Minimum possible sampling frequency for continuous time signal
This is one of those problems which ask for finding the sampling rate that avoids aliasing in the band of interest but don't care aliasing for the rest of the spectrum.

In this problem the real, baseband analog signal has a bandwith of $B = 1800$ Hz, which would require a minimum Nyquist sampling rate of $F_s = 2 B = 3600$ Hz when we are to avoid aliasing on the complete band of the signal; $-1800&lt; f &lt; 1800$ Hz which is depicted as the condition on figure-a below: (without amplitude scaling)



On the other hand the condition on the figure-b tells that if you want to preserve the signal only in the band of $-1000&lt; f &lt; 1000$ , then you can relax the aliasing condition to lower the sampling rate so that aliasing is prevented only in the band of ineterest; $-1000&lt; f &lt; 1000$ . That sampling rate you look is found by requiring on the shifted spectrum that, the leftmost cutoff frequency of right shifted replica is above $1000$ Hz of the unshifted replica, which is dispalyed on the figure-b.

$$ F_s - 1800 &gt; 1000 \implies F_s &gt; 2800$$

Hence the minimum sampling rate that will avoid aliasing in the band $-1000&lt; f &lt; 1000$ but otherwise will create aliasing in the range $ 1000&lt; |f| &lt; 1800 $ is $$ F_s = 2800 $$

Note that what you considered would be true if the signal was lowpass filtered to $1000$ Hz before being sampled. But in the problem no such antialiasing bandlimiting filtering is mentioned.