College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 1, Equations and Graphs - Section 1.7 - Solving Inequalities - 1.7 Exercises: 22

Answer

$(-\infty, \frac{1}{3}]$ Refer to the image below for the graph.
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Work Step by Step

Multiply the LCD of $12$ to both sides of the inequality to eliminate the fractions: $\begin{array}{ccc} &6\left(\dfrac{2}{3} - \dfrac{1}{2}x\right) &\ge &6\left(\dfrac{1}{6} +x\right) \\&6\left(\dfrac{2}{3}\right) - 6\left(\dfrac{1}{2}x\right) &\ge &6\left(\dfrac{1}{6}\right)+6(x) \\&4 - 3x &\ge &1+6x \end{array}$ Add $3x$ and subtract $1$to both sides of the inequality to obtain: $\begin{array}{ccc} &4-3x+3x -1&\ge & 1+6x+3x - 1 \\&3 &\ge &9x \end{array}$ Divide $9$ to both sides of the inequality to obtain: $\begin{array}{ccc} \\&\dfrac{3}{9} &\ge & \dfrac{9x}{9} \\&\dfrac{1}{3} &\ge &x \\&x &\le &\dfrac{1}{3} \end{array}$ Thus, the solution set is $(-\infty, \frac{1}{3}]$. To graph this solution set, plot a solid dot at $\dfrac{1}{3}$ then shade the region to its left. (refer to the attached image in the answer part above for the graph)
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