Nyquist rate is the sample rate that you need to have to sample a signal to avoid damaging it by aliasing
What that means is that for real-valued signals and real-valued sampling, the sampling rate must be more than two times the bandwidth of the analog signal.
That means that with a 6 kHz sampling rate, you can get a 100% representation of any 3 kHz-wide band.
It does not mean that the sampling rate needs to be twice the highest frequency in the signal. If your 3 kHz, for example, are the band between 9 kHz and 12 kHz, you do not have to sample at 2·12 kHz = 24 kHz; 6 kHz is totally enough to unambigously represent the signal digitally. You would still need to know that your 3 kHz were centered around 10.5 kHz, if you wanted to later relate it to other signals, but usually, that doesn’t matter.
We call this technique undersampling, and it works beautifully, and is a 100% standard technique with many technical applications. All you need to be sure is that everything your ADC (analog-to-digital converter) sees is bandlimited to half its sampling rate – that means, in the aforementioned example, that you must be sure that there’s no signal below 9 kHz and no signal above 12 kHz.
Notice that this is true for real-valued sampling only. If you used things like IQ demodulators (also known as direct conversion mixers, quadrature demodulators) to give you complex, equivalent baseband, you get two streams of synchronous samples. In that case, the factor of 2 falls away. This is a very important aspect for software defined radio.
If you’re in the later parts of a DSP course, your professor might have hinted at the fact that you can implement things like rational resamplers, where you’d normally have to upsample by a factor of M, then filter to erase all images (filter runs at input rate · M) , then filter to avoid all aliases (filter runs at input rate · M) before downsampling by N, with a single filter that effectively runs at 1/N of the input rate – which is actually sub-Nyquist sampling.
But that would basically be one of the highlights of a polyphase/multirate systems lecture, and I doubt he’d put that out there in a beginner’s course – it’s just too confusing.