1. Chapter Integers 5 5 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

2. 5-2Multiplication and Division of Integers Models Properties of Integer Multiplication Integer Division Order of Operations on Integers Ordering Integers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3. Integer Multiplication Models Pattern Model First find (3)( − 2) using repeated addition: (3)( − 2) = − 2 + − 2 + − 2 = − 6 Now use the commutative property to find ( − 2)(3): ( − 2)(3) = (3)( − 2) = − 6 To find ( − 3)( − 2) follow the pattern: 3( − 2) = − 6 2( − 2) = − 4 1( − 2) = − 2 0( − 2) = 0 − 1( − 2) = − 2( − 2) = − 3( − 2) = 2 4 6 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4. Chip Model Integer Multiplication Models Charged-Field Model Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

5. Chip Model To find ( − 3)( − 2) = 6, start with a value of 0 that includes at least 6 red chips, then remove 6 red chips. Integer Multiplication Models Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

6. Charged-Field Model Integer Multiplication Models Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

7. Number-Line Model Demonstrate multiplication by using a hiker moving along a number line. Traveling to the left (west) means moving in the negative direction, and traveling to the right (east) means moving in the positive direction. Time in the future is denoted by a positive value, and time in the past is denoted by a negative value. Integer Multiplication Models Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

8. Number-Line Model Integer Multiplication Models Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

9. Number-Line Model Integer Multiplication Models Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

10. Integer Multiplication For any integers a and b, 1. If a  0 and b  0 (or a  0 and b  0), then ab = |a||b|. 2. If one of a or b is less than 0 while the other is greater than or equal to 0, then ab = − |a||b|. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

11. Properties of Integer Multiplication For all integers a, b, c  I, the set of integers: Closure property of multiplication of integers ab is a unique integer. Commutative property of multiplication of integers ab = ba. Associative property of multiplication of integers (ab)c = a(bc). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

12. Properties of Integer Multiplication Distributive properties of multiplication over addition for integers a(b + c) = ab + ac and (b + c)a = ba + ca. Zero multiplication property of integers 0 is the unique integer such that for all integers a, 0 · a = 0 = a · 0. Identify property of multiplication 1 is the unique integer such that for all integers a, 1 · a = a = a · 1. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

13. Properties of Integer Multiplication For all integers a, b, and c, 1.( – 1)a = – a. 2.( – a)b = b( – a) = – (ab). 3.( – a)( – b) = ab. Distributive property of multiplication over subtraction for integers a(b – c) = ab – ac and (b – c)a = ba – ca. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

14. Example 5-10 Simplify each of the following so that there are no parentheses in the final answer: a. − 3(x − 2) − 3(x − 2) = − 3x − ( − 3)(2) = − 3x − ( − 6) = − 3x + 6 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

15. Example 5-10 (continued) This result is called the difference-of-squares formula. b.(a + b)(a − b) Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

16. Example 5-11 Use the difference-of-squares formula to simplify the following: a. (4 + b)(4 − b) (4 + b)(4 − b) = 4 2 − b 2 = 16 − b 2 b. ( − 4 + b)( − 4 − b) ( − 4 + b)( − 4 − b) = ( − 4) 2 − b 2 = 16 − b 2 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

17. Example 5-11 (continued) Use the difference-of-squares formula to simplify the following: c. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

18. Factoring When the distributive property of multiplication over subtraction is written in reverse order as ab – ac = a(b – c) and ba – ca = (b – c)a and similarly for addition, the expressions on the right of each equation are in factored form. The common factor a has been factored out. Both the difference-of-squares formula and the distributive properties of multiplication over addition and subtraction can be used for factoring. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

19. Example 5-12 Factor each of the following completely: a. b. c. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

20. Example 5-12 (continued) d. e. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

21. Integer Division Definition of Integer Division If a and b are any integers, then a ÷ b is the unique integer c, if it exists, such that a = bc. The quotient of two negative integers, if it exists, is a positive integer. The quotient of a positive and a negative integer, if it exists, is a negative integer. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

22. Example 5-13 Use the definition of integer division, if possible, to evaluate each of the following: a. 12 ÷ ( − 4) Let 12 ÷ ( − 4) = c. Then 12 = − 4c  c = − 3. 12 ÷ ( − 4) = − 3 b. − 12 ÷ 4 Let − 12 ÷ 4 = c. Then − 12 = 4c  c = − 3. − 12 ÷ 4 = − 3 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

23. Example 5-13 (continued) c. − 12 ÷ ( − 4) Let − 12 ÷ ( − 4) = c. Then − 12 = − 4c  c = 3. − 12 ÷ ( − 4) = 3 d. − 12 ÷ 5 Let − 12 ÷ 5 = c. Then − 12 = 5c. Because no integer c exists to satisfy this equation, − 12 ÷ 5 is undefined over the set of integers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

24. Example 5-13 (continued) e. (ab) ÷ b, b ≠ 0 Let (ab) ÷ b = x. Then ab = bx  x = a. (ab) ÷ b = a f. (ab) ÷ a, a ≠ 0 Let (ab) ÷ a = x. Then ab = ax  x = b. (ab) ÷ a = b Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

25. Order of Operations on Integers When addition, subtraction, multiplication, division, and exponentiation appear without parentheses: 1.Exponentiation is done first. 2.Multiplication and division in the order of their appearance from left to right. 3.Finally, addition and subtraction in the order of their appearance from left to right. Arithmetic operations inside parentheses must be done first according to rules 1–3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

26. Example 5-14 Evaluate each of the following: a. b. c. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

27. Example 5-14 (continued) d. e. f. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

28. Ordering Integers Definition of Less Than for Integers For any integers a and b, a is less than b, written a < b, if, and only if, there exists a positive integer k such that a + k = b. a a) if, and only if, b − a is equal to a positive integer; that is, b − a is greater than 0. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

29. Ordering Integers Properties of Inequalities of Integers 1.If x < y, and n is any integer, then x + n < y + n. 2.If x − y. 3.If x 0, then nx < ny. 4.If x ny. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

30. Example 5-15 Find all integers x that satisfy the following: a.x + 3 < − 2 b. − x − 3 < 5 x + 3 < − 2  x + 3 + − 3 < − 2 + − 3  x < − 5, x is an integer. − x − 3 − 8, x is an integer. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

31. Example 5-15 (continued) c.If x ≤ − 2, find the values of 5 − 3x. x ≤ − 2  − 3x ≥ ( − 3)( − 2)  − 3x ≥ 6 5 + − 3x ≥ 5 + 6  5 + − 3x ≥ 11 That is, all integers in the set {11, 12, 13, 14, …}. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.