Calculate absolute values with unknown constant I am to calculate all $x$ if $f(x) = g(x)$ and if

$$f(x)= |2x+2| + |3-2x|$$
$$g(x)= x + 14$$

How do I mix regular numbers with absolute values in such a sense? I thought I could calculate it like this:

$$|2x+2| + |3-2x| = 3x+2 + 3+2x = 3x+5$$

But then I realised that the end value of either of the absolute values is determined by $x$. It feels like a catch 22: how do I calculate this equation without knowing the value of $x$ until I've calculated it?
HINT: You can assume three cases:


$x \geq \dfrac 32$
$-1&lt;x&lt; \dfrac 32$
$x \leq -1$
1.
Find the roots of all |A| expressions(total of m):  

x=-1, x=3/2  


2.Partition the $\mathcal{R}$ (if you want to solve in Real numbers set) into m+1 sub domains  

I    x &lt;- 1  
II   -1 =&lt; x &lt;= 3/2  
III  x&gt;3/2 


3.Find the sign of |A| in each sub domain and solve for x
$I:$
$-(2x+2) + (3-2x) = x+14$
$-2x -2 + 3 -2x = x+14$
$-4x+1 = x+14$
$5x=-13$
$x=-13/5 \in I\checkmark$
$II:$
$(2x+2) + (3-2x) = x+14$
$+2x +2 + 3 -2x = x+14$
$5 = x+14$
$x=-9$
$x=-9 \notin II$
$III:$
$(2x+2)+ [-(3-2x)] = x+14$
$2x+2-3 +2x = x+14$
$4x-1 = x+14$
$3x=15$
$x=5 \in III\checkmark$

NOTE: in each sub domain check if the answer is in its corresponding sub domain