Determine maximum frequency of input signal to make system LTI
This question, and a number of similar others, from DTSP book can be a little tricky to recognise the fact that it's not actually asking alias free operation (which would require 8 khz cutoff signal as you expected), rather it is actually asking how much aliasing (due to initial sampling block) is tolerable: you can allow aliasing in those regions of the DT filter $H(e^{j\omega})$ whose stop band will help you get rid of the aliased spectrum, hence enable an LTI processing possible, but not with the samples of the complete original fullband signal $x_c(t)$ but with that of the reduced bandwidth input signal.

To see this you shall better consult to the spectral figure below: 
The mathematical condition on the maximum allowable cutoff frequency $\Omega_c$ of the input signal, whose aliased spectra will be deleted by the ideal lowpass filter with cutoff frequency $\pi/2$, is given as: 

$$\Omega_s - \Omega_c &gt; \frac{\pi/2}{T_s}$$
$$\frac{2\pi}{T_s} - \Omega_c &gt; \frac{\pi/2}{T_s}$$
$$\Omega_c &lt; \frac{2\pi}{T_s} - \frac{\pi/2}{T_s}$$
$$\Omega_c &lt; \frac{3\pi/2}{T_s}$$

Thats stated in hertz frequency as:
$$f_c &lt; \frac{3}{4}f_s$$

which gives a maximum allowable input signal bandwidth as $f_c=12,000$ Hz when the sampling rate is $f_s=16,000$ Hz.