Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 1 - Expressions, Equations, and Inequalities - 1-6 Absolute Value Equations and Inequalities - Practice and Problem-Solving Exercises - Page 46: 31

Answer

$x\lt-12\text{ OR }x\gt6$ Refer to the graph below.

Work Step by Step

Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the given inequality, $ |x+3|\gt9 ,$ implies \begin{align*}\require{cancel} x+3&\gt9 \\\\\text{OR}\\\\ x+3&\lt-9 .\end{align*} Using the properties of inequality, the inequality above is equivalent to \begin{align*}\require{cancel} x+3&\gt9 \\ x+3-3&\gt9-3 \\ x&\gt6 \\\\\text{OR}\\\\ x+3&\lt-9 \\ x+3-3&\lt-9-3 \\ x&\lt-12 .\end{align*} Hence, the solution is $ x\lt-12\text{ OR }x\gt6 .$ Since a hollowed dot is used for the symbols $\lt$ and $\gt,$ while a solid dot is used for the symbols $\le$ and $\ge,$ then the graph of the solution above is the set of numbers to the left of $ -12 $ and to the right of $ 6 $ with hollowed dots at $ -12 $ and $ 6 $.
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