#### Answer

$\left( -\dfrac{5}{2},-\dfrac{1}{2} \right]$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
4 \le -2x+3 \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}
4-3 \le -2x+3-3 \lt 8-3
\\\\
1 \le -2x \lt 5
.\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}{-2} \ge \dfrac{-2x}{-2} \gt \dfrac{5}{-2}
\\\\
-\dfrac{1}{2} \ge x \gt -\dfrac{5}{2}
\\\\
-\dfrac{5}{2} \lt x \le -\dfrac{1}{2}
.\end{array}
In interval notation, the solution set is $
\left( -\dfrac{5}{2},-\dfrac{1}{2} \right]
.$
The red graph is the graph of the solution set.