потому что если как вы говорите "невнимательно", то это будет "without paying any attention to"... а если как в примерном контексте, то будет иначе: "am reading a book but keep an eye on what's going on in that tv show", "I read a book yet try to follow that tv show"

а есть взять название темы - "одним глазом" то тут подхходит "barely looking"

или

"I barely watch TV, only the news and TV shows just to catch up with the plot"

что-то вроде этого :)

To keep one eye on somebody/something (Also: have one eye on somebody/something) --- to give part of your attention to one person or thing

e.g. Jim continued to talk to me, but he kept one eye on the clock to be sure he wouldn't be late.
Usage notes: sometimes used in the form "with one eye on someone or something":
e.g. She read her newspaper with one eye on arrivals board to see if his train had come in.
See also: eye, keep, one

Мне кажется, что to keep an eye on sth./ sb. и to keep one eye on sth. / sb. противопоставлены по смыслу

to keep an eye on sth./ sb. - следить, присматривать за кем-то/ чем-то
to keep one eye on sth. - смотреть одним глазом

Тогда можно перевести как:
I am reading a book while keeping one eye on a soap opera.