<s>[INST] Compund Inequality with Reciprocal $0<(2x-4)^{-1}<\frac{1}{2}$ I need help with this exercise: $0<(2x-4)^{-1}<\frac{1}{2}$.
$$0<(2x-4)^{-1}<\frac{1}{2}$$
$$0<\frac{1}{2x-4}<\frac{1}{2}$$
I separate this into two inequalities:

*

*$0<\frac{1}{2x-4}$


*$\frac{1}{2x-4}<\frac{1}{2}$
The first one I had no problem solving it.
$$0<\frac{1}{2x-4}$$
By the Reciprocal Property of inequalities
$$0<2x-4$$
$$4<2x$$
$$\frac{4}{2}<\frac{2x}{2}$$
$$2<x$$
$$x>2$$
Is the second that I have a problem with. Is there a way to solve it without using a sign chart/table? I do know how to do it with a sign chart/table. I just want to know if there is another way to do it that is not the sign table.
 [/INST] \begin{align*}
0<(2x-4)^{-1}<\frac{1}{2}\\ \\
\implies 
0<\frac{1}{2x-4}<\frac{1}{2}\\ \\
\implies  0< 2< 2x-4\\ \\
\implies  4< (2)+4< (2x-4) + 4 \\ \\
\implies  4< 6< 2x \\
\implies 2< 3< x\\
\implies 3< x\\
\end{align*}
</s>