Inequality with a quadratic factor

A quadratic factor with $\Delta<0$ has constant sign: here the denominator $x^2+1$ is always positive, so it has no effect and it is enough to study the numerator.

Example — Inequality with a quadratic factor

Solve $\dfrac{x^2-9}{x^2+1}>0$.

Numerator: $x^2-9 = (x-3)(x+3)$, zeros at $x=\pm 3$. Denominator: $x^2+1 > 0$ for every $x$ ($\Delta=-4<0$, $a=1>0$ → “always positive”).

So the denominator has no effect on the sign: it is enough to study the numerator.

	$x<-3$	$x=-3$	$-3<x<3$	$x=3$	$x>3$
$x+3$	$-$	$0$	$+$	$+$	$+$
$x-3$	$-$	$-$	$-$	$0$	$+$
$x^2+1$	$+$	$+$	$+$	$+$	$+$
$\dfrac{x^2-9}{x^2+1}$	$+$	$0$	$-$	$0$	$+$

Solution: $\boxed{x<-3 \;\text{or}\; x>3}$.

Links

Topics: Inequalities
Concepts: Fractional inequality · Factorisation · Sign analysis · Sign table
Skills: Solving inequalities · Factorising