If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think you do $4x+1<-3x$. Is this correct?
[Math] How to solve inequalities with absolute values on both sides
absolute valuealgebra-precalculusinequality
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Best Answer
You could also square everything $$ |f(x)| < |g(x)| \Leftrightarrow |f(x)|^2 < |g(x)|^2 \\ \Leftrightarrow f(x)^2 < g(x)^2 \\ \Leftrightarrow 0< g(x)^2-f(x)^2 \\ \Leftrightarrow 0< (g(x)-f(x))(g(x)+f(x)), \\ $$ which means that $g(x)-f(x)$ and $g(x)+f(x)$ have the same sign.