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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1. 4 - 1
Chapter 1 Review of the Real Number System Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1. 4 - 2
1. 4 Properties of Real Numbers Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1. 4 - 3
1. 4 Properties of Real Numbers Objectives 1. 2. 3. 4. Use the distributive property. Use the inverse properties. Use the identity properties. Use the commutative and associative properties. 5. Use the multiplication property of 0. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 4
1. 4 Properties of Real Numbers Using the Distributive Property The idea of the distributive property can be illustrated using rectangles. 3(2 + 5) = 3 • 2 + 3 • 5 2 5 3 3 Area of left part is 3 • 2 = 6 Area of right part is 3 • 5 = 15 Area of total rectangle is 3(2 + 5) = 21 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 5
1. 4 Properties of Real Numbers Use the Distributive Property For any real numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca. The distributive property can also be written as: ab + ac = a(b + c) ba + ca = (b + c)a Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 6
1. 4 Properties of Real Numbers Use the Distributive Property The distributive property allows us to rewrite a product as a sum: – 4(8 + (– 3)) = – 4(8) + (– 4) (– 3) or a sum as a product. – 6(3) + – 6(11) = – 6(3 + 11) Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 7
1. 4 Properties of Real Numbers Use the Distributive Property Product Sum – 6(x + 9) = – 6 x + (– 6)(9) = – 6 x + (– 54) = – 6 x – 54 4(a + b + c) = 4 a + 4 b + 4 c 7(3 x – 2 y + 13) = 7(3 x + (– 2 y) + 13) = 21 x + (– 14 y) + 91 = 21 x – 14 y + 91 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 8
1. 4 Properties of Real Numbers Use the Distributive Property Sum 8 c – 12 c = (8 c + (– 12 c)) = (8 + (– 12))c Product The distributive property can also be used for subtraction: a(b – c) = ab – ac = – 4 c 6 w – 2 w + 5 w = 6 w + (– 2)w + 5 w = (6 + (– 2) + 5)w = 9 w Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 9
1. 4 Properties of Real Numbers Use the Distributive Property The distributive property may be used to perform calculations mentally. Calculate 29 • 92 + 29 • 8 = 29(92 + 8) = 29(100) Combining the 92 and 8 makes the problem much easier! = 2900 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 10
1. 4 Properties of Real Numbers Using the Inverse Properties Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 11
1. 4 Properties of Real Numbers Using the Inverse Properties Complete the following statements. 5 – 19 3 11 – 5 Zero does not have a multiplicative inverse. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 12
1. 4 Properties of Real Numbers Use the Identity Properties For any real numbers a, a+0=0+a=a a · 1 = 1 · a = a. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 13
1. 4 Properties of Real Numbers Use the Identity Properties –(3 b + b – 7 b) = – 1(3 + 1 – 7)b = ((– 1)3 + (– 1)1 + (– 1)(– 7))b = (– 3 + (– 1) + 7)b = 3 b Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 14
1. 4 Properties of Real Numbers Terms and Like Terms consist of a number or a product of a number and one or more variables. 2 and 28 Like Terms 227 k and 2 k Like Terms 2 y and 4 y 2 Like Terms Like terms are numbers or numbers times variables raised to exactly the same power. Simplifying expressions is called combining like terms. Only like terms can be combined. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 15
1. 4 Properties of Real Numbers Use the Commutative Property Commutative Properties For any real numbers a and b, and a+b=b+a ab = ba. Interchange the order of the two terms or factors. The commutative properties are used to change the order of the terms or factors in an expression. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 16
1. 4 Properties of Real Numbers Use the Associative Properties For any real numbers a, b and c, a + (b + c) = (a + b) + c and a(bc) = (ab)c. Shift parentheses among three terms or factors; order stays the same. The associative properties are used to regroup (associate) the terms or factors in an expression, where the order stays the same. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 17
1. 4 Properties of Real Numbers Use the Commutative and Associative Properties Simplify. – 5 x + 8 x + 7 – 9 x + 3 Order of Operations = (– 5 x + 8 x) + 7 – 9 x + 3 Distributive Property = (– 5 + 8) x + 7 – 9 x + 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 18
1. 4 Properties of Real Numbers Use the Commutative and Associative Properties Continued: = [3 x + (7 – 9 x)] + 3 Commutative Property = [3 x + (– 9 x + 7)] + 3 Associative Property = [(3 x + [– 9 x]) + 7] + 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 19
1. 4 Properties of Real Numbers Use the Commutative and Associative Properties Continued: = [(3 x + [– 9 x]) + 7] + 3 Combine like terms = (– 6 x + 7) + 3 Associative Property = – 6 x + (7 + 3) Add like terms = – 6 x + 10 Copyright © 2010 Pearson Education, Inc. All rights reserved. In actual practice many of these steps are not actually written down, but you should mentally justify each step whether it is written down or not. Sec 1. 4 - 20
1. 4 Properties of Real Numbers Use the Commutative and Associative Properties Simplify. 4 – 1(3 g – 7) + 2 g(h) (– 3) + g Distributive Property = 4 – 3 g + 7 + 2 g(h)(– 3) + g Commutative and Associative Properties; Multiplying = 4 – 3 g + 7 + (– 6 gh) + g Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 21
1. 4 Properties of Real Numbers Use the Commutative and Associative Properties Continued: = 4 – 3 g + 7 + (– 6 gh) + g Commutative and Associative Properties = 4 + 7 – 3 g + (– 6 gh) Adding like terms =11 – 2 g – 6 gh Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 22
1. 4 Properties of Real Numbers Use the Distributive Property with Caution Contined — A Second Look: 4 – 1(3 g – 7) + 2 g(h) (– 3) + g Distributive property applies here since there is subtraction. = 4 – 3 g + 7 + 2 g(h)(– 3) + g Distributive property does not apply since there is no addition or subtraction. (2 g)(h) + (2 g)(– 3) Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1. 4 - 23
1. 4 Properties of Real Numbers Use the Multiplication Property of 0 For any real number a, a • 0=0 and 0 • a = 0. The product of any real number and 0 is 0. – 4 • 0 = 0 0 • 100 = 0 Copyright © 2010 Pearson Education, Inc. All rights reserved. 0 • 0=0 Sec 1. 4 - 24
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