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333371_0P01.qxp 12/27/06 9:28 AM 2 Chapter P Page 2 Prerequisites P.1 Real Numbers What you should learn Real Numbers Real numbers are used in everyday life to describe quantities such as age, miles per gallon, and population. Real numbers are represented by symbols such as ⫺5, 9, 0, 4 3, 0.666 . . . , 28.21, 冪2, , and 䊏 䊏 ⫺32. 3 冪 Here are some important subsets (each member of subset B is also a member of set A) of the set of real numbers. 再1, 2, 3, 4, . . .冎 䊏 䊏 䊏 Represent and classify real numbers. Order real numbers and use inequalities. Find the absolute values of real numbers and the distance between two real numbers. Evaluate algebraic expressions. Use the basic rules and properties of algebra. Set of natural numbers Why you should learn it 再0, 1, 2, 3, 4, . . .冎 Set of whole numbers 再. . . , ⫺3, ⫺2, ⫺1, 0, 1, 2, 3, . . .冎 Set of integers A real number is rational if it can be written as the ratio p兾q of two integers, where q ⫽ 0. For instance, the numbers Real numbers are used in every aspect of our lives, such as finding the surplus or deficit in the federal budget. See Exercises 83–88 on page 10. 1 1 125 ⫽ 0.3333 . . . ⫽ 0.3, ⫽ 0.125, and ⫽ 1.126126 . . . ⫽ 1.126 3 8 111 are rational. The decimal representation of a rational number either repeats 共as in ⫽ 3.145 兲 or terminates 共as in 12 ⫽ 0.5兲. A real number that cannot be written as the ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating decimal representations. For instance, the numbers 173 55 冪2 ⫽ 1.4142135 . . . ⬇ 1.41 ⫽ 3.1415926 . . . ⬇ 3.14 and are irrational. (The symbol ⬇ means “is approximately equal to.”) Figure P.1 shows subsets of real numbers and their relationships to each other. Real numbers are represented graphically by a real number line. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive and numbers to the left of 0 are negative, as shown in Figure P.2. The term nonnegative describes a number that is either positive or zero. © Alan Schein Photography/Corbis Real numbers Origin Negative direction Figure P.2 −4 −3 −2 −1 0 1 2 3 Positive direction 4 − 2.4 −2 − 53 2 −1 0 1 2 3 Every point on the real number line corresponds to exactly one real number. Figure P.3 Rational numbers The Real Number Line There is a one-to-one correspondence between real numbers and points on the real number line. That is, every point on the real number line corresponds to exactly one real number, called its coordinate, and every real number corresponds to exactly one point on the real number line, as shown in Figure P.3. −3 Irrational numbers One-to-One Correspondence −3 −2 π 0.75 −1 0 1 2 Integers Negative integers Noninteger fractions (positive and negative) Whole numbers 3 Every real number corresponds to exactly one point on the real number line. Natural numbers Figure P.1 Zero Subsets of Real Numbers 333371_0P01.qxp 12/27/06 9:28 AM Page 3 