Plotting with "Around"

Question

I have a number with certain uncertainty

a1 = Around[1, 0.1]

I try to plot a function with this number

Plot[a1*x, {x, 0, 1}]

However nothing happened.

I was looking for something similar to this (the colour is not necessary):

a = Plot[1.1*x, {x, 0, 1}, PlotStyle -> Red];
b = Plot[1*x, {x, 0, 1}, PlotStyle -> Black];
c = Plot[0.9*x, {x, 0, 1}, PlotStyle -> Blue];
Show[a, b, c]

Question 1

There is some way to make that the first plot appears? Or there is some more efficient method than plotting three times?

Question 2

If I have a1=Around[1,0.1] how can I get the Maximum value of a1? In this case is clear that is 1.1, however I have to obtain this value "manually"

Answer 1

Here's a way to get the plot programmatically:

a1 = Around[1, -0.1];
Plot[
  Evaluate@{a1["Value"] x, Thread[a1["Interval"] x, Interval]},
  {x, 0, 1},
  PlotStyle -> {Black, Blue, Red}
]

---

For question 2, either one of the following will give you the value + the uncertainty:

Plus @@ Thread@ a1[{"Value", "Uncertainty"}] 
a1["Value"] + a1["Uncertainty"] 

That, of course, may not be the maximum value of a1 if the uncertainty is e.g. a standard deviation, but that's up to you to interpret. You may also be interested in other properties that you can extract from Around objects (see the details section of its docs).