How do I add and subtract modules?

Modules are added according to the following rule (a and b are non-negative values):

1. | a | + | b | = a + b;

2. | -а | + | b | = a + b or | a | + | -b | = a + b;

3. | -а | + | -b | = a + b.

Modules are subtracted according to the following rule (a and b are non-negative values):

4. | a | – | b | = a – b;

5. | -а | – | b | = a – b or | a | – | -b | = a – b;

6. | -а | – | -b | = a – b.

For non-negative a and b, the following is true:

7. | a + b | = a + b;

8. | a – b | = a – b (if a is greater than b);

9. | a – b | = b – a (if b is greater than a);

10. | -a + (-b) | = a + b;

11. | -a – b | = a + b.