model building - regression in statistical analysis I have a theoretical question regarding model building. I have some real empirical data, but I will present the problem in more general terms.  
I have two groups. I have a dependent variable, and an independent variable.
For one group the relationship between continuous independent variable and the dependent variable  looks this way:

For the second group it looks this way:

Does it make sense to include both groups in one regression model (coded as dummy variables)?
a) Will the effects for each group not cancel each other out making it harder for the model to converge?
(my model actually does not converge, which is why I am asking this question - could this be the reason?)
b) as far as I understand, dummy variables change the intercept, and the slope remains the same. 
If I know that the slopes are so drastically different for the two groups, does it make sense to put them in the same regression model?
(I assume the slope would not be correct for either group)
c) I am aware that I can introduce interaction terms, to look at the effects for groups separately.  but... would just building two separate models, one for each group be wrong?
 A: Your independent variable is continuous, but you fail to mention if your dependent variable is also continuous? Because you are mentioning the lack of convergence of your model, I am surmising it may not be continuous? What type of regression model are you fitting? 
Assuming your dependent variable is continuous, when you fit a single regression model to the combined data from the two groups, you are in effect fitting two submodels - one submodel for the first group and the other submodel for the second group.  
If you don't include an interaction in your single regression model, you are essentially assuming that the relationship between the dependent and independent variables is the same in both submodels. Additionally, you are assuming that the variability of the model errors is the same in both submodels, which enables you to estimate it using the data from both groups.  
If you do include an interaction term in your single regression model, you are assuming that the relationship between the dependent and independent variables is different across the two submodels. Additionally, you are assuming that the variability of the model errors is the same in both submodels, which enables you to estimate it using the data from both groups.  
It's impossible to comment on your model convergence issues not knowing what kind of model you are fitting and not having access to your data.  Convergence issues can arise from a variety of reasons, which may be related to model specification (e.g., the model may be too complicated for the data at hand) and/or data issues. 
Usually, one fits the model which assumes different relationships/slopes across the groups and then tests whether the model can be simplified by dropping the interaction term between the independent variable and the group variable. 
If you are interested in comparing the relationship of interest across groups, then using a single model is recommended. 
