefore, x 3 9 is a conditional equation. Properties of Equality When two numerical or algebraic expressions are equal, it is reasonable to assume that if we change each in the same way, the resulting expressions will be equal. For example: 5 7 12 (5 7) 3 12 3 (5 7) 8 12 8 2(5 7) 2(12) 5 1 7 3 5 12 3 These examples suggest the following properties of equality: 118 First Degree Equations and Inequalities in One Variable Properties of Equality 1. The addition property of equality. If equals are added to equals, the sums are equal. 2. The subtraction property of equality. If equals are subtracted from equals, the differences are equal. 3. The multiplication property of equality. If equals are multiplied by equals, the products are equal. 4. The division property of equality. If equals are divided by nonzero equals, the quotients are equal. 5. The substitution principle. In a statement of equality, a quantity may be substituted for its equal. To solve an equation, you need to work backward or “undo” what has been done by using inverse operations. To undo the addition of a number, add its opposite. For example, to solve the equation x 7 19, use the addition property of equality. Add the opposite of 7 to both sides. x 1 7 5 19 27 27 x 5 12 The variable x is now alone on one side and it is easy to read the solution, x 12. To solve an equation in which the variable has been multiplied by a number, either divide by that number or multiply by its reciprocal. (Remember multiplying by the reciprocal is the same as dividing by the number.) To solve 1 6x 24, divide both sides by 6 or multiply both sides by. 6 6x 24 6x 6 5 24 6 x 4 or 6x 24 6(6x) 5 1 1 x 4 6(24) To solve x 3 5 5, multiply each side by the reciprocal of which is 3. 1
3 (3)x x 3 5 5 3 5 (3)5 x 15 In the equation 2x 3 15, there are two operations in the left side: multiplication and addition. In forming the left side of the equation, x was first multiplied by 2, and then 3 was added to the product. To solve this equation, we must undo these operations by using the inverse elements in the reverse order. Since the last operation was to add 3, the first step in solving the equation is to add its opposite, 3, to both sides of the equation or subtract 3 from both sides Solving Equations Using More Than One Operation 119 of the equation. Here we are using either the addition or the subtraction property of equality. 2x 1 3 5 15 2x 1 3 1 (23) 5 15 1 (23) 2x 5 12 or 2x 1 3 23 2x 5 5 15 23 12 Now we have a simpler equation that has the same solution set as the original and includes only multiplication by 2. To solve this simpler equation, we multiply both sides of the equation by, the reciprocal of 2, or divide both sides of the equation by 2. Here we can use either the multiplication or the division property of equality. 1 2 2x 5 12 2(2x) 5 1 1 x 5 6 2(12) or 2x 5 12 2x 2 5 12 2 x 5 6 After an equation has been solved, we check the equation, that is, we verify that the solution does in fact make the given equation true by replacing the variable with the solution and performing any computations. Check: 2x 3 15 2(6) 3 15 12 3 15 15 15 ✔ To find the solution of the equation 2x 3 15, we used several properties of the four basic operations and of equality. The solution below shows the mathematical principle that we used in each step. 2x 3 15 Given (2x 3) (3) 15 (3) Addition property of equality 2x [3 (3)] 15 (3) Associative property of addition 2x 0 12 2x 12 1 2(12) 1 2(12) 1 2(2x) 1 x 2(2) D 1x 6 x 6 C Additive inverse property Additive identity property Multiplication property of equality Associative property of multiplication Multiplicative inverse property Multiplicative identity property These steps and properties are necessary to justify the solution of an equation of this form. However, when solving an equation,
