Chapter 8 - XYZ Custom Plus

494Chapter 8 Real Numbers and Algebraic ExpressionsNoteAn algebraic expressiondoes not containan equal signIn this section we want to simplify expressions containing variables—that is,algebraic expressions. An algebraic expression is a combination of constants andvariables joined by arithmetic operations such as addition, subtraction, multiplicationand division.Ausing the Associative PropertyTo begin let’s review how we use the associative properties for addition and multiplicationto simplify expressions.Consider the expression 4(5x). We can apply the associative property of multiplicationto this expression to change the grouping so that the 4 and the 5 aregrouped together, instead of the 5 and the x. Here’s how it looks:4(5x) 5 (4 ?5)x Associative property5 20xMultiplication: 4 ∙ 5 = 20prACtiCE prOBlEmsMultiply.1. 5(7a)We have simplified the expression to 20x, which in most cases in algebra will beeasier to work with than the original expression.Here are some more examples.ExAmplE 17(3a) 5 (7 ? 3)a Associative property5 21a 7 times 3 is 212. 23(9x)ExAmplE 222(5x2(5x) 5 (22 ? 5)x Associative property5 210xxThe product of −2 and 5 is −103. 5(28y)ExAmplE 33(24y) 5 [3(24)] y Associative property5 212y3 times −4 is −12We can use the associative property of addition to simplify expressions also.Simplify.4. 6 1 (9 1 x)ExAmplE 43 1 (8 1 x) 5 (3 1 8) 1 x Associative property5 11 1 x The sum of 3 and 8 is 115. (3x1 7) 1 4ExAmplE 5(2x 1 5) 1 10 5 2x 1 (5 1 10) Associative property5 2x 1 15 AdditionB using the Distributive PropertyIn Chapter 1 we introduced the distributive property. In symbols it looks like this:a(b 1 c) 5 ab 1 acBecause subtraction is defined as addition of the opposite, the distributive propertyholds for subtraction as well as addition. That is,a(b 2 c) 5 ab 2 acApply the distributive property.6. 6(x1 4)Answers1. 35a 2. 227x27xx 3. 240yy4. 15 1 x 5. 3x1 11 6. 6x1 24We say that multiplication distributes over addition and subtraction. Here aresome examples that review how the distributive property is applied to expressionsthat contain variables.ExAmplE 6 4(x4( 1 5) 5 4(x) 1 4(5) Distributive property5 4x1 20 Multiplication