Section P.1 3 Real Numbers Ordering Real Numbers One important property of real numbers is that they are ordered. Definition of Order on the Real Number Line If a and b are real numbers, a is less than b if b ⫺ a is positive. This order is denoted by the inequality a < b. This relationship can also be described by saying that b is greater than a and writing b > a. The inequality a ≤ b means that a is less than or equal to b, and the inequality b ≥ a means that b is greater than or equal to a. The symbols <, >, ≤, and ≥, are inequality symbols. a −1 Geometrically, this definition implies that a < b if and only if a lies to the left of b on the real number line, as shown in Figure P.4. b 0 1 2 a < b if and only if a lies to the left of b. Figure P.4 x≤2 Example 1 Interpreting Inequalities x Describe the subset of real numbers represented by each inequality. a. x ≤ 2 b. x > ⫺1 0 1 2 3 4 Figure P.5 c. ⫺2 ≤ x < 3 Solution x > −1 a. The inequality x ≤ 2 denotes all real numbers less than or equal to 2, as shown in Figure P.5. b. The inequality x > ⫺1 denotes all real numbers greater than ⫺1, as shown in Figure P.6. c. The inequality ⫺2 ≤ x < 3 means that x ≥ ⫺2 and x < 3. The “double inequality” denotes all real numbers between ⫺2 and 3, including ⫺2 but not including 3, as shown in Figure P.7. Now try Exercise 31(a). x −2 −1 1 0 1 2 3 2 3 Figure P.6 −2 ≤ x < 3 x −2 −1 0 1 Figure P.7 Inequalities can be used to describe subsets of real numbers called intervals. In the bounded intervals below, the real numbers a and b are the endpoints of each interval. Bounded Intervals on the Real Number Line Notation 关a, b兴 Interval Type Closed Inequality Graph a ≤ x ≤ b x a 共a, b兲 Open b a < x < b x a 关a, b兲 a ≤ x < b 共a, b兴 a < x ≤ b b x a b a b x STUDY TIP The endpoints of a closed interval are included in the interval. The endpoints of an open interval are not included in the interval. 333371_0P01.qxp 12/27/06 9:28 AM 4 Chapter P Page 4 Prerequisites The symbols ⬁, positive infinity, and ⫺ ⬁, negative infinity, do not represent real numbers. They are simply convenient symbols used to describe the unboundedness of an interval such as 共1, ⬁兲 or 共⫺ ⬁, 3兴. Unbounded Intervals on the Real Number Line Notation 关a, ⬁兲 Interval Type Inequality x ≥ a STUDY TIP Graph x a 共a, ⬁兲 x > a Open An interval is unbounded when it continues indefinitely in one or both directions. x a 共⫺ ⬁, b兴 x ≤ b x b 共⫺ ⬁, b兲 x < b Open x b 共⫺ ⬁, ⬁兲 Entire real line ⫺⬁ < x < ⬁ x Example 2 Using Inequalities to Represent Intervals Use inequality notation to describe each of the following. a. c is at most 2. b. All x in the interval 共⫺3, 5兴 Solution a. The statement “c is at most 2” can be represented by c ≤ 2. b. “All x in the interval 共⫺3, 5兴” can be represented by ⫺3 < x ≤ 5. Now try Exercise 35. Example 3 Interpreting Intervals Give a verbal description of each interval. a. 共⫺1, 0兲 b. 关 2, ⬁兲 c. 