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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

Give an example of each of the following: (a) A natural number (b) An integer that is not a natural number (c) A rational number that is not an integer (d) An irrational numberComplete each statement and name the property of real numbers you have used. (a) ab = __________; __________ Property (b) a + (b + c) = __________; __________ Property (c) a(b + c) = __________; __________ PropertyExpress the set of real numbers between but not including 2 and 7 as follows. (a) In set-builder notation: ____________________ (b) In interval notation: ____________________The symbol | x | stands for the __________of the number x. If x is not 0, then the sign of | x | is always __________.The distance between a and b on the real line is d(a, b) = __________. So the distance between 5 and 2 is __________.Yes or No? If No, give a reason. Assume that a and b are nonzero real numbers. 6. (a) Is the sum of two rational numbers always a rational number? (b) Is the sum of two irrational numbers always an irrational number?Yes or No? If No, give a reason. Assume that a and b are nonzero real numbers. 7. (a) Is a b equal to b a? (b) Is 2(a 5) equal to 2a 10?Yes or No? If No, give a reason. Assume that a and b are nonzero real numbers. 8. (a) Is the distance between any two different real numbers always positive? (b) Is the distance between a and b the same as the distance between b and a?Real Numbers List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers 9. {1.5,0,52,7,2.71,,3.14,100,8}Real Numbers List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers 10. {1.3,1.3333,5,5.34,500,123,16,246579,205}Properties of Real Numbers State the property of real numbers being used. 11. 3 + 7 = 7 + 312EProperties of Real Numbers State the property of real numbers being used. 13. (x + 2y) + 3z = x + (2y + 3z)Properties of Real Numbers State the property of real numbers being used. 14. 2(A + B) = 2A + 2BProperties of Real Numbers State the property of real numbers being used. 15. (5x + 1)3 = 15x + 3Properties of Real Numbers State the property of real numbers being used. 16. (x + a)(x + b) = (x + a)x + (x + a)bProperties of Real Numbers State the property of real numbers being used. 17. 2x(3 + y) = (3 + y)2xProperties of Real Numbers State the property of real numbers being used. 18. 7(a + b + c) = 7(a + b) + 7cProperties of Real Numbers Rewrite the expression using the given property of real numbers. 19. Commutative Property of Addition, x + 3 = __________20EProperties of Real Numbers Rewrite the expression using the given property of real numbers. 21. Distributive Property, 4(A + B) = __________Properties of Real Numbers Rewrite the expression using the given property of real numbers. 22. Distributive Property, 5x + 5y = __________23E24EProperties of Real Numbers Use properties of real numbers to write the expression without parentheses. 25. 4(2m)Properties of Real Numbers Use properties of real numbers to write the expression without parentheses. 26. 43(6y)Properties of Real Numbers Use properties of real numbers to write the expression without parentheses. 27. 52(2x4y)28EArithmetic Operations Perform the indicated operations. 29. (a) 310+415 (b) 14+15Arithmetic Operations Perform the indicated operations. 30. (a) 2335 (b) 1+581631EArithmetic Operations Perform the indicated operations. 32. (a) 223322 (b) 25+12110+315Inequalities Place the correct symbol (, , or =) in the space. 33. (a) 372 (b) 372 (c) 3.57234EInequalities State whether each inequality is true or false. 35. (a) 3 4 (b) 3 4Inequalities State whether each inequality is true or false. 36. (a) 31.7325 (b) 1.7325337E38E(a) x is positive. (b) t is less than 4. (c) a is greater than or equal to . (d) x is less than 13 and is greater than 5. (e) The distance from p to 3 is at most 5.40ESets Find the indicated set if A=1,2,3,4,5,6,7B=2,4,6,8C=7,8,9,10 41. (a) A B (b) A B42E43ESets Find the indicated set if A=1,2,3,4,5,6,7B=2,4,6,8C=7,8,9,10 44. (a) A B C (b) A B CSets Find the indicated set if A=xx2B=xx4C=x1x5 45. (a) B C (b) B C46EIntervals Express the interval in terms of inequalities, and then graph the interval. 