Can we find the relationship between $I$, $J$ and $K$ if $f(x)\geq 0$ for $x\in I$, $f(x)>g(x)$ for $x\in J$, and $f^2(x)>g^2(x)$ for $x\in K$?

Question

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I am trying to solve inequalities with absolute values and/or radical operators. I know that

Non negative $a(x)$ and $b(x)$ and $a(x)\leq b(x)$ in a certain domain $I$ can be represented as $a^2(x)\leq b^2(x)$ without causing the changes in their domain.

For example, $|2x-1|>|x+1|$ can be solved by rewriting it as $(2x-1)^2>(x+1)^2$.

However, $|2x-1|>x-5$ for example, cannot be rewritten as $(2x-1)^2>(x-5)^2$ because the original domain has changed.

Can we find the relationship between $I$, $J$ and $K$ if $f(x)\geq 0$ for $x\in I$, $f(x)>g(x)$ for $x\in J$, and $f^2(x)>g^2(x)$ for $x\in K$ ? I am expecting I can find $J$ in terms of $I$ and $K$ to solve $f(x)>g(x)$.