What is the domain of sqrt(1/(|x-2|^2-|6-3x|+5/4)) ?

The domain of sqrt(1/(|x-2|^2-|6-3x|+5/4)) is x 9/2

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domain of sqrt(1/(|x-2|^2-|6-3x|+5/4))

domain

x < - \frac{1}{2} or \frac{3}{2} < x < \frac{5}{2} or x > \frac{9}{2}

Interval Notation

(- \infty, - \frac{1}{2}) \cup (\frac{3}{2}, \frac{5}{2}) \cup (\frac{9}{2}, \infty)

Solution steps

Find non-negative values for radicals:

x < - \frac{1}{2} or \frac{3}{2} < x < \frac{5}{2} or x > \frac{9}{2}

Find undefined (singularity) points:

x = - \frac{1}{2}, x = \frac{3}{2}, x = \frac{5}{2}, x = \frac{9}{2}

Combine real regions and undefined points for final function domain

x < - \frac{1}{2} or \frac{3}{2} < x < \frac{5}{2} or x > \frac{9}{2}

What is the domain of sqrt(1/(|x-2|^2-|6-3x|+5/4)) ?
The domain of sqrt(1/(|x-2|^2-|6-3x|+5/4)) is x<-1/2 \lor 3/2 <x< 5/2 \lor x> 9/2