1995 E300 : brake pedal hard to push, won't turn off

[Hillary]

Does anyone have any suggestions for diagnosing what to rule out before replacing the vacuum pump?

There is vacuum, reading about 10 or 11 at the little hose that comes directly off the pump that the 5-way connector goes into. There's also a reading of 10 or 11 at another hose (on the passenger side) that goes to a small black canister.

Also, the engine doesn't shut off for ten to twenty seconds. The brake pedal has become very hard to press. The door locks seem fine. Started having these symptoms about a week ago, and drove it a few miles then parked it.

The other thing it was doing is when backing out of the driveway, then shifting to forward, the transmission starts to slip right at that point. Otherwise, there seems to be no trouble with the transmission.

I have owned the car for two years, it has 290,000 miles on it and has been a great car.