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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 4 1.4 Properties of Real Numbers Objectives 1.Use the distributive property. 2.Use the inverse properties. 3.Use the identity properties. 4.Use the commutative and associative properties. 5.Use the multiplication property of 0.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 5 Using the Distributive Property The idea of the distributive property can be illustrated using rectangles. 1.4 Properties of Real Numbers 3(2 + 5) = 3 2 + 3 5 2 3 5 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 - 6 Use the Distributive Property 1.4 Properties of Real Numbers 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 ba + ca = a(b + c) = (b + c)a

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 7 Use the Distributive Property 1.4 Properties of Real Numbers The distributive property allows us to rewrite a product as a sum: or a sum as a product. –4(8 + (–3)) = –4(8) + (–4) (–3) –6(3) + –6(11) = –6(3 + 11)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 8 Use the Distributive Property 1.4 Properties of Real Numbers –6(x + 9) = 4(a + b + c) = 7(3x – 2y + 13) = –6x + (–6)(9) 4a + 4b + 4c 7(3x + (–2y) + 13) = 21x + (–14y) + 91 = 21x –14y + 91 = –6x + (–54) = –6x – 54 Product Sum

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 9 Use the Distributive Property 1.4 Properties of Real Numbers Sum Product 6w –2w + 5w = 6w + (–2)w + 5w = (6 + (–2) + 5)w = 9w 8c – 12c =(8c + (–12c)) = (8 + (–12))c = –4c The distributive property can also be used for subtraction: a(b – c) = ab – ac

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 10 Use the Distributive Property 1.4 Properties of Real Numbers The distributive property may be used to perform calculations mentally. Calculate 29 92 + 29 8. 29 92 + 29 8 =29(92 + 8) = 29(100) = 2900 Combining the 92 and 8 makes the problem much easier!

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 12 Using the Inverse Properties 1.4 Properties of Real Numbers 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 - 13 Use the Identity Properties 1.4 Properties of Real Numbers 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 - 14 Use the Identity Properties 1.4 Properties of Real Numbers –(3b + b – 7b) = –1(3 + 1 – 7)b = ((–1)3 + (–1)1 + (–1)(– 7))b = (–3 + (–1) + 7)b = 3b

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 15 Terms and Like Terms 1.4 Properties of Real Numbers Terms consist of a number or a product of a number and one or more variables. 2 and 28 227k and 2k y 2 and 4y 2 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. Like Terms

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 16 Use the Commutative Property 1.4 Properties of Real Numbers The commutative properties are used to change the order of the terms or factors in an expression. Commutative Properties For any real numbers a and b, a + b = b + a and ab = ba. Interchange the order of the two terms or factors.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 17 Use the Associative Properties 1.4 Properties of Real Numbers The associative properties are used to regroup (associate) the terms or factors in an expression, where the order stays the same. 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 18 Use the Commutative and Associative Properties 1.4 Properties of Real Numbers Simplify. –5x + 8x + 7 – 9x + 3 = (–5x + 8x) + 7 – 9x + 3 Order of Operations = (–5 + 8) x + 7 – 9x + 3 Distributive Property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 19 Use the Commutative and Associative Properties 1.4 Properties of Real Numbers Continued: Commutative Property Associative Property = [3x + (7 – 9x)] + 3 = [3x + (–9x + 7)] + 3 = [(3x + [–9x]) + 7] + 3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 20 Use the Commutative and Associative Properties 1.4 Properties of Real Numbers Continued: Combine like terms Associative Property = ( – 6x + 7) + 3 = [(3x + [–9x]) + 7] + 3 = – 6x + (7 + 3) = – 6x + 10 Add like terms 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 21 Use the Commutative and Associative Properties 1.4 Properties of Real Numbers Simplify. 4 –1(3g – 7) + 2g(h) (–3) + g = 4 –3g + 7 + 2g(h)(–3) + g = 4 –3g + 7 + (–6gh) + g Distributive Property Commutative and Associative Properties; Multiplying

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 22 Use the Commutative and Associative Properties 1.4 Properties of Real Numbers Continued: = 4 + 7 –3g + g + (–6gh) = 4 –3g + 7 + (–6gh) + g Commutative and Associative Properties Adding like terms =11 –2g – 6gh

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 23 Use the Distributive Property with Caution 1.4 Properties of Real Numbers Contined — A Second Look: 4 –1(3g – 7) + 2g(h) (–3) + g = 4 –3g + 7 + 2g(h)(–3) + g Distributive property does not apply since there is no addition or subtraction. (2g)(h) + (2g)(–3) Distributive property applies here since there is subtraction.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 24 Use the Multiplication Property of 0 1.4 Properties of Real Numbers The product of any real number and 0 is 0. Multiplication Property of 0 For any real number a, a 0 = 0 and 0 a = 0. –4 0 = 0 0 100 = 0 0 0 = 0