Petty and pedi are easily confused words, though I admit, the latter is more a slang abbreviation of a proper word.
The spell-check application of most word processing software programs would not catch a slip-up of these two words. Spell-check is looking for words that aren’t in its dictionary, and words that resemble words in its dictionary but are possibly spelled wrong. Spell-check isn’t perfect. It doesn’t know and can’t guess what word you wanted or what word you meant, it can only judge the words on the page. If you used words that are all spelled correctly, it gives you a pass anyway.
Autocorrect also tries to anticipate what word you want based on the first few letters. But words sharing those first few letters doesn’t mean the words are related.
Petty is an adjective. It means small things, or things of little meaning. A second meaning refers to someone making a big deal about a small matters. “making a mountain from a molehill.”
Pedi is a noun, it is modern slang for a “pedicure.” A pedicure means applying lacquer to one’s toenails, and all the moisturizing, trimming, shaping, etc. that precedes it. This service is typically performed by a professional in a nail salon for a fee. In English, words starting with “pedi-” or “pod-” often relate to the foot. Examples of these words include pedicabs (aka “bicycle cab”), podiatry (foot medicine), and podiatrist (foot doctor.)
The following story uses both words correctly:
Petula knew she had a novel business idea: Pedis in a Pedicab, see a city as you have your toenails painted. When friends asked about the inevitable bumps in the road and traffic, she dismissed these as “petty concerns.”