2. CHAPTER 9 Introduction to Real Numbers and Algebraic Expressions Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.1Introduction to Algebra 9.2The Real Numbers 9.3Addition of Real Numbers 9.4Subtraction of Real Numbers 9.5Multiplication of Real Numbers 9.6Division of Real Numbers 9.7Properties of Real Numbers 9.8Simplifying Expressions; Order of Operations

3. OBJECTIVES 9.7 Properties of Real Numbers Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind equivalent fraction expressions and simplify fraction expressions. bUse the commutative and associative laws to find equivalent expressions. cUse the distributive laws to multiply expressions like 8 and x – y. dUse the distributive laws to factor expressions like 4x – 12 + 24y. e Collect like terms.

4. 9.7 Properties of Real Numbers EQUIVALENT EXPRESSIONS Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two expressions that have the same value for all allowable replacements are called equivalent.

5. 9.7 Properties of Real Numbers THE IDENTITY PROPERTY OF 0 Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

6. 9.7 Properties of Real Numbers THE IDENTITY PROPERTY OF 1 Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

7. EXAMPLE 9.7 Properties of Real Numbers a Find equivalent fraction expressions and simplify fraction expressions. 2 Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8. 9.7 Properties of Real Numbers THE COMMUTATIVE LAWS Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Addition. For any numbers a and b, (We can change the order when adding without affecting the answer.) Multiplication. For any numbers a and b, (We can change the order when multiplying without affecting the answer.)

9. EXAMPLE 9.7 Properties of Real Numbers b Use the commutative and associative laws to find equivalent expressions. 6 Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Use the commutative laws to write an equivalent expression:

10. EXAMPLE 9.7 Properties of Real Numbers b Use the commutative and associative laws to find equivalent expressions. 6 Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. a) An expression equivalent to y + 5 is 5 + y by the commutative law of addition. b) An expression equivalent to mn is nm by the commutative law of multiplication. c) An expression equivalent to 7 + xy is xy + 7 by the commutative law of addition. Another expression equivalent to 7 + xy is 7 + yx by the commutative law of multiplication. Another equivalent expression is yx + 7.

11. 9.7 Properties of Real Numbers THE ASSOCIATIVE LAWS Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

12. EXAMPLE 9.7 Properties of Real Numbers b Use the commutative and associative laws to find equivalent expressions. 9 Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Use an associative law to write an equivalent expression: a) An expression equivalent to (y + z) + 3 is y + (z + 3) by the associative law of addition. b) An expression equivalent to 8(xy) is (8x)y by the associative law of multiplication.

13. EXAMPLE 9.7 Properties of Real Numbers b Use the commutative and associative laws to find equivalent expressions. 11 Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Use the commutative and associative laws to write at least three expressions equivalent to (3x)y.

14. EXAMPLE 9.7 Properties of Real Numbers b Use the commutative and associative laws to find equivalent expressions. 11 Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

15. 9.7 Properties of Real Numbers THE DISTRIBUTIVE LAW OF MULTIPLICATION OVER ADDITION Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

16. 9.7 Properties of Real Numbers THE DISTRIBUTIVE LAW OF MULTIPLICATION OVER SUBTRACTION Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

17. 9.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Terms are separated by addition signs. If there are subtraction signs, we can find an equivalent expression that uses addition signs.

18. EXAMPLE 9.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. 13 Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

19. EXAMPLE 9.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. 16 Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Multiply.

20. EXAMPLE 9.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Name the property or law illustrated by each equation.

21. EXAMPLE 9.7 Properties of Real Numbers c Use the distributive laws to multiply expressions like 8 and x – y. Slide 21Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Name the property or law illustrated by each equation.

22. 9.7 Properties of Real Numbers d Use the distributive laws to factor expressions like 4x – 12 + 24y. Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Factoring is the reverse of multiplying. To factor, we can use the distributive laws in reverse.

23. 9.7 Properties of Real Numbers FACTORING Slide 23Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. To factor an expression is to find an equivalent expression that is a product.

24. EXAMPLE 9.7 Properties of Real Numbers d Use the distributive laws to factor expressions like 4x – 12 + 24y. Slide 24Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Factor.

25. 9.7 Properties of Real Numbers e Collect like terms. Slide 25Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Terms whose variable factors are exactly the same, terms of constants, and terms whose variables are raised to the same power are called like terms. The process of collecting like terms is also based on the distributive laws. We can apply a distributive law when a factor is on the right because of the commutative law of multiplication.

26. EXAMPLE 9.7 Properties of Real Numbers e Collect like terms. Slide 26Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Collect like terms. Try to write just the answer, if you can.