47. (3, 0)Intervals Express the interval in terms of inequalities, and then graph the interval. 48. (2, 8]Intervals Express the interval in terms of inequalities, and then graph the interval. 49. [2, 8)50EIntervals Express the interval in terms of inequalities, and then graph the interval. 51. [2, )Intervals Express the interval in terms of inequalities, and then graph the interval. 52. (, 1)Intervals Express the inequality in interval notation, and then graph the corresponding interval. 53. x 154EIntervals Express the inequality in interval notation, and then graph the corresponding interval. 55. 2 x 156E57E58EIntervals Express each set in interval notation. 59. (a) (b)Intervals Express each set in interval notation. 60. (a) (b)Intervals Graph the set. 61. (2, 0) (1, 1)Intervals Graph the set. 62. (2, 0) (1, 1)Intervals Graph the set. 63. [4, 6] [0, 8)64EIntervals Graph the set. 65. (, 4) (4, )Intervals Graph the set. 66. (, 6] (2, 10)67E68E69E70EAbsolute Value Evaluate each expression. 71. (a) | ( 2) 6| (b) (13)(15)72E73E74EDistance Find the distance between the given numbers. 75. (a) 2 and 17 (b) 3 and 21 (c) 118and310(a) 715and121 (b) 38 and 57 (c) 2.6 and 1.8Repeating Decimal Express each repeating decimal as a fraction. (See the margin note on page 3.) 77. (a) 0.7 (b) 0.28 (c) 0.5778E79E80ESimplifying Absolute Value Express the quantity without using absolute value. 81. |a b|, where a b82ESigns of Numbers Let a, b, and c be real numbers such that a 0, b 0, and c 0. Find the sign of each expression. 83. (a) a (b) bc (c) a b (d) ab + ac84EArea of a Garden Marys backyard vegetable garden measures 20 ft by 30 ft, so its area is 20 30 = 600 ft2. She decides to make it longer, as shown in the figure, so that the area increases to A = 20(30 + x). Which property of real numbers tells us that the new area can also be written A = 600 + 20x?Temperature Variation The bar graph shows the daily high temperatures for Omak, Washington, and Geneseo, New York, during a certain week in June. Let TO represent the temperature in Omak and TG the temperature in Geneseo. Calculate TO TG and |TO TG| for each day shown. Which of these two values gives more information?Mailing a Package The post office will accept only packages for which the length plus the girth (distance around) is no more than 108 in. Thus for the package in the figure, we must have L+2(x+y)108 (a) Will the post office accept a package that is 6 in. wide, 8 in. deep, and 5 ft long? What about a package that measures 2 ft by 2 ft by 4 ft? (b) What is the greatest acceptable length for a package that has a square base measuring 9 in. by 9 in.?DISCUSS: Sums and Products of Rational and Irrational Numbers Explain why the sum, the difference, and the product of two rational numbers are rational numbers. Is the product of two irrational numbers necessarily irrational? What about the sum?DISCOVER PROVE: Combining Rational and Irrational Numbers Is 12+2 rational or irrational? Is 122rational or irrational? Experiment with sums and products of other rational and irrational numbers. Prove the following. (a) The sum of a rational number r and an irrational number t is irrational. (b) The product of a rational number r and an irrational number t is irrational. [Hint: For part (a), suppose that r + t is a rational number q, that is, r + t = q. Show that this leads to a contradiction. Use similar reasoning for part (b).]DISCOVER: Limiting Behavior of Reciprocals Complete the tables. What happens to the size of the fraction 1/x as x gets large? As x gets small? x 1/x 1 2 10 100 1000 x 1/x 1.0 0.5 0.1 0.01 0.001DISCOVER: Locating Irrational Numbers on the Real Line Using the figures below, explain how to locate the point 2 on a number line. Can you locate 5 by a similar method? How can the circle shown in the figure help us to locate on a number line? List some other irrational numbers that you can