Disequazione fratta $\dfrac{1}{x-2}<3$

Text

Solve the fractional inequality $\frac{1}{x-2}<3.$

Solution

Move everything to the left: $\frac{1}{x-2}-3<0\implies \frac{1-3(x-2)}{x-2}<0\implies \frac{7-3x}{x-2}<0.$ Key points: $x=\tfrac73$ (numerator) and $x=2$ (denominator, excluded). The ratio is negative:

• $x<2$: $(+)/(-)<0$;
• $2<x<\tfrac73$: $(+)/(+)>0$;
• $x>\tfrac73$: $(-)/(+)<0$. $\boxed{x<2\ \vee\ x>\tfrac73.}$