How to minimize a sum of absolute values using linear programming? I am having trouble understanding the logic behind optimization of cost function of the form $$\min (|x| + |y| + |z|) \,$$ subject to constraints $$Ax \le b \qquad Cx = d $$ such as $$ x + y \le 1 \qquad 2x + z = 3.$$
I have seen methods involving representing absolute values as a new variable and putting constraints on them, $i.e$ $$|x| = a \qquad -a \le x \le a$$ but I don't understand why should we represent an equality as inequality. Aren't we changing the equation itself? 
I have tried to think but am unable to grasp it. Please help.
The method is available on Wikipedia as a numerical example:- 
https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values 
EDIT:- 

On that page, they are solving another problem. They are trying to make the constraint |x| < b, so it's true that they can split that inequation in those two. In your case, you can't, because you need it to be either x, or -x, not any value in between.
What you can do, is using a bivalent variable. Those are also called logical variables. 
With them, you can define another variable, like u, and restrain them like this.
Let's say V is a bivalent variable.  And M a big number.
x - MV ≤ u ≤ x + MV
-x - M*(1-V) ≤ u ≤ -x + M*(1-V)
This way, when V is 0, the second constraint does nothing. And the first one forces u to be x.
When V is 1, the first constraint does nothing. And the second one forces u to be -x.
Then, when the problem is solved, u can only take one of those values.
