72 Chapter 1 Real Numbers and Algebraic Expressions Student Check 1 Find both the additive and multiplicative inverses of each number. Assume any variables are nonzero. a. -10 b. 7 8 c. 4y d. -9b e. a 3 The Commutative and Associative Properties Additional properties of the real numbers relate to how we add and multiply the numbers. These properties form the foundation of how we work with algebraic expressions. The commutative property of the real numbers states that the order in which we add real numbers or multiply real numbers doesn’t change the result. Commutative Property Examples (a = 2, b = 3) Addition For all real numbers a and b, a + b = b + a 2 + 3 = 3 + 2 = 5 Multiplication For all real numbers a and b, a · b = b · a 2 · 3 = 3 · 2 = 6 The associative property of the real numbers states that the way numbers are grouped when they are added or multiplied doesn’t change the outcome. Associative Property Examples (a = 2, b = 3, c = 4) Addition For all real numbers a, b, and c, a + (b + c) = (a + b) + c 2 + (3 + 4) = (2 + 3) + 4 2 + (7) = (5) + 4 9 = 9 Multiplication For all real numbers a, b, and c, (a · b) · c = a · (b · c) (2 · 3) · 4 = 2 · (3 · 4) (6) · 4 = 2 · (12) 24 = 24 �� ������������������������ ������������������ Apply the commutative and associative properties to rewrite each expression and simplify the result. 2a. 2 + y + 7 2b. 2( y)(7) 2c. (x + 6) + 4 2d. 4(6x) Solutions 2a. 2 + y + 7 = y + 2 + 7 Apply the commutative property of addition. = y + 9 Add the numbers. 2b. 2( y)(7) = 2(7)y Apply the commutative property of multiplication. = 14y Multiply the numbers. 2c. (x + 6) + 4 = x + (6 + 4) Apply the associative property of addition. = x + 10 Add the numbers. �� 2d. 4(6x) = (4 · 6)x Apply the associative property of multiplication. = 24x Multiply the numbers. Student Check 2 Apply the commutative or associative properties to rewrite each expression and simplify the result. a. 3 + x + 5 b. 3(x)(5) c. (b + 2) + 9 d. 2(9b) Objective 2 ▶ Apply the commutative and associative properties.
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