we do not need to write each of the steps, as shown in the examples that follow. 120 First Degree Equations and Inequalities in One Variable EXAMPLE 1 Solve and check: 7x 15 71 Solution How to Proceed (1) Write the equation: (2) Add 15, the opposite of 15 to each side: (3) Since multiplication and division are inverse operations, divide each side by 7: (4) Check the solution. Write the solution in place of x and perform the computations: 7x 7x 15 71 15 15 56 7x 56 7 7 x 8 7(8) 1 15 5? 56 1 15 5? 7x 15 71 71 71 71 71 ✔ Answer x 8 Note: The check is based on the substitution principle. EXAMPLE 2 Solution Find the solution set and check: 3 5x 2 6 18 3 5x 2 6 5 218 16 16 3 12 5x 5 5 3(212) 3 3 5x A B x 20 Addition property of equality Multiplication property of equality Check 6 18 3 5x 3 5(220) 2 6 5? 212 2 6 5? 218 218 18 18 ✔ Answer The solution set is {20}. EXAMPLE 3 Solve and check: 7 x 9 Solution METHOD 1. Think of 7 x as 7 (1x). Solving Equations Using More Than One Operation 121 27 71(2x) 5 9 27 2x 5 2 21x 21 5 2 21 x 2 Addition property of equality Division property of equality Check 7 x 9 7 2 (2) 5? 9 7 1 2 5? 9 9 9 ✔ METHOD 2. Add x to both sides of the equation so that the variable has a positive coefficient. How to Proceed (1) Write the equation: (2) Add x to each side of the equation: (3) Add 9 to each side of the equation: The check is the same as for Method 1. Answer {2} or EXERCISES Writing About Mathematics 1. Is it possible for the equation 2x 5 0 to have a solution in the set of positive real num- bers? Explain your answer. 2. Max wants to solve the equation 7x 15 71. He begins by multiplying both sides of the equation by, the reciprocal of the coefficient of x. 1 7 a. Is it possible for Max to solve the equation if he begins in this way? If so, what 1 would be the result of multiplying by and what would be
his next step? 7 b. In this section you learned to solve the equation 7x 15 71 by first adding the opposite of 15, 15, to both sides of the equation. Which method do you think is better? Explain your answer. Developing Skills In 3 and 4, write a complete solution for each equation, listing the property used in each step. 3. 3x 5 35 4. 1 2x21 15 122 First Degree Equations and Inequalities in One Variable 5. 55 6a 7 9. 15 a 3 In 5–32, solve and check each equation. 6. 17 8c 7 10. 11 6d 1 3 4y 5 14. 12 2 13. 8 3x 17. 7.2 21. 4a 0.2 5 18. 22. 4 3t 0.2 a 4 1 9 4m 5 25. 29. 47 4 5t 1 7 1 7 5 14 2 x 26. 0.04c 1.6 0 30. 0.8r 19 20 15. 7. 9 1x 7 11. 8 y = 1 4 5 45 5t 2 y 19. 2 5 1 3 1 5 4x 1 11 27. 15x 14 19 23. 16. 8. 11 15t 16 3a 12. 12 8 23 5m 9d 2 1 30 2 5 171 2 5 2 2 3y 28. 8 18c 1 20. 24. 13 31. 1 3w 1 6 5 22 32. 842 162m 616 Applying Skills 33. The formula F 9 5C132 gives the relationship between the Fahrenheit temperature F and the Celsius temperature C. Solve the equation 59 degrees Celsius when the Fahrenheit temperature is 59°. 9 5C132 to find the temperature in 34. When Kurt orders from a catalog, he pays $3.50 for shipping and handling in addition to the cost of the goods that he purchases. Kurt paid $33.20 when he ordered six pairs of socks. Solve the equation 6x 3.50 33.20 to find x, the price of one pair of socks. 35. When Mattie rents a car for one day, the cost is $29.00 plus $0.20 a mile. On her last trip, Mattie paid $66.40 for the car for one day. Find the number of miles, m, that Mattie drove by solving the equation 29 0.20x 66.40. 36. On his last trip to the post office, Hal
paid $4.30 to mail a package and bought some 39-cent stamps. He paid a total of $13.66. Find s, the number of stamps that he bought, by solving the equation 0.39s 4.30 13.66. 4-2 SIMPLIFYING EACH SIDE OF AN EQUATION An equation is often written in such a way that one or both sides are not in simplest form. Before starting to solve the equation by using additive and multiplicative inverses, you should simplify each side by removing parentheses if necessary and adding like terms. Recall that an algebraic expression that is a number, a variable, or a product or quotient of numbers and variables is called a term. First-degree equations in one variable contain two kinds of terms, terms that are constants and terms that contain the variable to the first power only. Simplifying Each Side of an Equation 123 Like and Unlike Terms Two or more terms that contain the same variable or variables, with corresponding variables having the same exponents, are called like terms or similar terms. For example, the following pairs are like terms. 6k and k 5x2 and 7x2 9ab and 0.4ab 9 2x2y3 and 211 3 x2y3 Two terms are unlike terms when they contain different variables, or the same variable or variables with different exponents. For example, the following pairs are unlike terms. 