How do I verify I have a broken drain pipe.

Our kitchen sink and washer share a drain that connects to the main drain under the concrete slab. Perodically the washer backs up in the kitchen sink. We've had plumbers out every 6 months (we've owned the house about 18 months) and they auger the drain then it works ok for about 6 months. This time, the plumber told us his auger couldn't break through so we must have a broken pipe under the slab. (He charged us $187 to tell us this) His solution is to install a pump and pump the kitchen and washer drainage to hook up to the upstairs drain. His estimate is $2850. Does this sound correct?. Is there a way of verifying the pipe is actually broken? Thanks in advance for you input.