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 - Test: Chapter 9 - Page 624: 20


$t \le -\dfrac{13}{5} \text{ or } t \ge \dfrac{7}{5}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $ |-5t-3| \ge 10 ,$ 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 $\ge.$ $\bf{\text{Solution Details:}}$ 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} -5t-3 \ge 10 \\\\\text{OR}\\\\ -5t-3 \le -10 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} -5t-3 \ge 10 \\\\ -5t \ge 10+3 \\\\ -5t \ge 13 \\\\\text{OR}\\\\ -5t-3 \le -10 \\\\ -5t \le -10+3 \\\\ -5t \le -7 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to \begin{array}{l}\require{cancel} -5t \ge 13 \\\\ t \le \dfrac{13}{-5} \\\\ t \le -\dfrac{13}{5} \\\\\text{OR}\\\\ -5t \le -7 \\\\ t \ge \dfrac{-7}{-5} \\\\ t \ge \dfrac{7}{5} .\end{array} Hence, the solution set is $ t \le -\dfrac{13}{5} \text{ or } t \ge \dfrac{7}{5} .$
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