Find the solution set of $\displaystyle x - 2 > \frac{x + 4}{x} $. The answer they gave was {x : x>4 or -1<x<0 } but my answer was {x : x>4 or x<-1}. Which is right?
This is how I did it.
$\displaystyle x - 2 > \frac{x + 4}{x} $
$\displaystyle x^2 - 2x > x + 4 $
$\displaystyle x^2 - 3x - 4 > 0 $
$\displaystyle (x - 4)(x + 1) > 0 $
Since the expression is > 0, hence, the x values are above the x -axis, right? And since the graph is a "smiling" graph, x > 4 or x < -1. That's how I got it.
You made a mistake here . You CANNOT multiply x-2 with x because u don know whether x is positive or negative . If its positive , the sign is not affected but if its negative , the sign will be affected.
So start like this :
$\displaystyle (x-2)-\frac{x+4}{x}>0$
$\displaystyle
\frac{x(x-2)}{x}-\frac{x+4}{x}>0
$