A) When considering a simple linear regression model, it is important to check the linearity assumption. Graphing the residuals vs the predictor variable can often give a good idea of whether or not this is true. A non-random pattern suggests that a simple linear model is not appropriate.
B) On the other hand, residuals have the property that the correlation between the residuals and the observations of the predictor variable is zero.
It sounds to me that B) contradicts A). (If the residuals and the predictor are not correlated, how can we see a non-random pattern in the plot?) Can you explain to me why this is not the case?