First, here's the easy way to do it, but your 
teacher won't accept it because it doesn't use 
algebra:

Their average will be their sum divided by 
3 or 171÷3 = 57,

The only 3 consecutive odd integers that average
57 are the three with 57 in the niddle, which are 

55,57, and 59.

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That's the easy way.  Here's the algebra way, that 
your teacher will accept:

The smallest odd integer = x
The middle sized odd integer = x+2
The largest odd integer = x+4

x + (x+2) + (x+4) = 171
x + x + 2 + x + 4 = 171
           3x + 6 = 171
               3x = 165
                x = 55

The smallest odd integer = x = 55
The middle sized odd integer = x+2 = 55+2 = 57
The largest odd integer = x+4 = 55+4 = 59
 
Edwin