共⫺ ⬁, 0兲 Solution a. This interval consists of all real numbers that are greater than ⫺1 and less than 0. b. This interval consists of all real numbers that are greater than or equal to 2. c. This interval consists of all negative real numbers. Now try Exercise 47. The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a ⫽ b, a < b, or a > b. Law of Trichotomy Additional Examples Use inequality notation to describe each of the following. a. m is at least ⫺3. b. All x in the interval [⫺2, 2] c. w is at least 1 and at most 5. d. All q in the interval 共⫺12, 0兲 Solution a. m ≥ ⫺3 b. ⫺2 ≤ x ≤ 2 c. 1 ≤ w ≤ 5 d. ⫺12 < q < 0 333371_0P01.qxp 12/27/06 9:28 AM Page 5 Section P.1 5 Real Numbers Absolute Value and Distance The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. Exploration Definition of Absolute Value If a is a real number, the absolute value of a is ⱍaⱍ ⫽ 冦⫺a, a, Absolute value expressions can be evaluated on a graphing utility. When evaluating an expression such as 3 ⫺ 8 , parentheses should surround the expression as shown below. ⱍ if a ≥ 0 . if a < 0 ⱍ Notice from this definition that the absolute value of a real number is never negative. For instance, if a ⫽ ⫺5, then ⫺5 ⫽ ⫺ 共⫺5兲 ⫽ 5. The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0. So, 0 ⫽ 0. ⱍ ⱍ ⱍⱍ Example 4 Evaluating the Absolute Value of a Number Evaluate ⱍxⱍ for (a) x > 0 and (b) x < 0. Evaluate each expression. What can you conclude? x Solution ⱍⱍ a. If x > 0, then x ⫽ x and ⱍxⱍ ⫽ x ⫽ 1. ⱍⱍ x b. If x < 0, then x ⫽ ⫺x and ⱍⱍ ⱍ5 ⫺ 2ⱍ ⱍ ⱍ ⱍ ⱍ a. 6 b. ⫺1 c. d. 2 ⫺ 5 x ⱍxⱍ ⫽ ⫺x ⫽ ⫺1. x x Now try Exercise 53. Properties of Absolute Value ⱍ ⱍ ⱍⱍ a a ⫽ ⱍ ⱍ, b ⫽ 0 b ⱍbⱍ ⱍⱍ 2. ⫺a ⫽ a ⱍ ⱍ ⱍ ⱍⱍ ⱍ 4. 1. a ≥ 0 3. ab ⫽ a b ⱍⱍ Absolute value can be used to define the distance between two points on the real number line. For instance, the distance between ⫺3 and 4 is ⱍ⫺3 ⫺ 4ⱍ ⫽ ⱍ⫺7ⱍ ⫽ 7 Distance Between Two Points on the Real Number Line Let a and b be real numbers. The distance between a and b is ⱍ ⱍ −3 −2 −1 Figure P.8 as shown in Figure P.8. ⱍ 7 ⱍ d共a, b兲 ⫽ b ⫺ a ⫽ a ⫺ b . 0 1 2 3 4 The distance between ⴚ3 and 4 is 7. 333371_0P01.qxp 12/27/06 6 Chapter P 9:28 AM Page 6 Prerequisites Algebraic Expressions One characteristic of algebra is the use of letters to represent numbers. The letters are variables, and combinations of letters and numbers are algebraic expressions. Here are a few examples of algebraic expressions. 2x ⫺ 3, 5x, 4 , x ⫹2 7x ⫹ y 2 Definition of an Algebraic Expression An algebraic expression is a combination of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. The terms of an algebraic expression are those parts that are separated by addition. For example, x 2 ⫺ 5x ⫹ 8 ⫽ x 2 ⫹ 共⫺5x兲 ⫹ 8 has three terms: x 2 and ⫺5x are the variable terms and 8 is the constant term. The numerical factor of a variable term is the coefficient of the variable term. For instance, the coefficient of ⫺5x is ⫺5, and the coefficient