3x and 4y 5x2 and 5x3 9ab and 0.4a 8 3x3y2 and 4 7x2y3 To add like terms, we use the distributive property of multiplication over addition. 9x 2x (9 2)x 11x 16d 3d (–16 3)d 13d Note that in the above examples, when like terms are added: 1. The sum has the same variable factor as the original terms. 2. The numerical coefficient of the sum is the sum of the numerical coeffi- cients of the terms that were added. The sum of like terms can be expressed as a single term. The sum of unlike terms cannot be expressed as a single term. For example, the sum of 2x and 3 cannot be written as a single term but is written 2x 3. EXAMPLE 1 Solve and check: 2x 3x 4 6 Solution How to Proceed Check (1) Write the equation: (2) Simplify the left side by 2x 3x 4 6 5x
4 6 2x 3x 4 6 26 2(22) 1 3(22) 1 4 5? combining like terms: (3) Add 4, the additive inverse of 4, to each side: (4) Multiply by, the 1 5 multiplicative inverse of 5: (5) Simplify each side. Answer 2 24 2 6 1 4 5? 26 6 6 ✔ 4 4 10 5x 5(5x) 5 1 1 5(210) x 2 124 First Degree Equations and Inequalities in One Variable Note: When solving equations, remember to check the answer in the original equation and not in the simplified one. The algebraic expression that is on one side of an equation may contain parentheses. Use the distributive property to remove the parentheses solving the equation. The following examples illustrate how the distributive and associative properties are used to do this. EXAMPLE 2 Solve and check: 27x 3(x 6) 6 Solution Since 3(x 6) means that (x 6) is to be multiplied by 3, we will use the distributive property to remove parentheses and then combine like terms. Note that for this solution, in the first three steps the left side is being simplified. These steps apply only to the left side and only change the form but not the numerical value. The next two steps undo the operations of addition and multiplication that make up the expression 24x 18. Since adding 18 and dividing by 24 will change the value of the left side, the right side must be changed in the same way to retain the equality. How to Proceed (1) Write the equation: (2) Use the distributive property: (3) Combine like terms: (4) Use the addition property of equality. Add 18, the additive inverse of 18, to each side: (5) Use the division property of equality. Divide each side by 24: (6) Simplify each side: Check (1) Write the equation: (2) Replace x by 21 2 (3) Perform the indicated computation: Answer x 21 2 27x 3(x 6) 6 27x 3x 18 6 24x 18 6 18 18 12 24x 24x 24 5 212 24 21 x 2 27 21 2 A 27 A B 21 2 3 B 21 2 27x 3(x 6) 6 5? 6 2 2 6 B A 261 5? 6 2 3 2 B A 2 1 183 227 2 5? 6 2 1 39 2 5? 6 227 12 2
5? 6 6 6 ✔ Simplifying Each Side of an Equation 125 Representing Two Numbers with the Same Variable Problems often involve finding two or more different numbers. It is useful to express these numbers in terms of the same variable. For example, if you know the sum of two numbers, you can express the second in terms of the sum and the first number. • If the sum of two numbers is 12 and one of the numbers is 5, then the other number is 12 5 or 7. • If the sum of two numbers is 12 and one of the numbers is 9, then the other number is 12 9 or 3. • If the sum of two numbers is 12 and one of the numbers is x, then the other number is 12 x. A problem can often be solved algebraically in more than one way by writing and solving different equations, as shown in the example that follows. The methods used to obtain the solution are different, but both use the facts stated in the problem and arrive at the same solution. EXAMPLE 3 The sum of two numbers is 43. The larger number minus the smaller number is 5. Find the numbers. Solution This problem states two facts: FACT 1 The sum of the numbers is 43. FACT 2 larger number is 5 more than the smaller. The larger number minus the smaller number is 5. In other words, the (1) Represent each number in terms of the same variable using Fact 1: the sum of the numbers is 43. Let x the larger number. Then, 43 x the smaller number. (2) Write an equation using Fact 2: The larger number minus the smaller number is 5. \|_________________\| ↓ ↓ ↓ 5 x \|__________________\| ↓ (43 x) ↓ 126 First Degree Equations and Inequalities in One Variable (3) Solve the equation. (a) Write the equation: (b) To subtract (43 x), add its opposite: (c) Combine like terms: (d) Add the opposite of 43 to each side: (e) Divide each side by 2: (4) Find the numbers. The larger number x 24. The smaller number 43 x 43 24 19. x (43 x) 5 x (43 x) 5 2x 43 5 43 43 48 2x 2 5 48 2x 2 x 24 Check A word problem is checked by comparing the proposed solution with the facts stated in the original wording of the problem. Substituting numbers in the equation is not
