Find the domain of the function

Question

x≤0.5 or x>12 is the answer but I don't understand how. The denominator should not be equal to zero and a square root cannot take negative numbers. So how is the answer x≤0.5 ? Shouldn't it be x>=0.5 and x is not equal to 12?

Thanks for the help:)

Answer 1

If $x\lt0.5$, then both numerator and denominator of $(8x-4)/(x-12)$ are negative, so the quotient is positive, hence the square root can be taken. If $0.5\lt x\lt12$, then the numerator is positive but the denominator is negative.