1.4 Properties of Real Numbers and Algebraic Expressions

Transcription

1 0 CHAPTER Real Numbers and Algebraic Expressions.4 Properties of Real Numbers and Algebraic Expressions S Use Operation and Order Symbols to Write Mathematical Sentences. 2 Identify Identity Numbers and Inverses. Identify and Use the Commutative, Associative, and Distributive Properties. 4 Write Algebraic Expressions. 5 Simplify Algebraic Expressions. Using Symbols to Write Mathematical Sentences In Section.2, we used the symbol = to mean is equal to. All of the following key words and phrases also imply equality. Equality equals is/was represents is the same as gives yields amounts to is equal to EXAMPLES Write each sentence as an equation.. The sum of x and 5 is 20. (+++)+++* T x + 5 = Two times the sum of and y amounts to 4. (+)+* (+++)+++* (+)+* T T T 2 + y2 = 4. The difference of 8 and x is the same as the product of 2 and x. (++++)++++* (++)++* (+++)+++* T T T 8 - x = 2# x 4. The quotient of z and 9 amounts to 9 plus z. (++++)++++* (+)+* T T z, 9 = 9 + z or z 9 = 9 + z 4 Write each sentence using mathematical symbols.. The product of -4 and x is Three times the difference of z and equals 9.. The sum of x and 5 is the same as less than twice x. 4. The sum of y and 2 is 4 more than the quotient of z and 8. If we want to write in symbols that two numbers are not equal, we can use the symbol, which means is not equal to. For example, 2 Graphing two numbers on a number line gives us a way to compare two numbers. For two real numbers a and b, we say a is less than b if on the number line a lies to the left of b. Also, if b is to the right of a on the number line, then b is greater than a. The symbol 6 means is less than. Since a is less than b, we write Helpful Hint Notice that if a 6 b, then b 7 a. For example, since - 6 7, then a b b 6 a The symbol 7 means is greater than. Since b is greater than a, we write b 7 a

2 Section.4 Properties of Real Numbers and Algebraic Expressions EXAMPLE 5 statement. Insert 6, 7, or = between each pair of numbers to form a true a b. 2 4 c d e f. 2 4 a since - lies to the right of - 2 on the number line. b. 2 4 =. 2 0 c since -5 lies to the left of 0 on the number line d since -.5 lies to the left of -.05 on the number line e f The denominators are the same, so since By dividing, we see that 4 = 0.75 and 2 = c Thus 2 6 since 0.666c Insert 6, 7, or = between each pair of numbers to form a true statement. a b c. 0-7 d e f Helpful Hint When inserting the 7 or 6 symbol, think of the symbols as arrowheads that point toward the smaller number when the statement is true. In addition to 6 and 7, there are the inequality symbols and Ú. The symbol means ;is less than or equal to< and the symbol Ú means ;is greater than or equal to< For example, the following are true statements. 0 0 since 0 = 0-8 since Ú -5 since -5 = -5-7 Ú -9 since EXAMPLE 6 Write each sentence using mathematical symbols. a. The sum of 5 and y is greater than or equal to 7. b. is not equal to z. c. 20 is less than the difference of 5 and twice x.

3 2 CHAPTER Real Numbers and Algebraic Expressions a. 5 + y Ú 7 b. z c x 6 Write each sentence using mathematical symbols. a. The difference of x and is less than or equal to 5. b. y is not equal to -4. c. Two is less than the sum of 4 and one-half z. 2 Identifying Identities and Inverses Of all the real numbers, two of them stand out as extraordinary: 0 and. Zero is the only number that, when added to any real number, results in the same real number. Zero is thus called the additive identity. Also, one is the only number that, when multiplied by any real number, results in the same real number. One is thus called the multiplicative identity. Identity Properties Addition The additive identity is 0. a + 0 = 0 + a = a Multiplication The multiplicative identity is. a # = # a = a In Section.2, we learned that a and -a are opposites. Another name for opposite is additive inverse. For example, the additive inverse of is -. Notice that the sum of a number and its opposite is always 0. In Section., we learned that, for a nonzero number, b and are reciprocals. b Another name for reciprocal is multiplicative inverse. For example, the multiplicative inverse of - 2 is -. Notice that the product of a number and its reciprocal is always. 2 Inverse Properties Opposite or Additive Inverse For each number a, there is a unique number -a called the additive inverse or opposite of a such that a + -a2 = -a2 + a = 0 Reciprocal or Multiplicative Inverse For each nonzero a, there is a unique number called the multiplicative a inverse or reciprocal of a such that a # a = a # a = EXAMPLE 7 Write the additive inverse, or opposite, of each. a. 4 b. 7 c. -.2 a. The opposite of 4 is -4. b. The opposite of 7 is - 7. c. The opposite of -.2 is =.2. 7 Write the additive inverse, or opposite, of each. a. -7 b. 4.7 c. - 8