locate on a number line.PROVE: Maximum and Minimum Formulas Let max(a, b) denote the maximum and min(a, b) denote the minimum of the real numbers a and b. For example, max(2,5) = 5 and min(1, 2) = 2. (a) Prove that max(a,b)=a+b+|ab|2 (b) Prove that min(a,b)=a+b|ab|2WRITE: Real Numbers in the Real World Write a paragraph describing different real-world situations in which you would use natural numbers, integers, rational numbers, and irrational numbers. Give examples for each type of situation.DISCUSS: Commutative and Noncommutative Operations We have learned that addition and multiplication are both commutative operations. (a) Is subtraction commutative? (b) Is division of nonzero real numbers commutative? (c) Are the actions of putting on your socks and putting on your shoes commutative ? (d) Are the actions of putting on your hat and putting on your coat commutative? (e) Are the actions of washing laundry and drying it commutative?PROVE: Triangle Inequality We prove Property 5 of absolute values, the Triangle Inequality: |x+y||x|+|y| (a) Verify that the Triangle Inequality holds for x = 2 and y = 3, for x = 2 and y = 3, and for x = 2 and y = 3. (b) Prove that the Triangle Inequality is true for all real numbers x and y. [Hint: Take cases.](a) Using exponential notation, we can write the product 5 5 5 5 5 5 as __________. (b) In the expression 34 the number 3 is called the __________, and the number 4 is called the __________.(a) When we multiply two powers with the same base, we __________ the exponents. So 34 35 = __________. (b) When we divide two powers with the same base, we __________ the exponents. So 3532=.3EExplain what 43/2 means, then calculate 43/2 in two different ways: (41/2)=or(43)=Explain how we rationalize a denominator, then complete the following steps to rationalize 13: 13=13=Find the missing power in the following calculation: 51/35=5.Yes or No? If No, give a reason. 7. (a) Is the expression (23)2 equal to 34? (b) Is there a difference between (5)4 and 54?8E9E10E11E12E13E14E15E16ERadicals and Exponents Evaluate each expression. 17. (a) 26 (b) (2)6 (c) (15)2(3)318E19E20E21E22E23E24ERadicals and Exponents Evaluate each expression. 25. (a) 315 (b) 483 (c) 24318326ERadicals and Exponents Evaluate each expression. 27. (a) 1323 (b) 23323 (c) 144164428E29E30E31E32EExponents Simplify each expression, and eliminate any negative exponents. 33. (a) a9a2a (b) (a2a4)3 (c) (x2)3(5x6)34E35E36E37E38E39EExponents Simplify each expression, and eliminate any negative exponents. 40. (a) (x4z24y5)(2x3y2z3)2 (b) (rs2)3(r3s2)2Exponents Simplify each expression, and eliminate any negative exponents. 41. (a) 8a3b42a5b5 (b) (y5x2)342E43E44E45E46E47E48E49E50E51E52E53ERadical Expressions Simplify the expression. 54. (a) 27a2+63a (b) 75t+100t255E56ERational Exponents Evaluate each expression. 57. (a) 322/5 (b) (49)1/2 (c) (1681)3/4Rational Exponents Evaluate each expression. 58. (a) 1252/3 (b) (2564)3/2 (c) 274/359E60E61E62E63E64ERational Exponents Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. 65. (a) (8a6b3/2)2/3 (b) (4a6b8)3/266E67E68E69E70E71E72ERadicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. 73. (a) y56y23 (b) (5x3)(2x4)74E75E76E77E78ERationalize Put each fractional expression into standard form by rationalizing the denominator. 79. (a) 16 (b) 32 (c) 92480E81E82E83E84E85E86EScientific Notation Write the number indicated in each statement in scientific notation. 87. (a) A light-year, the distance that light travels in one year, is about 5,900,000,000,000 mi. (b) The diameter of an electron is about 0.0000000000004 cm. (c) A drop of water contains more than 33 billion billion molecules.88E89E90E91E92E93EScientific Notation Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. 