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?

First, partition of range of $x$ into 3 parts:


*

*$x\geq\frac{3}{2}$, in this case, $f(x)=(2x+2)+(2x-3)=4x-1$. Then solve the eqation $4x-1=x+14$ and find that $x=5$.

*$-1\leq x\leq\frac{3}{2}$, then $f(x)=(2x+2)+(3-2x)=5$. Then you can find that the solution of $5=x+14$ is $x=9$, which out of the range $-1\leq x\leq\frac{3}{2}$. Therefore, this is not a solution.

*$x\leq-1$, then $f(x)=(-2x-2)+(3-2x)=1-4x$. The solution of $1-4x=x+14$ is $x=-\frac{13}{5}$, which is in the range. 
Therefore, the solution of $f(x)=g(x)$ is $x=5$ and $-\frac{13}{5}$.
