Telegraph machines couldn’t type dots, so X is the next best thing
The use of X at the end of abbreviations comes from the days of the telegraph, where sending an actual dot (period/full-stop) required more effort than just sending another letter.
TX, RX and PAX all come from telegraphy.
There’s also a strong argument for RX originating from the prescription symbol, ℞ (short for the latin “recipere” - “take…”), and although that’s not actually X, but rather a line through the leg of the R, it was often typed as Rx.
… but sometimes X is just phonetic
“Fax” is a phonetic rendering of the first syllable of “facsimile”, tix is a shortening of “tics”, which itself is the plural of tic, short for “ticket”
What’s a telegraph?
A telegraph was originally a five-wire connection that allowed messages to be sent between stations. By placing a voltage on one or more of the wires, a message could be sent nearly instantly over long distances. Later, the telegraph companies came up with clever signalling systems that let them carry the same signals on just one wire, but the original system was five parallel wires. That limit of five “wires” is important, because it created a limitation on what you can send by telegraph.
In order to have the messages make any sense, you need to agree what each combination of on or off lines means, and the system used in the USA and most other countries until the mid-1960s was what is known as International Telegraphic Alphabet No. 2
Letters or figures, but not both
Using five wires with just “on or off” signalling meant that there were only 32 possible combinations, and one of those had to be kept for “nothing is happening right now” (all lines off). But the remaining 31 possible signals are barely enough for all 26 letters of English (let alone other alphabets), and they leave no way of sending numbers or punctuation marks.
To get around this problem, the system gave a unique code for the letters A to Z and the space character, but also defined one of the remain given the meaning “I will now switch to using numbers”. When a receiving station got that code, they would stop looking up the incoming codes on the “letters” page of their code book, and instead used the “figures” list. Sending the “switch” code while in “figures” mode would tell the receiver to interpret the characters as letters again.
Time is money
The cost of a telegraph is fixed: an operator, a printer and a telegraph-line between stations. So, the more you can send along that wire, the more money you can make. This lead to the use of abbreviations for common words, especially longer ones.
So, instead of writing “Transmission” the operator would prefer to just send “T.”, knowing that the receiver would understand the meaning.
But: there’s no period (full-stop) symbol available while the telegraph system is in “letters mode”. So, rather than send “[switch-to-numbers][DOT][switch-back-to-letters]” every time a dot was needed, the operators would use X as a substitute: very few English words end in X, so it was an ideal marker for an abbreviated word. It was two less keystrokes, but importantly it was far faster to send.
Air travel needed fast information
When people began to travel by air, the telegraph became essential. This is where abbreviations like WX (“weather”) and PAX (“passengers”) originated. Both were essential things for airports to know.
From air-travel PAX spread into hotel reservations, and though you can occasionally still see it today on occasion, the hotel people I know consider it to be an affectation.
