Section 1.3 Properties of Real Numbers and Simplifying Algebraic Expressions 35 Distributive Property Example (a = 2, b = 3, c = 4) For all real numbers a, b, and c, a(b + c) = ab + ac 2(3 + 4) = 2(3) + 2(4) 2(7) = 6 + 8 14 = 14 Objective 2 Examples Apply the commutative, associative, or distributive properties to rewrite each expression as an equivalent expression and then simplify the equivalent expression. 2a. 2 + y + 7 2b. 2 (y)(7) 2c. (x + 6) + 4 2d. 4(6x) 2e. 4(x + 6) 2f. -2(3a + 5) 2g. -(6x - 9) 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. 2e. 4(x + 6) = 4(x) + 4(6) Apply the distributive property. = 4x + 24 Simplify each product. 2f. -2(3a + 5) = -2(3a) + (-2)(5) Apply the distributive property. =-6a - 10 Simplify each product. 2g. -(6x - 9) = -1(6x - 9) Write as a product of -1 and 6x - 9. = -1(6x) + (-1)(-9) Apply the distributive property. =-6x + 9 Simplify each product. Student Check 2 Apply the commutative, associative, or distributive properties to rewrite each expression as an equivalent expression and then simplify the equivalent expression. a. 3 + x + 5 b. 3(x)(5) c. (b + 2) + 9 d. 2(9b) e. 8(y +3) f. -7(6a + 4) g. -(2y - 1) Simplifying Algebraic Expressions The properties just discussed provide the framework for simplifying algebraic expressions. To simplify an expression, we may have to clear parentheses by applying the distributive property and then we combine any like terms. The terms of an expression are the addends of the expression. For example, in the expression 6x + 2, the terms are 6x and 2. Recall like terms are terms with identical variables raised to the same exponents. To combine like terms, we combine their numerical coefficients and keep the variable the same. The coefficient of a term is the number that is multiplied by the variable. Recall 4x + 2x = x + x + x + x 4x + x + x 2x =6x ('')''* ()* Objective 3 ▶ Simplify algebraic expressions.
hendricks_intermediate_algebra_1e_ch1_3
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