of x 2 is 1. To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression. Here are two examples. Expression ⫺3x ⫹ 5 3x 2 ⫹ 2x ⫺ 1 Value of Variable Substitute Value of Expression x⫽3 ⫺3共3兲 ⫹ 5 ⫺9 ⫹ 5 ⫽ ⫺4 x ⫽ ⫺1 3共⫺1兲 ⫹ 2共⫺1兲 ⫺ 1 3⫺2⫺1⫽0 2 When an algebraic expression is evaluated, the Substitution Principle is used. It states, “If a ⫽ b, then a can be replaced by b in any expression involving a.” In the first evaluation shown above, for instance, 3 is substituted for x in the expression ⫺3x ⫹ 5. Basic Rules of Algebra There are four arithmetic operations with real numbers: addition, multiplication, subtraction, and division, denoted by the symbols ⫹, ⫻ or ⭈ , ⫺, and ⫼ or 兾. Of these, addition and multiplication are the two primary operations. Subtraction and division are the inverse operations of addition and multiplication, respectively. Subtraction: Add the opposite of b. a ⫺ b ⫽ a ⫹ 共⫺b兲 Division: Multiply by the reciprocal of b. If b ⫽ 0, then a兾b ⫽ a 冢b冣 ⫽ b . 1 a In these definitions, ⫺b is the additive inverse (or opposite) of b, and 1兾b is the multiplicative inverse (or reciprocal) of b. In the fractional form a兾b, a is the numerator of the fraction and b is the denominator. Common Error A common error is to use the wrong order of operations when evaluating expressions. Remind students that the order of operations is as follows. 1. First do operations that occur within symbols of grouping. 2. Then evaluate exponential expressions. 3. Then do multiplications and divisions from left to right. 4. Finally do additions and subtractions from left to right. 333371_0P01.qxp 12/27/06 9:28 AM Page 7 Section P.1 Real Numbers Because the properties of real numbers below are true for variables and algebraic expressions, as well as for real numbers, they are often called the Basic Rules of Algebra. Try to formulate a verbal description of each property. For instance, the Commutative Property of Addition states that the order in which two real numbers are added does not affect their sum. Basic Rules of Algebra Let a, b, and c be real numbers, variables, or algebraic expressions. Property Example Commutative Property of Addition: a⫹b⫽b⫹a 4x ⫹ Commutative Property of Multiplication: ab ⫽ ba 共1 ⫺ x兲 x 2 ⫽ x 2共1 ⫺ x兲 Associative Property of Addition: 共a ⫹ b兲 ⫹ c ⫽ a ⫹ 共b ⫹ c兲 共x ⫹ 5兲 ⫹ x 2 ⫽ x ⫹ 共5 ⫹ x 2兲 Associative Property of Multiplication: 共ab兲 c ⫽ a共bc兲 共2x ⭈ 3y兲共8兲 ⫽ 共2x兲共3y ⭈ 8兲 Distributive Properties: a共b ⫹ c兲 ⫽ ab ⫹ ac 3x共5 ⫹ 2x兲 ⫽ 3x ⭈ 5 ⫹ 3x ⭈ 2x x2 ⫽ x 2 ⫹ 4x 共 y ⫹ 8兲 y ⫽ y ⭈ y ⫹ 8 ⭈ y 共a ⫹ b兲c ⫽ ac ⫹ bc Additive Identity Property: a⫹0⫽a 5y 2 ⫹ 0 ⫽ 5y 2 Multiplicative Identity Property: a ⭈1⫽a 共4x 2兲共1兲 ⫽ 4x 2 Additive Inverse Property: a ⫹ 共⫺a兲 ⫽ 0 Multiplicative Inverse Property: a 1 ⭈ a ⫽ 1, a⫽0 6x 3 ⫹ 共⫺6x 3兲 ⫽ 0 共x 2 ⫹ 4兲 冢x 2 冣 1 ⫽1 ⫹4 Because subtraction is defined as “adding the opposite,” the Distributive Properties are also true for subtraction. For instance, the “subtraction form” of a共b ⫹ c兲 ⫽ ab ⫹ ac is a共b ⫺ c兲 ⫽ ab ⫺ ac. Properties of Negation and Equality Let a, b, and c be real numbers, variables, or algebraic expressions. Property 1. 