#### Answer

$\left( -3,\dfrac{1}{9} \right]$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
4 \le -9x+5 \lt 8
,$ use 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 properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
4-5 \le -9x+5-5 \lt 8-5
\\\\
-1 \le -9x \lt 3
.\end{array}
Dividing all sides by a negative number (and consequently reversing the sign), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-1}{-9} \ge \dfrac{-9x}{-9} \gt \dfrac{3}{-9}
\\\\
\dfrac{1}{9} \ge x \gt -3
\\\\
-3\lt x \le \dfrac{1}{9}
.\end{array}
In interval notation, the solution set is $
\left( -3,\dfrac{1}{9} \right]
.$
The red graph is the graph of the solution set.