The reality is that it isn't rich vs poor. It's media attention. Most of the time a rich murder gets more media attention which causes higher ups to commit more resources. But if the murder of a poor person got the same media attention, it would get more resources too. The issue is they don't have the funding to commit the reasources to investigate all murders at what you and I would consider a reasonable amount. So they have to hold back until it is a high profile case.
So while the overall effect is the same. The root cause is different. It's really that the people accept it and don't vote for decision makers who will fix the funding issue. Of course it would take more than funding to convert the police force into something that could perform at a level where they were thoroughly investigating all crimes. But noone has the stomach for the bill, so it doesn't matter.