Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set - Page 600: 82


$t\le6 \text{ or } t\ge8$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inqeuality, $ |t-7|+3\ge4 ,$ isolate first the absolute value expression. Then use the definition of a greater than (greater than or equal to) absolute value inequality and solve each resulting inequality. Finally, graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Using the properties of inequality to isolate the absolute value expression results to \begin{array}{l}\require{cancel} |t-7|+3\ge4 \\\\ |t-7|\ge4-3 \\\\ |t-7|\ge1 .\end{array} 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 inequality above is equivalent to \begin{array}{l}\require{cancel} t-7\ge1 \\\\\text{OR}\\\\ t-7\le-1 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} t-7\ge1 \\\\ t\ge1+7 \\\\ t\ge8 \\\\\text{OR}\\\\ t-7\le-1 \\\\ t\le-1+7 \\\\ t\le6 .\end{array} Hence, the solution set is $ t\le6 \text{ or } t\ge8 .$
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