Reformulate an optimization problem with absolute values as a linear program

Reformulate the following optimization problem as a linear program.

$$\begin{array}{ll} \text{minimize} & 2 x_1 + 3 |x_2-10|\\ \text{subject to} & |x_1+2|+|x_2|\leq 5\end{array}$$

First, I tried the following. If $$y=x_1+2=y^+-y^-$$, then $$|x_1+2|=y^++y^-$$, where $$y^+, y^-\geq 0$$. Now, if $$z=x_2-10=z^+-z^-$$, then $$|x_2-10|=z^++z^-$$, $$z^+, z^- \geq 0$$.

But the problem appear when I try to express $$|x_2|$$ in terms of $$z^+, z^-$$, because I get an inequality of $$|x_2|$$ and not an equality, i.e, $$|x_2|=|z^+-z^-+10|\leq |z^+-z^-|+10= z^++z^-+10$$.

Could you give me a suggestion?

If $$w = x_2 = w^+ - w^-$$ then $$|x_2| = w^+ + w^-$$, $$w^+, w^- \ge 0$$. Or even simpler just use $$x_2 = x_2^+ - x_2^-$$, then $$|x_2| = x_2^+ + x_2^-$$, $$x_2^+, x_2^- \ge 0$$.