The Tenet walkout is one of several strikes and organizing efforts nationwide as unions work to rebuild from a steep membership decline that began 50 years ago. Many are focusing on white-collar, female-dominated and service-sector industries such as health care, teaching and the media and not just blue-collar, male-dominated industries like manufacturing, where the United Auto Workers is striking against General Motors. A recent Gallup poll showed Americans support unions by a 2-to-1 margin, up from a near split 10 years ago and nearly the highest level since the 1960s.