のに: a result that contrasts with expectations
A result that follows from a reason differs from one that defies expectations. With のに, compare what the first fact led you to expect with the outcome in the later clause.
Linking a fact to a surprising outcome
早く家を出たのに、電車に間に合いませんでした。 says that, despite leaving home early, the speaker did not make the train. The result contrasts with the expectation that leaving early would be enough.
前日に予約したのに、席がありませんでした。 contrasts having booked the day before with finding no seat. It may convey dissatisfaction, but のに is not always a strong accusation; surprise or unexpectedness also depends on context.
Using なのに with nouns and な-adjectives
Attach のに to plain verbs and い-adjectives, as in 行ったのに and 安いのに. For affirmative non-past nouns and な-adjectives, use な: 日曜日なのに and 静かなのに. Do not mechanically form 日曜日だのに.
日曜日なのに、会社へ行きます。 says that someone is going to work even though it is Sunday. It differs from 日曜日なので, which presents Sunday as a reason. Compare the relationship made by なので with that made by なのに.
Keeping different uses of のに separate
このはさみは紙を切るのに便利です。 means that these scissors are useful for cutting paper. It does not present an unexpected result despite cutting. Distinguish のに for use or purpose from the contrastive のに.
For a contrastive sentence, note the first fact, the expectation, and the actual outcome. The expectation may be implicit, but do not invent extra events to fill it in. For a use-related sentence, first ask what the item is useful for.
Check: What relationship does 安いのに、丈夫です。 present between price and durability?
Show the answer and why
It presents durability as contrary to an expectation associated with the low price. It does not say that the low price causes durability or establish a rule that all inexpensive things are weak.