sufficient since the equation formed may not be correct. The sum of the numbers is 43: 24 19 43. The larger number minus the smaller number is 5: 24 19 5. Alternate Solution Reverse the way in which the facts are used. (1) Represent each number in terms of the same variable using Fact 2: the larger number is 5 more than the smaller. Let x the smaller number. Then, x 5 the larger number. (2) Write an equation using the first fact. The sum of the numbers is 43. \|______________________\| ↓ ↓ ↓ 43 x (x 5) (3) Solve the equation. (a) Write the equation: (b) Combine like terms: (c) Add the opposite of 5 to each side: (d) Divide each side by 2: (4) Find the numbers. The smaller number x 19. The larger number x 5 19 5 24. (5) Check. (See the first solution.) x (x 5) 43 2x 5 43 5 5 38 2 5 38 2x 2 x 19 2x Answer The numbers are 24 and 19. Simplifying Each Side of an Equation 127 EXERCISES Writing About Mathematics 1. Two students are each solving a problem that states that the difference between two num- bers is 12. Irene represents one number by x and the other number by x 12. Henry represents one number by x and the other number by x 12. Explain why both students are correct. 2. A problem states that the sum of two numbers is 27. The numbers can be represented by x and 27 x. Is it possible to determine which is the larger number and which is the smaller number? Explain your answer. Developing Skills In 3–28, solve and check each equation. 3. x (x 6) 20 5. (15x 7) 12 4 7. x (4x 32) 12 9. 5(x 2) 20 11. 8(2c 1) 56 13. 30 2(10 y) 15. 25 2(t 5) 19 17. 55 4 3(m 2) 19. 3(2b 1) 7 50 21. 7r (6r 5) 7 23. 5m 2(m 5) 17 25. 3(a 5) 2(2a 1) 0 4. x (12 x) 38 6. (14 3c) 7c 94 8. 7x (4x 39) 0 10. 3(y
9) 30 12. 6(3c 1) 42 14. 4(c 1) 32 16. 18 6x 4(2x 3) 18. 5(x 3) 30 10 20. 5(3c 2) 8 43 22. 8b 4(b 2) 24 24. 28y 6(3y 5) 40 26. 0.04(2r 1) 0.03(2r 5) 0.29 27. 0.3a (0.2a 0.5) 0.2(a 2) 1.3 28. 3 4(8 1 4x) 2 1 3(6x 1 3) 5 9 Applying Skills In 29–33, write and solve an equation for each problem. Follow these steps: a. List two facts in the problem. b. Choose a variable to represent one of the numbers to be determined. c. Use one of the facts to write any other unknown numbers in terms of the chosen variable. d. Use the second fact to write an equation. e. Solve the equation. 128 First Degree Equations and Inequalities in One Variable f. Answer the question. g. Check your answer using the words of the problem. 29. Sandi bought 6 yards of material. She wants to cut it into two pieces so that the difference between the lengths of the two pieces will be 1.5 yards. What should be the length of each piece? 30. The Tigers won eight games more than they lost, and there were no ties. If the Tigers played 78 games, how many games did they lose? 31. This month Erica saved $20 more than last month. For the two months, she saved a total of $70. How much did she save each month? 32. On a bus tour, there are 100 passengers on three buses. Two of the buses each carry four fewer passengers than the third bus. How many passengers are on each bus? 33. For a football game, of the seats in the stadium were filled. There were 31,000 empty seats 4 5 at the game. What is the stadium’s seating capacity? 4-3 SOLVING EQUATIONS THAT HAVE THE VARIABLE IN BOTH SIDES A variable represents a number. As you know, any number may be added to both sides of an equation without changing the solution set. Therefore, the same variable (or the same multiple of the same variable) may be added to or subtracted from both sides of
