There is a small inaccuracy in my positioning of the points in the graph above. The lower x-coordinate should be at -4.52 and the upper one at 5.52.
+118687 Hi again Gibsonj338,
\(|5|>|\sqrt{-x+x^2}|\)
\(5>\sqrt{-x+x^2}\\ \qquad \mbox{For real solutions }\\ \qquad -x+x^2\ge0\\ \qquad x(x-1)\ge0\\ \qquad x\le0\quad or \quad x\ge1\\~\\ 25>x^2-x\\ 0>x^2-x-25\\ x^2-x-25<0\\ \mbox{This is a concave up parabola so this will be true between the roots}\\~\\ roots:\\ x=\frac{1\pm\sqrt{1+100}}{2}\\ required \;\;region:\\ \frac{1-\sqrt{101}}{2}<x<\frac{1+\sqrt{101}}{2}\\ BUT\\ \mbox{The region is also restrained by }\quad x\le0\quad and \quad x\ge 1\\ so\\~\\ \mbox{The region where this is true is }\\ \frac{1-\sqrt{101}}{2}<x\le0 \quad \cup \quad 1\le x < \frac{1+\sqrt{101}}{2}\\~\\ \mbox{This can also be expressed as} \\ (\frac{1-\sqrt{101}}{2},\; 0 ] \quad \cup \quad [1,\; \frac{1+\sqrt{101}}{2}) \)
Another look
+129928 Yeah.......this was a tricky one.......a graph definitely helps, IMHO....!!!!
+118687 Hi Chris,
I use graphs a lot but I was not even tempted for this one.
We all develop our own methods for doing things. :)
My answer for this one is really quite comical
http://web2.0calc.com/questions/mg-to-grams
This method that I developed independently is excellent for difficult rates questions - they just fall out.
But this is not such a difficult question and my method is rediculously involved.
I am sure asinus's method is better and I strongly suspect that the last answering guest just shock his head at the rigmarole I went through.
His/her answer is probably excellent although I have not really looked at it I will admit.
We all have our quirks :))
+118687 I developed my rates method when I was teaching maths for nurses.
They have really horrible rates questions.
drops per minute, dose per weight, ml/drop, and a million other rates all have to worked out in combination with each other. It gets really tricky.
My method deals with these hard questions really effectively and it can be done with no understanding of algebra!
Of course nurses ultimately use stupid 'formulas'. They are stupid because they don't make sense. But people learn to use them by rote, they 'formulas' work and they have been used for a long time, so I guess my method will never seriously catch on :(( What a pity :((
Not to worry 