共⫺1兲 a ⫽ ⫺a Example 共⫺1兲7 ⫽ ⫺7 2. ⫺ 共⫺a兲 ⫽ a ⫺ 共⫺6兲 ⫽ 6 3. 共⫺a兲b ⫽ ⫺ 共ab兲 ⫽ a共⫺b兲 共⫺5兲3 ⫽ ⫺ 共5 ⭈ 3兲 ⫽ 5共⫺3兲 4. 共⫺a兲共⫺b兲 ⫽ ab 共⫺2兲共⫺x兲 ⫽ 2x 5. ⫺ 共a ⫹ b兲 ⫽ 共⫺a兲 ⫹ 共⫺b兲 ⫺ 共x ⫹ 8兲 ⫽ 共⫺x兲 ⫹ 共⫺8兲 ⫽ ⫺x ⫺ 8 6. If a ⫽ b, then a ⫹ c ⫽ b ⫹ c. 1 2 ⫹ 3 ⫽ 0.5 ⫹ 3 共2兲 ⫽ 16共2兲 7 8. If a ⫹ c ⫽ b ⫹ c, then a ⫽ b. 1.4 ⫺ 1 ⫽ 5 ⫺ 1 7. If a ⫽ b, then ac ⫽ bc. 42 9. If ac ⫽ bc and c ⫽ 0, then a ⫽ b. 3 冪9 ⫽ 4 4 STUDY TIP Be sure you see the difference between the opposite of a number and a negative number. If a is already negative, then its opposite, ⫺a, is positive. For instance, if a ⫽ ⫺2, then ⫺a ⫽ ⫺ 共⫺2兲 ⫽ 2. 7 333371_0P01.qxp 12/27/06 9:29 AM 8 Chapter P Page 8 Prerequisites Properties of Zero STUDY TIP Let a and b be real numbers, variables, or algebraic expressions. 1. a ⫹ 0 ⫽ a and 3. 0 ⫽ 0, a a⫺0⫽a 2. a a⫽0 4. ⭈0⫽0 a is undefined. 0 5. Zero-Factor Property: If ab ⫽ 0, then a ⫽ 0 or b ⫽ 0. The “or” in the Zero-Factor Property includes the possibility that either or both factors may be zero. This is an inclusive or, and it is the way the word “or” is generally used in mathematics. Properties and Operations of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that b ⫽ 0 and d ⫽ 0. 1. Equivalent Fractions: a c ⫽ b d if and only if a ⫺a a ⫽ 2. Rules of Signs: ⫺ ⫽ b b ⫺b 3. Generate Equivalent Fractions: and ad ⫽ bc. ⫺a a ⫽ ⫺b b a ac ⫽ , c⫽0 b bc a ±c a c 4. Add or Subtract with Like Denominators: ± ⫽ b b b 5. Add or Subtract with Unlike Denominators: 6. Multiply Fractions: 7. Divide Fractions: a b c x 2x 5 ⫹ ⫽ 3 5 ac ⭈ d ⫽ bd a c a ⫼ ⫽ b d b b. 7 3 7 ⫼ ⫽ x 2 x ⭈ x ⫹ 3 ⭈ 2x ⫽ 11x 15 2 ⱍ ⱍ a c ad ± bc ± ⫽ b d bd d ad ⭈ c ⫽ bc , c⫽0 Example 5 Properties and Operations of Fractions a. Activities 1. Use inequality notation to describe the set of nonnegative numbers. Answer: x ≥ 0 2. Use interval notation to describe the inequality ⫺6 < x ≤ 13. Answer: 共⫺6, 13兴 3. Evaluate: 4 ⫺ ⫺3 . Answer: 1 4. Find the distance between ⫺41 and 16. Answer: 57 15 14 ⭈ 3 ⫽ 3x Add fractions with unlike denominators. Divide fractions. STUDY TIP In Property 1 of fractions, the phrase “if and only if” implies two statements. One statement is: If a兾b ⫽ c兾d, then ad ⫽ bc. The other statement is: If ad ⫽ bc, where b ⫽ 0 and d ⫽ 0, then a兾b ⫽ c兾d. Now try Exercise 113. If a, b, and c are integers such that ab ⫽ c, then a and b are factors or divisors of c. A prime number is an integer that has exactly two positive factors: itself and 1. For example, 2, 3, 5, 7, and 11 are prime numbers. The numbers 4, 6, 8, 9, and 10 are composite because they can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as the product of prime numbers. For instance, the prime factorization of 24 is 24 ⫽ 2 ⭈ 2 ⭈ 2 ⭈ 3. Point out to students that to add or subtract fractions with unlike denominators, they can either use Property 5 of fractions as in Example 5(a), or they can rewrite the fractions with like denominators using the least common denominator (LCD) of the fractions. 