an equation without changing the solution set. For instance, to solve 8x 30 5x, write an equivalent equation that has only a constant in the right side. To do this, eliminate 5x from the right side by adding its opposite, 5x, to each side of the equation. METHOD 1 8x 30 5x 5x 5x 3x 30 x 10 METHOD 2 8x 30 5x 8x (5x) 30 5x (5x) 3x 30 x 10 Check 8x 30 5x 5? 30 1 5(10) 5? 30 1 50 8(10) 80 80 80 ✔ Answer: x 10 To solve an equation that has the variable in both sides, transform it into an equivalent equation in which the variable appears in only one side. Then, solve the equation. Solving Equations That Have the Variable in Both Sides 129 EXAMPLE 1 Solve and check: 7x 63 2x Solution How to Proceed (1) Write the equation: (2) Add 2x to each side of the equation: 7x 63 2x 2x 2x 9x 63 Check 7x 63 2x 5? 7(7) 5? 49 49 49 ✔ 63 2 2(7) 63 2 14 (3) Divide each side of the equation by 9: (4) Simplify each side: 9x 9 5 63 9 x 7 Answer x 7 EXAMPLE 2 Solution To solve an equation that has both a variable and a constant in both sides, first write an equivalent equation with only a variable term on one side. Then solve the simplified equation. The following example shows how this can be done. Solve and check: 3y 7 5y 3 METHOD 1 3y 7 5y 3 5y 5y 2y 7 7 2y 3 7 10 3y METHOD 2 3y 7 5y 3 3y 7 2y 3 3 3 10 2y Check 3y 7 5y 3 3(5) 7 15 7 5? 5(5) 2 3 5? 25 2 3 22 22 ✔ 22y 22 210 22 y 5 2y 10 2 5 2 y 5 Answer y 5 A graphing calculator can be used to check an equation. The calculator can determine whether a given statement of equality or inequality is true or false. If the statement is true, the calculator will display 1; if the statement is false, the calculator will display 0. The symbols for equality and inequality are found in the menu. TEST 130 First Degree
Equations and Inequalities in One Variable To check that y 5 is the solution to the equation 3y 7 5y 3, first store 5 as the value of y. then enter the equation to be checked. ENTER: 5 STO ALPHA Y ENTER 3 ALPHA Y 7 2nd TEST ENTER 5 ALPHA Y 3 ENTER DISPLAY The calculator displays 1 which indicates that the statement of equality is true for the value that has been stored for y. 5 1 EXAMPLE 3 The larger of two numbers is 4 times the smaller. If the larger number exceeds the smaller number by 15, find the numbers. Note: When s represents the smaller number and 4s represents the larger number, “the larger number exceeds the smaller by 15” has the following meanings. Use any one of them. 1. The larger equals 15 more than the smaller, written as 4s = 15 s. 2. The larger decreased by 15 equals the smaller, written as 4s 15 s. 3. The larger decreased by the smaller is 15, written as 4s s 15. Solution Let s = the smaller number. Then 4s = the larger number. The larger is 15 more than the smaller. \|_________\| \|__________\| \|__________\| ↓ 4s ↓ ↓ s ↓ ↓ 15 4s 15 s 4s 15 s s s 3s 15 s 5 4s 4(5) 20 Check The larger number, 20, is 4 times the smaller number, 5. The larger number, 20, exceeds the smaller number, 5, by 15. Answer The larger number is 20; the smaller number is 5. Solving Equations That Have the Variable in Both Sides 131 EXAMPLE 4 In his will, Uncle Clarence left $5,000 to his two nieces. Emma’s share is to be $500 more than Clara’s. How much should each niece receive? Solution (1) Use the fact that the sum of the two shares is $5,000 to express each share in terms of a variable. Let x Clara’s share. Then 5,000 x Emma’s share. (2) Use the fact that Emma’s share is $500 more than Clara’s share to write an equation. Emma’s share is $500 more than Clara’s share. \|____________\| \|_____________\| ↓ ↓ x ↓ ↓ 5,000 x 500 \|________
__\| ↓ (3) Solve the equation to find Clara’s share. x 5,000 x 500 x x 500 2x 500 2x 5,000 500 4,500 2,250 x Clara’s share is x $2,250. (4) Find Emma’s share: 5,000 x 5,000 2,250 $2,750. Alternate Solution (1) Use the fact that Emma’s share is $500 more than Clara’s share to express each share in terms of a variable. Let x Clara’s share. Then x 500 Emma’s share. (2) Use the fact that the sum of the two shares is $5,000 to write an equation. Clara’s share plus Emma’s share is $5,000. \|____________\| ↓ x (x 500) 5,000 \|______________\| ↓ ↓ ↓ ↓ (3) Solve the equation to find Clara’s share x (x 500) 5,000 2x 500 5,000 500 500 4,500 2,250 2x x Clara’s share is x $2,250. 132 First Degree Equations and Inequalities in One Variable (4) Find Emma’s share: x 500 2250 500 $2,750. Check $2,750 is $500 more than $2,250, and $2,750 $2,250 $5,000. Answer Clara’s share is $2,250, and Emma’s share is $2,750. EXERCISES Writing About Mathematics 1. Milus said that he finds it easier to work with integers than with fractions. Therefore, in 2a 1 3, he began by multiplying both sides of the 3 4a 2 7 5 1 order to solve the equation equation by 4. 