Confidence intervals for factors+continuous variables

[mtal]

I have

y_{ij} = \mu_{i} + \alpha x_{ij} + e_{ij}

where i = 1, 2,3 and j = 1, \ldots , r .
\mu_{i} represents the mean of the data set plus factor levels i , \alpha x_{ij} is a continuous variable.

So, the problem is the following:

Construct confidence intervals for \mu_1 , \mu_2 - \mu_1 , and (\mu_3 - \mu_2) - (\mu_2 - \mu_1) .

The original problem consisted of finding conf.intervals for the same things except the continuous variable wasn't in the model.
I have looked around alot, but I can't find any instructions on confidence intervals when both factors and continuous variables are included.

Help appreciated!

[Enuma_Elish]

You can think of it as follows:

Without the x, y = m + e implies m = mean(y).

When you have an x and the equation becomes y = m + ax + e, m = mean(y - ax), where a is the estimated slope coefficient.

EnumaElish