4 Section.4 Properties of Real Numbers and Algebraic Expressions EXAMPLE 8 Write the multiplicative inverse, or reciprocal, of each. a. b. -9 c. 7 4 a. The reciprocal of is. b. The reciprocal of -9 is - 9. c. The reciprocal of 7 4 is 4 7 because 7 # =. 8 Write the multiplicative inverse, or reciprocal, of each. a. - 5 b. 4 c. -2 Helpful Hint The number 0 has no reciprocal. Why? There is no number that when multiplied by 0 gives a product of. CONCEPT CHECK Can a number s additive inverse and multiplicative inverse ever be the same? Explain. Using the Commutative, Associative, and Distributive Properties In addition to these special real numbers, all real numbers have certain properties that allow us to write equivalent expressions that is, expressions that have the same value. These properties will be especially useful in Chapter 2 when we solve equations. The commutative properties state that the order in which two real numbers are added or multiplied does not affect their sum or product. Commutative Properties For real numbers a and b, Addition: a + b = b + a Multiplication: a # b = b # a The associative properties state that regrouping numbers that are added or multiplied does not affect their sum or product. Answer to Concept Check: no; answers may vary Associative Properties For real numbers a, b, and c, Addition: a + b2 + c = a + b + c2 Multiplication: a # b2 # c = a # b # c2

5 4 CHAPTER Real Numbers and Algebraic Expressions EXAMPLE 9 Use the commutative property of addition to write an expression equivalent to 7x x + 5 = 5 + 7x. 9 Use the commutative property of addition to write an expression equivalent to 8 + x. EXAMPLE 0 Use the associative property of multiplication to write an expression equivalent to 4 # 9y2. Then simplify this equivalent expression. 4# 9y2 = 4 # 92y = 6y. 0 Use the associative property of multiplication to write an expression equivalent to # b2. Then simplify the equivalent expression. The distributive property states that multiplication distributes over addition. Distributive Property For real numbers a, b, and c, Also, ab + c2 = ab + ac ab - c2 = ab - ac EXAMPLE Use the distributive property to multiply. a. 2x + y2 b. -x - 2 c. 0.7ab - 22 a. (2x+y) = # 2x + # y Apply the distributive property. = 6x + y Apply the associative property of multiplication. b. Recall that -x - 2 means -x - 2. (x-) = -x = -x + c. 0.7a(b-2) = 0.7a # b - 0.7a # 2 = 0.7ab -.4a Answer to Concept Check: no; 62a2b2 = 2ab2 = 6ab Use the distributive property to multiply. a. 4x + 5y2 b z2 c. 0.xy - 2 CONCEPT CHECK Is the statement below true? Why or why not? 62a2b2 = 62a2 # 6b2

6 Section.4 Properties of Real Numbers and Algebraic Expressions 5 4 Writing Algebraic Expressions As mentioned earlier, an important step in problem solving is to be able to write algebraic expressions from word phrases. Sometimes this involves a direct translation, but often an indicated operation is not directly stated but rather implied. EXAMPLE 2 Write each as an algebraic expression. a. A vending machine contains x quarters. Write an expression for the value of the quarters. b. The number of grams of fat in x pieces of bread if each piece of bread contains 2 grams of fat. c. The cost of x desks if each desk costs $56. d. Sales tax on a purchase of x dollars if the tax rate is 9%. Each of these examples implies finding a product. a. The value of the quarters is found by multiplying the value of a quarter (0.25 dollar) by the number of quarters. In words: value of a quarter # number of quarters T # T Translate: 0.25 # x, or 0.25x b. In words: number of grams of fat in # number of pieces one piece of bread of bread T T Translate: 2 # x, or 2x c. In words cost of a desk # number of desks T T Translate: 56 # x, or 56x d. In words: sales tax rate # purchase price T T Translate: 0.09 # x, or 0.09x (Here, we wrote 9% as a decimal, 0.09.) 2 Write each as an algebraic expression. a. A parking meter contains x dimes. Write an expression for the value of the dimes. b. The grams of carbohydrates in y cookies if each cookie has 26 g of carbohydrates. c. The cost of z birthday cards if each birthday card costs $.75. d. The amount of money you save on a new cell phone costing t dollars if it has a 5% discount. Let s continue writing phrases as algebraic expressions. Two or more unknown numbers in a problem may sometimes be related. If so, try letting a variable represent one unknown number and then represent the other unknown number or numbers as expressions containing the same variable. EXAMPLE Write each as an algebraic expression. a. Two numbers have a sum of 20. If one number is x, represent the other number as an expression in x. b. The older sister is 8 years older than her younger sister. If the age of the younger sister is x, represent the age of the older sister as an expression in x.

