Copyright © 2016, 2013, and 2010, Pearson Education, Inc. 5 Chapter Integers Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

5-2 Multiplication and Division of Integers Models for multiplication of integers. Properties of multiplication of integers. Integer division. Order of operations on integers. Inequalities with integers.

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

Integer Multiplication Models Chip Model Charged-Field Model

Integer Multiplication Models 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 Charged-Field Model

Integer Multiplication Models 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 Number-Line Model

Integer Multiplication Models Number-Line Model

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|.

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).

Properties of Integer Multiplication Identify property of multiplication 1 is the unique integer such that for all integers a, 1 · a = a = a · 1. 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.

Properties of Integer Multiplication For all integers a, b, and c, (–1)a = –a. (–a)b = b(–a) = –(ab). (–a)(–b) = ab. Distributive property of multiplication over subtraction for integers a(b – c) = ab – ac and (b – c)a = ba – ca.

Example 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

Example(continued) b. (a + b)(a − b) This result is called the difference-of-squares formula.

Example Use the difference-of-squares formula to simplify the following: a. (4 + b)(4 − b) (4 + b)(4 − b) = 42 − b2 = 16 − b2 b. (−4 + b)(−4 − b) (−4 + b)(− 4 − b) = (−4)2 − b2 = 16 − b2

Example (continued) Use the difference-of-squares formula to simplify the following: c.

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.

Example Factor each of the following completely: a. b. c.

Example (continued) d. e.

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.

Example 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

Example (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.

Example(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

Order of Operations on Integers When addition, subtraction, multiplication, division, and exponentiation appear without parentheses: Exponentiation is done first. Multiplication and division in the order of their appearance from left to right. 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.

Example Evaluate each of the following: a. b. c.

Example (continued) d. e. f.

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 < b (or equivalently, b > a) if, and only if, b − a is equal to a positive integer; that is, b − a is greater than 0.

Ordering Integers Properties of Inequalities of Integers If x < y, and n is any integer, then x + n < y + n. If x < y, then −x > −y. If x < y and n > 0, then nx < ny. If x < y and n < 0, then nx > ny.

Example Find all integers x that satisfy the following: a. x + 3 < −2 x + 3 < −2  x + 3 + −3 < −2 + −3  x < −5, x is an integer. b. −x − 3 < 5 −x − 3 < 5  −x − 3 + 3 < 5 + 3  −x < 8  x > −8, x is an integer.

Example(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, …}.