Algebraically solving the inequality $\frac{1}{x} - 1 > 0$ $$\frac{1}{x}-1&gt;0$$


$$\therefore \frac{1}{x} &gt; 1$$

$$\therefore 1 &gt; x$$

However, as evident from the graph (as well as common sense), the right answer should be $1&gt;x&gt;0$. Typically, I wouldn't multiple the x on both sides as I don't know its sign, but as I was unable to factories the LHS, I did so. How can I get this result algebraically?
You have 
$$\frac{1}{x} - 1 &gt; 0$$
Forming a common denominator yields
$$\frac{1 - x}{x} &gt; 0$$
The inequality is true if the numerator and denominator have the same sign.
Hint: $x(1-x)&gt;0$.Can you continue?