1. Chapter 1 Section 6

2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Multiplying and Dividing Real Numbers Find the product of a positive number and a negative number. Find the product of two negative numbers. Identify factors of integers. Use the reciprocal of a number to apply the definition of division. Use the rules for order of operations when multiplying and dividing signed numbers. Evaluate expressions involving variables. Translate words and phrases involving multiplication and division. Translate simple sentences into equations. 1.6 2 3 5 4 6 7 8

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The result of multiplication is called the product. We already know how to multiply positive numbers, and we know that the product of two positive numbers is positive. We also know that the product of 0 and any positive number is 0, so we extend that property to all real numbers. Multiplication by Zero For any real number x, Slide 1.6-3 Definition

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the product of a positive and a negative number. Objective 1 Slide 1.6-4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The product of a 3(−1) represents the sum Also, and These results maintain the pattern which suggests a new rule. Multiplying Numbers with Different Signs For any positive real numbers x and y, and. That is, the product of two numbers with opposite signs is negative. Slide 1.6-5 Find the product of a positive number and a negative number.

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Find the product. Slide 1.6-6 EXAMPLE 1 Multiplying a Positive Number and a Negative Number

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the product of two negative numbers. Objective 2 Slide 1.6-7

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Multiplying Two Negative Numbers For any positive real numbers x and y, That is, the product of two negative numbers is positive. Example: Slide 1.6-8 Find the product of two negative numbers.

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Find the product. Slide 1.6-9 EXAMPLE 2 Multiplying Two Negative Numbers

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Identify factors of integers. Objective 3 Slide 1.6-10

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. In Section 1.1, the definition of a factor was given for whole numbers. The definition can be extended to integers. If the product of two integers is a third integer, then each of the two integers is a factor of the third. The table below show several examples of integers and factors of those integers. Slide 1.6-11 Identify factors of integers.

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the reciprocal of a number to apply the definition of division. Objective 4 Slide 1.6-12

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The quotient, of two numbers is found by multiplying by the reciprocal, or multiplicative inverse, of the second number. The following table shows several numbers and their multiplicative inverses. Reciprocal or Multiplicative Inverse Pairs of numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other. Slide 1.6-13 Use the reciprocal of a number to apply the definition of division.

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Definition of Division For any real numbers x and y, with y ≠ 0, That is, to divide two numbers, multiply the first by the reciprocal, or multiplicative inverse, of the second. Slide 1.6-14 Use the reciprocal of a number to apply the definition of division. (cont’d) Recall that an equivalent form of is where x is called the dividend and y is called the divisor.

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Find each quotient, using the definition of division. Slide 1.6-15 EXAMPLE 3 Using the Definition of Division

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. When dividing fractions, multiplying by the reciprocal works well. However, using the definition of division directly with integers is awkward. It is easier to divide in the usual way and then determine the sign of the answer. Slide 1.6-16 Use the reciprocal of a number to apply the definition of division. (cont’d) Division Involving 0 For any real numbers x, with x ≠ 0, Dividing Signed Numbers The quotient of two numbers having the same sign is positive. The quotient of two numbers having different signs is negative.

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each quotient. Solution: Slide 1.6-17 EXAMPLE 4 Dividing Signed Numbers

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Equivalent Forms For any positive real numbers x and y, Slide 1.6-18 Use the reciprocal of a number to apply the definition of division. (cont’d) Equivalent Forms For any positive numbers x and y,

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the rules for order of operations when multiplying and dividing signed numbers. Objective 5 Slide 1.6-19

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Perform each indicated operation. Slide 1.6-20 EXAMPLE 5 Using the Rules for Order of Operations

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate expressions involving variables. Objective 6 Slide 1.6-21

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate if and. Solution: Slide 1.6-22 EXAMPLE 6 Evaluating Expressions for Numerical Values

23. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Translate words and phrases involving multiplication and division. Objective 7 Slide 1.6-23

24. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The word product refers to multiplication. The table gives other key words and phrases that indicate multiplication in problem solving. Slide 1.6-24 Translate words and phrases involving multiplication and division.

25. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write a numerical expression for the phrase, and simplify the expression. Three times the difference between 4 and −11. Three-fifths of the sum of 2 and −7. Solution: Slide 1.6-25 EXAMPLE 7 Translating Words and Phrases (Multiplication)

26. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The word quotient refers to division. In algebra, quotients are usually represented with a fraction bar; the symbol ÷ is seldom used. The table gives some key phrases associated with division. Slide 1.6-26 Translate words and phrases involving multiplication and division.

27. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write a numerical expression for the phrase, and simplify the expression. The product of −9 and 2, divided by the difference between 5 and −1. Solution: Slide 1.6-27 EXAMPLE 8 Interpreting Words and Phrases Involving Division

28. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Translate simple sentences into equations. Objective 8 Slide 1.6-28

29. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write each sentence as an equation, using x as the variable. Then find the solution from the list of integers between −12 and 12, inclusive. The quotient of a number and −2 is 6. Twice a number is −6. Solution: Slide 1.6-29 EXAMPLE 9 Translating Sentences into Equations