Disequazione fratta $\dfrac{x^2-9}{x+1}\le 0$

Text

Solve the fractional inequality $\frac{x^2-9}{x+1}\le 0.$

Solution

Factor the numerator: $\dfrac{(x-3)(x+3)}{x+1}\le 0$. Key points: $x=-3,\ 3$ (numerator, included) and $x=-1$ (denominator, excluded). Sign of the ratio:

• $x<-3$: $(-)(-)/(-)<0$;
• $-3<x<-1$: $(-)(+)/(-)>0$;
• $-1<x<3$: $(-)(+)/(+)<0$;
• $x>3$: $(+)(+)/(+)>0$. $\boxed{x\le -3\ \vee\ -1<x\le 3.}$