The verb tenere translates, much of the time, as "to hold," "to keep." But we also use the verb to talk about things or people we care about, that matter to us, and consequently do not want to lose.
We use it intransitively with the preposition a (to, in, about...) to mean to care about, to consider important. We can use it with things or people.
Io ci tengo al mi [mio] lavoro. E il mi [mio] capo nun [non] vuole grane.
I care about my job. And my boss doesn't want trouble.
Captions 21-22, Il Commissario Manara S1EP7 - Sogni di Vetro - Part 10
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In a way, the person wants to keep his or her job, so tenere makes sense. When you care about a person, but it's not the moment for talking about actual love, tenere is a good verb to use. You care about someone and you don't want to lose them.
Io ci tengo a te.
I care about you.
Caption 8, Il Commissario Manara S2EP2 - L'addio di Lara - Part 1
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Another way to think about it is that "it matters."
Oh, mi raccomando, non mi fate fare cattive figure perché ci tengo, capito?
Oh, and I mean it — don't make me look bad, because it really matters to me, you get it?
Caption 41, Il Commissario Manara S2EP4 - Miss Maremma - Part 11
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We can also use tenere when we want to make sure to mention something. So we can follow the preposition a with either a noun or a verb.
Ci tengo a dire una cosa,
I feel the need to say one thing,
Caption 17, Il Commissario Manara S2EP9 - L'amica ritrovata - Part 3
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So when something or someone means something to you, try saying, Ci tengo (it matters to me).
You can turn it into a question:
Ci tieni davvero tanto a mangiare al ristorante stasera? Perché io sono molto stanca” (do you really care about going out to eat tonight? Because I am really tired).
Since tenere is used so much in various contexts, it may be hard to search for examples, but the more you watch and listen, the more you will notice that Italians use this turn of phrase all the time.
What are some of the things a cui tieni (that matter to you)?
È una fotografia alla quale tengo molto.
It's a photograph I'm very attached to.
Caption 27, L'arte della cucina La Prima Identitá - Part 9
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