total population = 132,000
11 voting districts named D1, D2, D3, ..., D11

we know that D1 + D2 + D3 + ... + D11 = 132,000 (a)
and that the maximum gap between all districts is 110% in terms of population count, i.e. Di <= 1.1 Dj (b)

the minimum possible population for a given district is the one for which all the others districts have the same population (thus maximizing the gap)

hence if D1 is the least populated district, D2 = D3 = ... = D11
(a) can then be written as D1 + 10 D2 = 132,000
and when the gap is maximum, (b) translates into D2 = 1.1 D1

combining the two gives us D1 + 11 D1 = 132,000
and D1 = 11,000

You can verify that equations (a) and (b) above are respected

That is a very slick explanation. Thanks a ton!!! And yes, your answer is correct.