333371_0P01.qxp 12/27/06 9:29 AM Page 9 Section P.1 P.1 Exercises Real Numbers 9 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check Fill in the blanks. 1. A real number is _______ if it can be written as the ratio p of two integers, where q ⫽ 0. q 2. _______ numbers have infinite nonrepeating decimal representations. 3. The distance between a point on the real number line and the origin is the _______ of the real number. 4. Numbers that can be written as the product of two or more prime numbers are called _______ numbers. 5. Integers that have exactly two positive factors, the integer itself and 1, are called _______ numbers. 6. An algebraic expression is a combination of letters called _______ and real numbers called _______ . 7. The _______ of an algebraic expression are those parts separated by addition. 8. The numerical factor of a variable term is the _______ of the variable term. 9. The _______ states: If ab ⫽ 0, then a ⫽ 0 or b ⫽ 0. In Exercises 1– 6, determine which numbers are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers. 1. 再 ⫺9, ⫺ 72, 5, 23, 冪2, 0, 1, ⫺4, ⫺1冎 2. 再 冪5, ⫺7, ⫺ 73, 0, 3.12, 54, ⫺2, ⫺8, 3冎 In Exercises 19–24, plot the two real numbers on the real number line. Then place the correct inequality symbol (< or >) between them. 19. ⫺4, ⫺8 20. ⫺3.5, 1 3 21. 2, 7 16 22. 1, 3 3. 再2.01, 0.666 . . . , ⫺13, 0.010110111 . . . , 1, ⫺10, 20冎 23. 4. 再2.3030030003 . . . , 0.7575, ⫺4.63, 冪10, ⫺2, 0.03, ⫺10冎 In Exercises 25–32, (a) verbally describe the subset of real numbers represented by the inequality, (b) sketch the subset on the real number line, and (c) state whether the interval is bounded or unbounded. 1 6 1 5. 再 ⫺ , ⫺ 3, 3, 2冪2, ⫺7.5, ⫺2, 3, ⫺3冎 12 1 6. 再 25, ⫺17, ⫺ 5 , 冪9, 3.12, 2, 6, ⫺4, 18冎 In Exercises 7–12, use a calculator to find the decimal form of the rational number. If it is a nonterminating decimal, write the repeating pattern. 7. 9. 11. 5 16 41 333 ⫺ 100 11 10. 17 4 3 7 12. 218 ⫺ 33 8. 5 2 6, 3 8 3 24. ⫺ 7, ⫺ 7 25. x ≤ 5 26. x > 3 27. x < 0 28. x ≥ 4 29. ⫺2 < x < 2 30. 0 ≤ x ≤ 5 31. ⫺1 ≤ x < 0 32. 0 < x ≤ 6 In Exercises 33 –38, use inequality and interval notation to describe the set. In Exercises 13 –16, use a calculator to rewrite the rational number as the ratio of two integers. 13. 4.6 14. 12.3 15. ⫺6.5 16. ⫺1.83 33. x is negative. 34. z is at least 10. 35. y is nonnegative. 36. y is no more than 25. 37. p is less than 9 but no less than ⫺1. 38. The annual rate of inflation r is expected to be at least 2.5%, but no more than 5%. In Exercises 17 and 18, approximate the numbers and place the correct inequality symbol (< or >) between them. In Exercises 39– 44, use interval notation to describe the graph. 