7 6 CHAPTER Real Numbers and Algebraic Expressions c. Two angles are complementary if the sum of their measures is 90. If the measure of one angle is x degrees, represent the measure of the other angle as an expression in x. d. If x is the first of two consecutive integers, represent the second integer as an expression in x. a. If two numbers have a sum of 20 and one number is x, the other number is the rest of 20. In words: twenty minus x T T T Translate: 20 - x b. The older sister s age is In words: eight years added to younger sister>s age T T T Translate: 8 + x c. In words: ninety minus x T T T Translate: 90 - x d. The next consecutive integer is always one more than the previous integer. In words: the first integer plus one T T T Translate: x + Write each as an algebraic expression. a. Two numbers have a sum of 6. If one number is x, represent the other number as an expression in x. b. Two angles are supplementary if the sum of their measures is 80. If the measure of one angle is x degrees, represent the measure of the other angle as an expression in x. c. If x is the first of two consecutive even integers, represent the next even integer as an expression in x. d. One brother is 9 years younger than another brother. If the age of the younger brother is x, represent the age of the older brother as an expression in x. 5 Simplifying Algebraic Expressions Often, an expression may be simplified by removing grouping symbols and combining any like terms. The terms of an expression are the addends of the expression. For example, in the expression x 2 + 4x, the terms are x 2 and 4x. Expression Terms -2x + y -2x, y x 2 - y x2, - y 5, 7 Terms with the same variable(s) raised to the same power are called like terms. We can add or subtract like terms by using the distributive property. This process is called combining like terms.

8 Section.4 Properties of Real Numbers and Algebraic Expressions 7 EXAMPLE 4 Use the distributive property to simplify each expression. a. x - 5x + 4 b. 7yz + yz c. 4z + 6. a. x - 5x + 4 = - 52x + 4 Apply the distributive property. = -2x + 4 b. 7yz + yz = 7 + 2yz = 8yz c. 4z + 6. cannot be simplified further since 4z and 6. are not like terms. 4 Use the distributive property to simplify. a. 6ab - ab b. 4x x c. 7p - 9 Let s continue to use properties of real numbers to simplify expressions. Recall that the distributive property can also be used to multiply. For example, 2(x+) = -2x = -2x - 6 The associative and commutative properties may sometimes be needed to rearrange and group like terms when we simplify expressions. -7x x 2-2 = -7x 2 + x = x = -4x 2 + EXAMPLE 5 Simplify each expression. a. xy - 2xy xy b. 7x x 2-42 c. 2.x x d. 2 4a - 6b2-9a + 2b a. xy - 2xy xy = xy - 2xy + xy Apply the commutative property. = xy Apply the distributive property. = 2xy - 2 Simplify. b. 7x 2 +-5(x 2-4) = 7x x Apply the distributive property. = 2x Simplify. c. Think of - -x as - -x and use the distributive property. 2.x x = 2.x x + 5. =.x Combine like terms d. a4a-6bb- a9a+2b-b+ 2 4 = 2a - b - a - 4b Use the distributive property. = -a - 7b Combine like terms.

9 8 CHAPTER Real Numbers and Algebraic Expressions 5 Simplify each expression. a. 5pq - 2pq - - 4pq + 8 b. x x 2-62 c..7x x -.2 Answer to Concept Check: x - 4x - 52 = x - 4x + 20 = -x + 20 d. 5 5c - 25d2-2 8c + 6d CONCEPT CHECK Find and correct the error in the following. x - 4x - 52 = x - 4x - 20 = -x - 20 Vocabulary, Readiness & Video Check Complete the table by filling in the symbols. Symbol Meaning.. 5. is less than is not equal to is greater than or equal to Symbol Meaning is greater than is equal to is less than or equal to Use the choices below to fill in each blank. Not all choices will be used. like terms distributive - a commutative unlike combining associative a 7. The opposite of nonzero number a is. 8. The reciprocal of nonzero number a is. 9. The property has to do with order. 0. The property has to do with grouping.. ab + c2 = ab + ac illustrates the property. 2. The of an expression are the addends of the expression. Martin-Gay Interactive Videos See Video.4 Watch the section lecture video and answer the following questions In the lecture before Example, what 6 symbols are discussed that can be used to compare two numbers? 4. Complete these statements based on the lecture given before Example 4. Reciprocal is the same as inverse and opposite is the same as inverse. 5. The commutative and associative properties are discussed in Examples 7 and 8 and the lecture before. What s the one word used again and again to describe the commutative property? The associative property? 6. In Example 0, what important point are you told to keep in mind when working with applications that have to do with money? 7. From Examples 2 4, how do we simplify algebraic expressions? If the expression contains parentheses, what property might we apply first?

