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 - Page 148: 31

Answer

$(2.5, 5.5]$ Refer to the image below for the graph.
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Work Step by Step

Multiply $12$ to each part of the inequality to eliminate the fractions: $\begin{array}{ccccc} &12\left(\dfrac{2}{3}\right) &\ge &12\left(\dfrac{2x-3}{12}\right) &\gt &12\left(\dfrac{1}{6}\right) \\&8 &\ge &2x-3 &\gt &2 \end{array}$ Add $3$ to each part of the inequality to obtain: $\begin{array}{ccccc} &8+3 &\ge &2x-3+3 & \gt &2+3 \\&11 &\ge &2x &\gt &5 \end{array}$ Divide each part by $2$ to obtain: $\begin{array}{ccccc} &\dfrac{11}{2} &\ge &\dfrac{2x}{2} &\gt &\dfrac{5}{2} \\&5.5 &\ge &x &\gt &2.5 \end{array}$ This inequality is equivalent to: $2.5 \lt x \le 5.5$ Thus, the solution set is $\bf(2.5 , 5.5]$. To graph this solution set, a hole (hollow dot) at $2.5$ and a solid dot at $5.5$ then shade the region in between. (refer to the attached image in the answer part above for the graph)
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