In an exam, I encountered the question:
2x > |x-1|
I made the mistake of squaring both sides to:
(2x)² > (x-1)²
But that added extraneous solutions: x<-2, x> 1/3
In my second attempt, I removed the modulus like this:
2x < -(x-1), 2x > x-1
But then my answer is -1<x<1/3 which is wrong.
In my third attempt I drew the graph, and realised that the modulus means that |x-1| is always positive (how stupid am I) so clearly 2x > 0.
Then 2x > -(x-1),
x>1/3, which is the right answer.
My question is how do you approach modulus inequalities?
Do you have to draw the graph all the time?
Because there are some questions that don't need that to answer them, like |x-5|>6 and |x|<|3x-2|.