Are hospitals in the habit of releasing patients with mental issues and admitting they tried to kill themselves?

My sister tried to kill herself by taking a bunch of sleeping pills and some other pills last night. I couldn’t get her on the phone so I went over there to check on her and found her on the floor in and out of consciousness. I called 911 and the police and paramedics showed up. The medic asked her if she tried to kill herself and she said yes because she was tired of living. She is depressed every day so he said that they are going to transport her to the hospital for help. I was very worried about my sister so I asked the medic if they would keep her there at the hospital. He said yes. Since she was admitted that she tried to kill herself, they would hold her for 72 hours to talk to counselors. I was relieved that she would be in good care now that she was taken to the hospital for the help that she needed. They took her to the hospital and I went home. My mom calls me early in the morning hours upset because she called the hospital to check on my sister to find out that they had treated and released her saying that she had signed herself out. I was mortified to find this out we could not find my sister for hours after finding out that they had released her. We didn’t know if she had went and tried to kill herself again. I mean isn’t this negligence on the hospitals part? What if my sister would have walked right out of there and tried again and been successful this time?