Inequality (absolute value)

Question

$$|x-4|^2 -5|x-4| +6 > 0$$

How can I get rid of the absolute value? Does it work the same way equations with absolute value work?

Answer 1

Hint:
Substitute $u=|x-4|$, and you get the quadratic inequality $$u^2-5u+6\gt0.$$ Do you know how to proceed further?

Answer 2

You will probably want to factor first. That is $$|x-4|^2 -5|x-4| +6 = (|x-4|-2)(|x-4|-3)>0.$$ From this, we can see $|x-4|-2>0,$ and $|x-4|-3>0,$ $\mathbf{or}$ $|x-4|-2<0$ and $|x-4|-3<0.$

Thus $x<1$, or $x>7$ (for our first case); but also $x>2$ or $x<6$, that is $2<x<6$ (for the second).