It’s correct that the Las Vegas Strip is not within the City of Las Vegas, whose southern boundary is Sahara Avenue. Las Vegas is one of five incorporated cities in Clark County, which covers most of southern Nevada; the others are North Las Vegas, Henderson, Boulder City and Mesquite. All the rest of southern Nevada is officially unincorporated and the county provides government and services to the 680,000 residents, compared to the 1.02 million who live in the cities.
Clark, size-wise, is one of the largest counties in the country, encompassing 8,000 square miles, larger than the state of New Jersey. The Las Vegas Strip is located in the unincorporated county; thus its public services -- everything (except the police, which we’ll get to below) from air quality to fire protection, from building inspections to parks and recreation -- are provided by the county government.
The police department is an exception to the rule that Las Vegas residents are served by the city and unincorporated residents by the county. From the early 1900s until 1973, there were two separate law-enforcement agencies: the Clark County Sheriff’s Department and the Las Vegas Police Department. But in 1973, the two departments merged into the Las Vegas Metropolitan Police Department (LVMPD), which patrols all of Clark County today, from Primm to Mesquite, from Mt. Charleston to Laughlin.
You can read all about the history of the two departments and their merger into the LVMPD in our book Policing Las Vegas.