17. 39. 18. −2 −7 −1 −6 0 −5 1 −4 2 −3 −2 3 4 −1 0 x −2 −1 0 1 2 3 4 5 6 −3 −2 −1 0 1 2 3 4 5 40. x 333371_0P01.qxp 12/27/06 10 9:29 AM Chapter P Page 10 Prerequisites 41. 75. y is at least six units from 0. x −4 −2 0 −4 −2 2 4 6 8 10 12 42. 76. y is at most two units from a. x 0 2 4 77. While traveling on the Pennsylvania Turnpike, you pass milepost 57 near Pittsburgh, then milepost 236 near Gettysburg. How many miles do you travel between these two mileposts? 6 43. x −a a+4 44. 78. The temperature in Bismarck, North Dakota was 60⬚F at noon, then 23⬚F at midnight. What was the change in temperature over the 12-hour period? x −c + 2 c+1 In Exercises 45 – 48, give a verbal description of the interval. 47. 共⫺ ⬁, 2兴 46. 共⫺ ⬁, 4兴 48. 关1, ⬁兲 In Exercises 49– 54, evaluate the expression. ⱍ ⱍ ⱍⱍ ⱍ ⱍ ⱍ ⱍ ⱍx ⫺ 1ⱍ 49. ⫺10 50. 0 51. ⫺3 ⫺3 52. ⫺1 ⫺ ⫺2 53. 54. ⱍ ⱍ ⱍx ⫹ 2ⱍ x⫹2 x⫺1 In Exercises 55 – 60, evaluate the expression for the given values of x and y. Then use a graphing utility to verify your result. ⱍ ⱍ ⱍy ⫺ 4xⱍ for x ⫽ 2 and y ⫽ ⫺3 ⱍxⱍ ⫺ 2ⱍyⱍ for x ⫽ 2 and y ⫽ ⫺1 ⱍ2xⱍ ⫺ 3ⱍyⱍ for x ⫽ ⫺4 and y ⫽ 1 55. 2x ⫹ y for x ⫽ ⫺2 and y ⫽ 3 56. 57. 58. 59. 60. ⱍ ⱍ ⱍⱍ ⱍ ⱍ ⱍ ⱍ 3 x ⫺ 2y for x ⫽ ⫺2 and y ⫽ 2 2x ⫹ y In Exercises 61–66, place the correct symbol 冇<, >, or ⴝ冈 between the pair of real numbers. ⱍ ⱍ䊏⫺ ⱍ⫺3ⱍ ⫺5䊏⫺ ⱍ5ⱍ ⫺ ⱍ⫺2ⱍ䊏⫺ ⱍ2ⱍ ⱍ ⱍ䊏ⱍ4ⱍ ⫺ ⱍ⫺6ⱍ䊏ⱍ⫺6ⱍ 62. ⫺4 63. 64. 65. Budgeted Expense, b 79. Wages $112,700 $113,356 80. Utilities $9400 $9772 81. Taxes $37,640 $37,335 82. Insurance $2575 $2613 69. a ⫽ 71. a ⫽ ⫺ 52, b ⫽ 0 16 112 5 , b ⫽ 75 66. ⫺(⫺2)䊏⫺2 2000 1800 䊏 䊏 䊏 䊏 0.05b 䊏 䊏 䊏 䊏 1579.3 1600 1400 1351.8 1200 1000 1995 1 4, b⫽ Receipts 11 4 72. a ⫽ 9.34, b ⫽ ⫺5.65 1997 1999 2001 2003 2005 Year 68. a ⫽ ⫺126, b ⫽ ⫺75 70. a ⫽ ⱍa ⫺ bⱍ 2153.9 1991.2 1827.5 1782.3 2200 In Exercises 67– 72, find the distance between a and b. 67. a ⫽ 126, b ⫽ 75 Actual Expense, a Federal Deficit In Exercises 83–88, use the bar graph, which shows the receipts of the federal government (in billions of dollars) for selected years from 1995 through 2005. In each exercise you are given the expenditures of the federal government. Find the magnitude of the surplus or deficit for the year. (Source: U.S. Office of Management and Budget) 3x ⫹ 2y for x ⫽ 4 and y ⫽ 1 x 61. ⫺3 Budget Variance In Exercises 79–82, the accounting department of a company is checking to determine whether the actual expenses of a department differ from the budgeted expenses by more than $500 or by more than 5%. Fill in the missing parts of the table, and determine whether the actual expense passes the “budget variance test.” Receipts (in billions of dollars) 45. 共⫺6, ⬁兲 83. 1995 84. 1997 In Exercises 73 – 76, use absolute value notation to describe the situation. 85. 1999 73. The distance between x and 5 is no more than 3. 87. 2003 74. The distance between x and ⫺10 is at least 6. 88. 2005 86. 