In the exercise you provided, the strict inequality vs. non-strict inequality issue and the fact that x is an integer are the reasons you are avoiding the lower bound (though I can understand what you mean by "superior limit", it is actually a totally unrelated topic, usually abbreviated "limsup"; that's why I am avoiding that terminology).
When you solve the inequality |2x + 3| =< 12, you will obtain that x is between -7.5 and 4.5, inclusive. Since we are also given that x is an integer, we conclude that x is between -7 and 4, inclusive. So, x is greater than or equal to -7, which is the same thing (since x is an integer) as saying x is strictly greater than -8. Notice that if we were not given that x is an integer, the sentences "x is greater than or equal to -7" and "x is strictly greater than -8" would be very different; for instance, x = -7.5 would have been excluded in the former but included in the latter.
Long story short, there is no rule to tell us to avoid the upper or lower bounds in a non-strict inequality; we just have to keep in mind the pieces of information the question has provided us (such as the information that x is an integer) and do our analysis in the light of that information to find the correct answer.
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