Convert abstract value function to linear programming model

Two problems of optimization are given as below:

Problem 1:

$Max\space z=|2x_1-3x_2|$
$s.t$
$4x_1+x_2 \le 4$
$2x_1-x_2 \le 0.5$
$x_1,x_2 \ge 0$
Problem 2:

$Min\space z=|2x_1-3x_2|$
$s.t$
$4x_1+x_2 \le 4$
$2x_1-x_2 \le 0.5$
$x_1,x_2 \ge 0$

According to the question, one of them can be rewritten as a LP, and the other one cannot. The question wants to determine which one can be rewritten as a LP problem, and Why the other one cannot.

Note1 : My problem is that i've never converted an absolute objective function to a LP objective function. So, I think I have no way of determining the one which can be rewritten as a LP. But, I know the simplex method and i can solve a LP problem using it.