Answer
$\left( -\infty,\dfrac{23}{6} \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
\dfrac{2}{3}(3x-1)\ge\dfrac{3}{2}(2x-3)
,$ use the Distributive Property and the properties of inequality.
For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$
For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Using the Distributive Property and the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2}{3}(3x)+\dfrac{2}{3}(-1)\ge\dfrac{3}{2}(2x)+\dfrac{3}{2}(-3)
\\\\
2x-\dfrac{2}{3}\ge3x-\dfrac{9}{2}
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
6\left( 2x-\dfrac{2}{3} \right)\ge6\left( 3x-\dfrac{9}{2} \right)
\\\\
12x-4\ge18x-27
\\\\
12x-18x\ge-27+4
\\\\
-6x\ge-23
.\end{array}
Dividing both sides by a negative number (and consequently reversing the sign), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
x\le\dfrac{-23}{-6}
\\\\
x\le\dfrac{23}{6}
.\end{array}
The red graph is the graph of the solution set.
In interval notation, the solution set is $
\left( -\infty,\dfrac{23}{6} \right]
.$
