#### Answer

$x=\left\{ -1,-\dfrac{1}{2} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|4x+3|-2=-1
,$ use the properties of equality to isolate the absolute value expression. Then use the properties of absolute value equality.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
|4x+3|=-1+2
\\\\
|4x+3|=1
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
4x+3=1
\\\\\text{OR}\\\\
4x+3=-1
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
4x+3=1
\\\\
4x=1-3
\\\\
4x=-2
\\\\
x=-\dfrac{2}{4}
\\\\
x=-\dfrac{1}{2}
\\\\\text{OR}\\\\
4x+3=-1
\\\\
4x=-1-3
\\\\
4x=-4
\\\\
x=-\dfrac{4}{4}
\\\\
x=-1
.\end{array}
Hence, $
x=\left\{ -1,-\dfrac{1}{2} \right\}
.$