Answer
$x=-1\text{ OR }x=-\dfrac{3}{2}$
Work Step by Step
Using the properties of equality, the given equation, $
4|3x+4|=4x+8
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4|3x+4|}{4}=\dfrac{4x+8}{4}
\\\\
|3x+4|=x+2
.\end{array}
Removing the absolute value sign, the expression above is equivalent to
\begin{array}{l}\require{cancel}
3x+4=x+2
\\\\\text{ OR }\\\\
3x+4=-(x+2)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x+4=x+2
\\
3x-x=2-4
\\
2x=-2
\\
\dfrac{2x}{2}=-\dfrac{2}{2}
\\
x=-1
\\\\\text{ OR }\\\\
3x+4=-(x+2)
\\
3x+4=-x-2
\\
3x+x=-2-4
\\
4x=-6
\\
\dfrac{4x}{4}=-\dfrac{6}{4}
\\
x=-\dfrac{3}{2}
.\end{array}
Since the right side of the original equation is not a constant, then checking of solutions is required.
Substituting $
x=-1
$ in the original equation results to
\begin{array}{l}\require{cancel}
4|3x+4|=4x+8
\\
4|3(-1)+4|=4(-1)+8
\\
4|-3+4|=-4+8
\\
4|1|=4
\\
4(1)=4
\\
4=4
\text{ (TRUE)}
.\end{array}
Substituting $
x=-\dfrac{3}{2}
$ in the original equation results to
\begin{array}{l}\require{cancel}
4|3x+4|=4x+8
\\
4\left|3\left( -\dfrac{3}{2} \right)+4\right|=4\left( -\dfrac{3}{2} \right)+8
\\
4\left|-\dfrac{9}{2} +4\right|=2(-3)+8
\\
4\left|-\dfrac{9}{2} +\dfrac{8}{2}\right|=-6+8
\\
4\left|-\dfrac{1}{2}\right|=2
\\
4\left(\dfrac{1}{2}\right)=2
\\
2=2
\text{ (TRUE)}
.\end{array}
Hence, the solutions are $
x=-1\text{ OR }x=-\dfrac{3}{2}
.$
