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

$-\dfrac{1}{8}\lt x \lt \dfrac{1}{2}$
$\bf{\text{Solution Outline:}}$ 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).