2001 䊏 䊏 䊏 䊏 䊏 䊏 Expenditures $1515.8 billion $1601.3 billion $1701.9 billion $1863.9 billion $2157.6 billion $2472.2 billion ⱍ Receipts ⫺ Expenditures 䊏 䊏 䊏 䊏 䊏 䊏 ⱍ 333371_0P01.qxp 12/27/06 9:29 AM Page 11 Section P.1 In Exercises 89 – 94, identify the terms. Then identify the coefficients of the variable terms of the expression. 1 90. 2x ⫺ 9 n 91. 冪3x2 ⫺ 8x ⫺ 11 92. 7冪5x2 ⫹ 3 5兾n x ⫺5 2 5 In Exercises 95–98, evaluate the expression for each value of x. (If not possible, state the reason.) Values (a) x ⫽ 3 (b) x ⫽ ⫺ 12 96. 4 ⫺ 3x (a) x ⫽ 2 (b) x ⫽ ⫺ 56 97. x2 ⫺ 4 x2 98. x⫹4 (a) x ⫽ 2 (b) x ⫽ ⫺2 (a) x ⫽ 1 (b) x ⫽ ⫺4 0.000001 124. (a) Use a calculator to complete the table. 1 10 100 10,000 100,000 5兾n (b) Use the result from part (a) to make a conjecture about the value of 5兾n as n increases without bound. True or False? In Exercises 125 and 126, determine whether the statement is true or false. Justify your answer. 99. x ⫹ 9 ⫽ 9 ⫹ x 100. 2共 12 兲 ⫽ 1 125. Let a > b, then 1 共h ⫹ 6兲 ⫽ 1, h ⫽ ⫺6 h⫹6 126. Because 102. 共x ⫹ 3兲 ⫺ 共x ⫹ 3兲 ⫽ 0 103. 2共x ⫹ 3兲 ⫽ 2x ⫹ 6 1 1 > , where a ⫽ 0 and b ⫽ 0. a b a⫹b a b c c c ⫽ ⫹ , then ⫽ ⫹ . c c c a⫹b a b In Exercises 127 and 128, use the real numbers A, B, and C shown on the number line. Determine the sign of each expression. 104. 共z ⫺ 2兲 ⫹ 0 ⫽ z ⫺ 2 105. x ⫹ 共 y ⫹ 10兲 ⫽ 共x ⫹ y兲 ⫹ 10 1 0.0001 Synthesis In Exercises 99–106, identify the rule(s) of algebra illustrated by the statement. 106. 7共7 0.01 (b) Use the result from part (a) to make a conjecture about the value of 5兾n as n approaches 0. n 95. 2x ⫺ 5 101. 0.5 2x3 94. 3x 4 ⫹ Expression 11 123. (a) Use a calculator to complete the table. 89. 7x ⫹ 4 93. 4x 3 ⫹ Real Numbers ⭈ 12兲 ⫽ 共 17 ⭈ 7兲12 ⫽ 1 ⭈ 12 ⫽ 12 C B A 0 In Exercises 107–116, perform the operation(s). (Write fractional answers in simplest form.) 107. 109. 111. 5 3 16 ⫹ 16 5 1 5 8 ⫺ 12 ⫹ 6 108. 110. x 3x ⫹ 6 4 112. 12 1 ⫼ 113. x 8 115. 6 4 7 ⫺ 7 10 6 11 ⫹ 33 ⫺ 13 66 2x x ⫹ 5 10 11 3 ⫼ 114. x 4 共25 ⫼ 4兲 ⫺ 共4 ⭈ 38 兲 116. 共 35 ⫼ 3兲 ⫺ 共6 ⭈ 48 兲 In Exercises 117–122, use a calculator to evaluate the expression. (Round your answer to two decimal places.) 117. 14共⫺3 ⫹ 3 7 兲 11.46 ⫺ 5.37 119. 3.91 121. 2 3 共⫺2 ⫺ 6兲 ⫺25 118. 3共 5 ⫺ 12 ⫹ 3 8 兲 12.24 ⫺ 8.4 120. 2.5 122. 1 5 共⫺8 ⫺ 9兲 ⫺ 13 127. (a) ⫺A 128. (a) ⫺C (b) B ⫺ A 129. Exploration (b) A ⫺ C ⱍ ⱍ ⱍⱍ ⱍⱍ Consider u ⫹ v and u ⫹ v . (a) Are the values of the expressions always equal? If not, under what conditions are they not equal? (b) If the two expressions are not equal for certain values of u and v, is one of the expressions always greater than the other? Explain. 130. Think About It Is there a difference between saying that a real number is positive and saying that a real number is nonnegative? Explain. 131. Writing Describe the differences among the sets of whole numbers, natural numbers, integers, rational numbers, and irrational numbers. ⱍⱍ 132. Writing Can it ever be true that a ⫽ ⫺a for any real number a? Explain.