4 A 3 1 2a 1 3 5 4 4a 2 7 3a 28 2a 12 A B B Do you agree with Milus that this is a correct way of obtaining the solution? If so, what mathematical principle is Milus using? 2. Katie said that Example 3 could be solved by letting equal the smaller number and x equal the larger number. Is Katie correct? If so, what equation would she write to solve the problem? x 4 Developing Skills In 3–36, solve and check each equation. 3. 7x 10 2x 6. y 4y 30
9. 0.8m 0.2m 24 23 4x 1 24 5 3x 12. 15. x 9x 72 18. 7r 10 3r 50 21. x 4 9x 4 24. c 20 55 4c 27. 3m (m 1) 6m 1 3t 2 11 5 4(16 2 t) 2 1 2 3t 30. 32. 8c 1 7c 2(7 c) 34. 4(3x 5) 5x 2( x 15) 36. 5 3(a 6) a 1 8a 4. 9x 44 2x 7. 2d 36 5d 10. 8y 90 2y 13. 5a 40 3a 16. 0.5m 30 1.1m 19. 4y 20 5y 9 22. 9x 3 2x 46 25. 2d 36 3d 54 28. x 3(1 x) 47 x 4y 2 8 5. 5c 28 c 4y 5 11 21 8. 11. 2.3x 36 0.3x 14. 5c 2c 81 4c 5 93 41 4c 1 44 17. 20. 7x 8 6x 1 23. y 30 12y 14 26. 7y 5 9y 29 29. 3b 8 10 (4 8b) 31. 18 4n 8 2(1 8n) 33. 8a 3(5 2a) 85 3a 35. 3m 5m 12 7m 88 5 Solving Equations That Have the Variable in Both Sides 133 In 37–42, a. write an equation to represent each problem, and b. solve the equation to find each number. 37. Eight times a number equals 35 more than the number. Find the number. 38. Six times a number equals 3 times the number, increased by 24. Find the number. 39. If 3 times a number is increased by 22, the result is 14 less than 7 times the number. Find the number. 40. The greater of two numbers is 1 more than twice the smaller. Three times the greater exceeds 5 times the smaller by 10. Find the numbers. 41. The second of three numbers is 6 more than the first. The third number is twice the first. The sum of the three numbers is 26. Find the three numbers. 42. The second of three numbers is 1 less than the first. The third number is 5 less than the sec- ond. If the first number is twice as large
as the third, find the three numbers. Applying Skills In 43–50, use an algebraic solution to solve each problem. 43. It took the Gibbons family 2 days to travel 925 miles to their vacation home. They traveled 75 miles more on the first day than on the second. How many miles did they travel each day? 44. During the first 6 month of last year, the interest on an investment was $130 less than dur- ing the second 6 months. The total interest for the year was $1,450. What was the interest for each 6-month period? 45. Gemma has 7 more five-dollar bills than ten-dollar bills. The value of the five-dollar bills equals the value of the ten-dollar bills. How many five-dollar bills and ten-dollar bills does she have? 46. Leonard wants to save $100 in the next 2 months. He knows that in the second month he will be able to save $20 more than during the first month. How much should he save each month? 47. The ABC Company charges $75 a day plus $0.05 a mile to rent a car. How many miles did Mrs. Kiley drive if she paid $92.40 to rent a car for one day? 48. Kesha drove from Buffalo to Syracuse at an average rate of 48 miles per hour. On the return trip along the same road she was able to travel at an average rate of 60 miles per hour. The trip from Buffalo to Syracuse took one-half hour longer than the return trip. How long did the return trip take? 49. Carrie and Crystal live at equal distances from school. Carries walks to school at an average rate of 3 miles per hour and Crystal rides her bicycle at an average rate of 9 miles per hour. It takes Carrie 20 minutes longer than Crystal to get to school. How far from school do Crystal and Carrie live? 50. Emmanuel and Anthony contributed equal amounts to the purchase of a gift for a friend. Emmanuel contributed his share in five-dollar bills and Anthony gave his share in onedollar bills. Anthony needed 12 more bills than Emmanuel. How much did each contribute toward the gift? 134 First Degree Equations and Inequalities in One Variable 4-4 USING FORMULAS TO SOLVE PROBLEMS To solve for the subject of a formula, substitute the known values in the formula and perform the required computation. For example, to find the area of a triangle when b 4.
