All of the commiserate tiers of pay have to increase too when the minimum increases
Currently, I get paid $12/hr, I've been working at the same place for the past 4 years. When I started, I was paid $8 per hour, and I've gotten raises on account on both my increasing age and my very good performance and reputation for working hard over that time. If the minimum wage was increased to $15, sure I would get 3 more dollars per hour but now I'm making minimum wage again, and there's no difference in pay between my job and a new hire at mcdonalds. The company I work at (a biotech startup) isn't doing amazingly well right now because we need more investors, we're not selling products commercially. Both the president and the vice president have had to take several pay cuts throughout the past 5 years, so I don't think I would ask for more than $15, let alone get it.
The point is, increasing the minimum wage is not a good solution because it further distributes the problem. It sucks that single mothers working low-paying jobs can't support their families, but this is not the correct solution, it will just make things worse for everyone. The only thing that I could see working is a government stipend to those who cannot support themselves. Teenagers working their first job don't deserve $15 an hour. Making the pay rate a function of age would not work because there would be age discrimination from companies who would only hire teenagers.