Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.5 Real Numbers and Their Properties; Clock Addition - Exercise Set 5.5 - Page 309: 56

Answer

Properties used are distributive, commutative, associative and commutative respectively.

Work Step by Step

Consider the expression and evaluate it: \[7+2\left( x+9 \right)=7+\left( 2x+18 \right)\ \ \ (why?)\] \[2\]is distributed over\[x\]and\[9\]. Thus, distributive property is used. \[\begin{align} & 7+2\left( x+9 \right)=7+\left( 2x+18 \right)\ \ \ (\text{distributive property}) \\ & =7+\left( 18+2x \right)\ \ \ (why?) \end{align}\] Order of addition is changed in \[\left( 2x+18 \right)\]. Thus, commutative property of addition is used. \[\begin{align} & 7+2\left( x+9 \right)=7+\left( 2x+18 \right)\ \ \ (\text{distributive property}) \\ & =7+\left( 18+2x \right)\ \ \ (commutative\ property) \\ & =\left( 7+18 \right)+2x\ \ \ (why?) \end{align}\] Associative property of addition is used as \[\left( 7+18 \right)\]is added first. \[\begin{align} & \left( 7+18 \right)+2x=25+2x \\ & =2x+25\ \ \ (why?) \end{align}\] Order of addition is changed in \[25+2x\]. Thus, commutative property of addition is used. Hence, the properties used are distributive, commutative, associative and commutative respectively.
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