10 Section.4 Properties of Real Numbers and Algebraic Expressions 9.4 Exercise Set MIXED Translating. Write each sentence using mathematical symbols. See Examples through 4 and 6 through 8.. The sum of 0 and x is The difference of y and amounts to 2.. Twice x, plus 5, is the same as Three more than the product of 4 and c is The quotient of n and 5 is 4 times n. 6. The quotient of 8 and y is more than y. 7. The difference of z and one-half is the same as the product of z and one-half. 8. Five added to one-fourth q is the same as 4 more than q. 9. The product of 7 and x is less than or equal to subtracted from the reciprocal of x is greater than 0.. Twice the difference of x and 6 is greater than the reciprocal of. 2. Four times the sum of 5 and x is not equal to the opposite of 5.. Twice the difference of x and 6 is times the sum of 6 and y is -5. Insert 6, 7, or = between each pair of numbers to form a true statement. See Example Fill in the chart. See Examples 7 and Number Opposite Reciprocal Use a commutative property to write an equivalent expression. See Example x + y 6. a + 2b 7. z # w 8. r # s 9. # x x 2 # 9 0 Use an associative property to write an equivalent expression. See Example # 7x2 42. # 0z2 4. x y 44. 5q + 2r + s z2 # y x2 # y Use the distributive property to find the product. See Example. 47. x y a + b c + 7d x + 5y + 2z a + b + 9c x - 2y a - b x6y m9n - 42 Complete the statement to illustrate the given property. 57. x + 6 = Commutative property of addition = Additive identity property a - 2 b = Additive inverse property 60. 4x + 2 = Distributive property 6. 7 # = Multiplicative identity property # = Multiplication property of zero 6. 02y2 = Associative property 64. 9y + x + z2 = Associative property Translating Write each of the following as an algebraic expression. See Examples 2 and. 65. Write an expression for the amount of money (in dollars) in d dimes. 66. Write an expression for the amount of money (in dollars) in n nickels. 67. Two numbers have a sum of 2. If one number is x, represent the other number as an expression in x. 68. Two numbers have a sum of 25. If one number is x, represent the other number as an expression in x. 69. Two angles are supplementary if the sum of their measures is 80. If the measure of one angle is x degrees, represent the measure of the other angle as an expression in x. 70. If the measure of an angle is 5x degrees, represent the measure of its complement as an expression in x. 7. The cost of x compact discs if each compact disc costs $6.49.

11 40 CHAPTER Real Numbers and Algebraic Expressions 72. The cost of y books if each book costs $ If x is an odd integer, represent the next odd integer as an expression in x. 74. If 2x is an even integer, represent the next even integer as an expression in x. MIXED Simplify each expression. See Examples, 4, and x + 8-0x 76. 5y y - 20y 77. 5k - k c - 4-2c2 79. x x y2-4 + y2 8. xy xy x yz + 2-7yz + + y n n t2 + 2t y2 = 5 # 725 # y a27b2 = 5a2 # 57b2 To demonstrate the distributive property geometrically, represent the area of the larger rectangle in two ways: first as width times length and second as the sum of the areas of the smaller rectangles. Example: x y Area: xy 0. b c a y z z Area: xz Area of larger rectangle: x(y z) Area of larger rectangle: xy xz Thus: x(y z) xy xz n n z z x - 2x x 88. 7n + 2n x x y y b b a a x y x x - 5y x x - 6y a - 8b2-7 7a - 2b x - y2-8 24x - 40y CONCEPT EXTENSIONS In each statement, a property of real numbers has been incorrectly applied. Correct the right-hand side of each statement. See the second Concept Check in this section. 04. b c b c b c a 05. Name the only real number that is its own opposite, and explain why this is so. 06. Name the only real number that has no reciprocal, and explain why this is so. 07. Is division commutative? Explain why or why not. 08. Is subtraction commutative? Explain why or why not. 09. Evaluate 24, 6, 2 and 24, 62,. Use these two expressions and discuss whether division is associative. 0. Evaluate and Use these two expressions and discuss whether subtraction is associative. Simplify each expression.. 8.z + 7.z y x x + 42 = x y = 4y + 8