Okay, are you doing Maths extension II? If you do so or not note that the absolute values represent the length of distance from 0 to a certain equation so in this case,
what we have here is that the value has to be either bigger than
![](https://latex.codecogs.com/png.latex?\bg_white \frac{1}{x})
or smaller than
![](https://latex.codecogs.com/png.latex?\bg_white -\frac{1}{x})
. With that knowledge in hand, the next step is to state that where is
![](https://latex.codecogs.com/png.latex?\bg_white |x-2|=0)
because this is the point where the line reflects itself off a wall.
Do you see it?
If so congrats because that serves a vital role in what we are doing the reason is that when
![](https://latex.codecogs.com/png.latex?\bg_white |x-2|)
is just simply when is
![](https://latex.codecogs.com/png.latex?\bg_white x-2 > \frac{1}{x})
and when
![](https://latex.codecogs.com/png.latex?\bg_white |x-2|)
is simply when is
![](https://latex.codecogs.com/png.latex?\bg_white -x+2 < -\frac{1}{x})
.
Now what I am about to do will attract sceptics but hey the reason why it works will be revealed in a while.
To start off
![](https://latex.codecogs.com/png.latex?\bg_white x-2 > \frac{1}{x})
. Now meet the critics. "Sir, why are you allowed to say that
![](https://latex.codecogs.com/png.latex?\bg_white x^{2}-2x-1>0)
when normally our maths teacher always tells us to multiply by an even power before doing this?"
The reason is that at
![](https://latex.codecogs.com/png.latex?\bg_white x > 2)
all the values that come after it are positive as thus, it is safe to multiply by x because multiplying by a positive value does not change the result in a fraction. The next step is to find the roots in
![](https://latex.codecogs.com/png.latex?\bg_white x^{2}-2x-1>0)
. If you have found the roots a little challenge awaits you find the root that is greater than 2. If you have done than Horray you are halfway done.
The next step is this find where
![](https://latex.codecogs.com/png.latex?\bg_white -x+2 < -\frac{1}{x})
. The next step will be
![](https://latex.codecogs.com/png.latex?\bg_white -x^{3}+2x^{2} < -x)
. Bring everything to one side and it becomes
![](https://latex.codecogs.com/png.latex?\bg_white -x^{3}+2x^{2}+x < 0, x(-x^{2}+2x+1) < 0)
have a look at this
![](https://latex.codecogs.com/png.latex?\bg_white (-x^{2}+2x+1) < 0)
is simply
![](https://latex.codecogs.com/png.latex?\bg_white (x+1-\sqrt{2})(x+1+\sqrt{2}))
. The rest is history. There you can find the values that satisfy
![](https://latex.codecogs.com/png.latex?\bg_white |x-2| > \frac{1}{x})
. If you had to do this question by hand my method will work.