3.4.7.3. x pronounced like a unvoiced sh [ʃ], a voiced s or a voiced ks
The english voiceless sh sounds like the Portuguese ch. The x in caixa (box) and the ch in chamar (to call) have the same phonetic value.
caixa
chamar
This is due to historic reasons. Different sounds vanished away in the course of history, but the graphems representing these sounds remained. These led to a situation where the same sound is represented by different graphems. The education systems all over the world and in all languages make a great effort to explain to the pupils why the same sound is represented by different spellings or graphems. A more rational approach woud be to abolish these different spellings. The sound itself is not problematic, it corresponds to the English sh in words like to shine, shoulder, to shrink.
x pronounced like the unvoiced English sh [ʃ]
at the beginning of a word
xingar
to insult
at the beginning of a syllable; after a vowel x always introduces a new syllable
lixo
garbage
In the combination ex, a combination we have in a lot of words, more than 500 to be precise, exceção (exception), execrável (dreadful), exercer (to practice), exíguo (little), exato (exact) etc. it is pronounced voiced.
x pronounced voiced
in the combination ex
exemplo (example)
Some people have problems to hear the difference between voiced and unvoiced, what is not a big problem, because one can feel it as well. The s in decision is a voiced /sh/ and in sheep it is unvoiced. Voiced means, that the vocal chords vibrate and and unvoiced means, that the vocal chords doesn't vibrate. That is something you can feel by putting your fingers on your larynx. In the case of decision you will feel that your vocal chords vibrate when you pronounce the s and that they doesn't vibrate, when you pronounce the sh in sheep. This said voiced is relative. In most cases the voiced sh is between vowels and vowels are always voiced. If the flow of air is not interrupted by the tongue, the lips, the teeth, the uvular something must produce a vibration, otherwise there is no sound at all. The body of guitar doesn't sound by itself. Only if someone touches the chords there is a vibration. We need a vibration to produce a sound and in the case of the vowels there is non and therefore the vocal chords have to vibrate to produce a sound. This means that it is a little bit unclear in words like decision, ambition etc. whether the s is voiced or whether only the vowels sounds. However in other languages the difference between voiced and unvoiced is clearer and they can show up as well after an unvoiced consonant.
x pronounced as a unvoiced s [s]
In words that have been introduced to Portuguese more recently and that are only used by more educated people x is pronounced unvoiced.