70 centimeters and h 3.20 centimeters, substitute the given values in the formula for the area of a triangle: 1 2bh 1 2(4.70 cm) A 7.52 cm2 (3.20 cm) A is the subject of the formula. Now that you can solve equations, you will be able to find the value of any variable in a formula when the values of the other variables are known. To do this: 1. Write the formula. 2. Substitute the given values in the formula. 3. Solve the resulting equation. The values assigned to the variables in a formula often have a unit of measure. It is convenient to solve the equation without writing the unit of measure, but the answer should always be given in terms of the correct unit of measure. EXAMPLE 1 The perimeter of a rectangle is 48 centimeters. If the length of the rectangle is 16 centimeters, find the width to the nearest centimeter. Solution You know that the perimeter of a geometric figure is the sum of the lengths of all of its sides. When solving a perimeter problem, it is helpful to draw and label a figure to model the region. Use the formula P 2l 2w. P 2l 2w 48 2(16) 2w 48 32 2w 32 32 16 16 2w 8 w 2w Answer 8 centimeters Check P 2l 2w 2(16) 1 2(8) 48 5? 5? 48 48 48 ✔ 32 1 16 16 cm w w 16 cm Using Formulas to Solve Problems 135 EXAMPLE 2 A garden is in the shape of an isosceles triangle, a triangle that has two sides of equal measure. The length of the third side of the triangle is 2 feet greater than the length of each of the equal sides. If the perimeter of the garden is 86 feet, find the length of each side of the garden. Solution Let x the length of each of the two equal sides. Then, x 2 the length of the third side. The perimeter is the sum of the lengths of the sides. \|__________________________________\| \|_____________\| ↓ ↓ x x (x 2) 86 ↓ x x x + 2 86 3x 2 84 3x 28 x The length of each of the equal sides x 28. The length of the third side x 2 28 2 30. Check Perimeter 28 28 30 86 ✔ Answer The length of each of the equal sides is 28 feet. The length of the third side (the base) is 30
feet. EXAMPLE 3 The perimeter of a rectangle is 52 feet. The length is 2 feet more than 5 times the width. Find the dimensions of the rectangle. Solution Use the formula for the perimeter of a rectangle, P 2l 2w, to solve this problem. Let w the width, in feet, of the rectangle. Then 5w 2 the length, in feet, of the rectangle. P 2l 2w 52 2(5w 2) 2w 52 10w 4 2w 52 12w 4 48 12w 4 w 5? 5? 44 1 8 52 52 52 ✔ 2(22) 1 2(4) 52 Check The length, 22, is 2 more than 5 times the width, 4. ✔ P 2l 2w Answer The width is 4 feet; the length is 5(w) 2 5(4) 2 22 feet. 136 First Degree Equations and Inequalities in One Variable EXAMPLE 4 Sabrina drove from her home to her mother’s home which is 150 miles away. For the first half hour, she drove on local roads. For the next two hours she drove on an interstate highway and increased her average speed by 15 miles per hour. Find Sabrina’s average speed on the local roads and on the interstate highway. Solution List the facts stated by this problem: FACT 1 Sabrina drove on local roads for hour or 0.5 hour. 1 2 FACT 2 Sabrina drove on the interstate highway for 2 hours. FACT 3 Sabrina’s rate or speed on the interstate highway was 15 mph more than her rate on local roads. This problem involves rate, time, and distance. Use the distance formula, d rt, where r is the rate, or speed, in miles per hour, t is time in hours, and d is distance in miles. (1) Represent Sabrina’s speed for each part of the trip in terms of r. Let r Sabrina’s speed on the local roads. Then r 15 Sabrina’s speed on the interstate highway. (2) Organize the facts in a table, using the distance formula. Rate Time Distance Local Roads Interstate highway r r 15 0.5 2 0.5r 2(r 15) (3) Write an equation. The distance on the local roads plus the distance on the highway is 150 miles. \|________\| \|_____________________________\| ↓ ↓ 150 \|____________________________\| ↓ 2(r 15

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