94. (3.542106)9(5.05104)1295E96E97E98EVolume of the Oceans The average ocean depth is 3.7 103 m, and the area of the oceans is 3.6 1014 m2. What is the total volume of the ocean in liters? (One cubic meter contains 1000 liters.)National Debt As of July 2013, the population of the United States was 3.164 108, and the national debt was 1.674 1013 dollars. How much was each persons share of the debt? [Source: U.S. Census Bureau and U.S. Department of Treasury]Number of Molecules A sealed room in a hospital, measuring 5 m wide, 10 m long, and 3 m high, is filled with pure oxygen. One cubic meter contains 1000 L, and 22.4 L of any gas contains 6.02 1023 molecules (Avogadros number). How many molecules of oxygen are there in the room?102ESpeed of a Skidding Car Police use the formula s=30fd to estimate the speed s (in mi/h) at which a car is traveling if it skids d feet after the brakes are applied suddenly. The number f is the coefficient of friction of the road, which is a measure of the slipperiness of the road. The table gives some typical estimates for f. (a) If a car skids 65 ft on wet concrete, how fast was it moving when the brakes were applied? (b) If a car is traveling at 50 mi/h, how far will it skid on wet tar?Distance from the Earth to the Sun It follows from Keplers Third Law of planetary motion that the average distance from a planet to the sun (in meters) is d=(GM42)1/3T2/3 where M = 1.99 1030 kg is the mass of the sun, G = 6.67 1011 N m2/kg2 is the gravitational constant, and T is the period of the planets orbit (in seconds). Use the fact that the period of the earths orbit is about 365.25 days to find the distance from the earth to the sun.105E106E107E108E109EConsider the polynomial 2x5 + 6x4 + 4x3. (a) How many terms does this polynomial have? __________ List the terms: _______________. (b) What factor is common to each term? __________ Factor the polynomial: 2x5 + 6x4 + 4x3 = __________.To factor the trinomial x2 + 7x + 10, we look for two integers whose product is __________and whose sum is __________. These integers are __________and __________, so the trinomial factors as _______________3EThe Special Product Formula for the square of a sum is (A + B)2 = _______________. So (2x + 3)2 = _______________.The Special Product Formula for the product of the sum and difference of terms is (A + B)(A B) = _______________. So (5 + x)(5 x) = _______________.6E7E8EPolynomials Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial; then list its terms and state its degree. 9. 5x3 + 6 ___ _____ ___10E11E12EPolynomials Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial; then list its terms and state its degree. 13. x x2 + x3 x4 ___ _____ ___14E15EPolynomials Find the sum, difference, or product. 16. (5 3x) + (2x 8)Polynomials Find the sum, difference, or product. 17. (2x2 3x + 1) + (3x2 + 5x 4)18E19E20E21E22E23E24EUsing FOIL Multiply the algebraic expressions using the FOIL method and simplify. 25. (3t 2)(7t 4)26E27E28E29E30E31E32EUsing Special Product Formulas Multiply the algebraic expressions using a Special Product Formula and simplify. 33. (2u + v)234E35E36E37E38E39EUsing Special Product Formulas Multiply the algebraic expressions using a Special Product Formula and simplify. 40. (2y + 5)(2y 5)Using Special Product Formulas Multiply the algebraic expressions using a Special Product Formula and simplify. 41. (x+2)(x2)42EUsing Special Product Formulas Multiply the algebraic expressions using a Special Product Formula and simplify. 43. (y + 2)344E45E46E47EMultiplying Algebraic Expressions Perform the indicated operations and simplify. 48. (x + 1)(2x2 x + 1)Multiplying Algebraic Expressions Perform the indicated operations and simplify. 49. (2x 5)(x2 x + 1)50E51E52E53E54E55EMultiplying Algebraic Expressions Perform the indicated operations and simplify. 56. (x1/2 + y1/2)(x1/2 y1/2)Multiplying Algebraic Expressions Perform the indicated operations and simplify. 57. (ab)(a+b)58E59E60E61E62E63E64EFactoring Common Factor Factor out the common factor. 65. y(y 6) + 9(y 6)66E67E68E69E70E71E72EFactoring Trinomials Factor the trinomial. 73. 3x2 16x + 574E75E76E77E78E79E80EUsing Special Factoring Formulas Use a Special Factoring Formula to factor the expression. 81. 8s3 125t382E83E84E85E86E87E88EFactoring by Grouping Factor the expression by grouping terms. 89. x3 + x2 + x + 190E91E92E93E94E95E96E