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 - 9.2 Intersections, Unions, and Compound Inequalities - 9.2 Exercise Set - Page 590: 68


$-4 \lt t \le -\dfrac{10}{3}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $ 4 \gt g(t) \ge 2 ,$ replace the function with the given, $ g(t)=-3t-8 .$ Then use the properties of inequality to isolate the variable. Finally, graph the solution set. $\bf{\text{Solution Details:}}$ Replacing the inequality with the given function, then \begin{array}{l}\require{cancel} 4 \gt -3t-8 \ge 2 .\end{array} Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} 4 \gt -3t-8 \ge 2 \\\\ 4+8 \gt -3t-8+8 \ge 2+8 \\\\ 12 \gt -3t \ge 10 .\end{array} Multiplying both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to \begin{array}{l}\require{cancel} 12 \gt -3t \ge 10 \\\\ \dfrac{12}{-3} \gt \dfrac{-3t}{-3} \ge \dfrac{10}{-3} \\\\ -4 \lt t \le -\dfrac{10}{3} .\end{array} The graph includes the points from $-4$ (exclusive) to $-\dfrac{10}{3}$ (inclusive).
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