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 592: 111


$-\dfrac{1}{8}\lt x \lt \dfrac{1}{2}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the properties of inequality to solve the given inequality, $ 3x\lt4-5x\lt5+3x .$ Then 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 equality, the given is equivalent to \begin{array}{l}\require{cancel} 3x\lt4-5x\lt5+3x \\\\ 3x-3x\lt4-5x-3x\lt5+3x-3x \\\\ 0\lt4-8x\lt5 \\\\ 0-4\lt4-8x-4\lt5-4 \\\\ -4\lt-8x\lt1 .\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} -4\lt-8x\lt1 \\\\ \dfrac{-4}{-8}\lt\dfrac{-8x}{-8}\lt\dfrac{1}{-8} \\\\ \dfrac{1}{2}\gt x \gt -\dfrac{1}{8} \\\\ -\dfrac{1}{8}\lt x \lt \dfrac{1}{2} .\end{array} The graph consists of all points from $-\dfrac{1}{8}$ (exclusive) to $\dfrac{1}{2}$ (exclusive).
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