Definition
The complementary slackness conditions for inequality constraint requires that either the constraint be satisfied as an equality, or the corresponding multiplier be zero.
Explain
The possibility that both conditions hold simultaneously is allowed. In that case, the problem is to decide whether to keep the constraint in L* (as active, because 
) or not (as inactive, because 
).
Test
For non-binding constraint, check if the constraint equals to zero at solution as well.
If so, SCS does not hold.
Remember
Active / binding constraint: 
Inactive / non-binding constraint: 
Strick complementary slackness can be satisfied with 
if 
(
) is not the constraint corresponding to 
.
Note
- Reference: Chapter 6 of Mathematics for Economists by Novshek