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Choice A is correct. It's given that the cleaning service cleans both offices and homes, where $f$ is the number of offices and $h$ is the number of homes the cleaning service can clean per day. Therefore, the expression $f+h$ represents the number of places the cleaning service can clean per day. It's also given that the cleaning service can clean at most $14$ places per day. Since $f+h$ represents the number of places the cleaning service can clean per day and the service can clean at most $14$ places per day, it follows that the inequality $f+h\le 14$ represents this situation.

Choice B is incorrect. This inequality represents a cleaning service that cleans at least $14$ places per day.

Choice C is incorrect. This inequality represents a cleaning service that cleans at most $14$ more offices than homes per day.

Choice D is incorrect. This inequality represents a cleaning service that cleans at least $14$ more offices than homes per day.