Two models or one? Significant effect of predictor - in two treatments

Say you have a response variable that you suspect is linearly affected by your predictor variable, but only under treatment A and not treatment B.

I was tempted to model this as a linear model using the treatment in interaction with the predictor, as in response ~ predictor*treatment. This analysis gives you the following parameters and their p-values.

• β0 : The intercept at Treatment A
• β1 : The slope at Treatment A
• β0-B : The intercept change in Treatment B
• β1-B : The slope change in Treatment B

Say β0 and β1 are significant (i.e, not zero), but not β0-B and B-1B.

I'd interpret this as "The slope and intercept - in treatment B - is not different from treatment A", but not necessarily "The slope is not significantly different from zero in treatment B". Thus, this analysis is not answering my question, and I'm starting to wonder if it is necessary to run two different linear models instead (one using data points from treatment A and one from B).

Are two separate models the only way to solve this, or have I misunderstood something important here?