Simple fractional inequality

In fractional inequalities the zero of the denominator is a point of non-existence: it must always be excluded from the solution. In the sign table we mark it with a double bar (“d”).

Example — Simple fractional inequality

Solve $\dfrac{(x+1)(x-2)}{x-4} \ge 0$.

Cornerstones: $x=-1$, $x=2$ (zeros of the numerator), $x=4$ (zero of the denominator — and a point of non-existence).

	$x<-1$	$x=-1$	$-1<x<2$	$x=2$	$2<x<4$	$x=4$	$x>4$
$x+1$	$-$	$0$	$+$	$+$	$+$	$+$	$+$
$x-2$	$-$	$-$	$-$	$0$	$+$	$+$	$+$
$x-4$	$-$	$-$	$-$	$-$	$-$	$\nexists$	$+$
$\dfrac{(x+1)(x-2)}{x-4}$	$-$	$0$	$+$	$0$	$-$	$\nexists$	$+$

The column $x=4$ (marked with $\nexists$, “does not exist”) reminds us that at that point the fraction is not defined.

Solution: we look for where $\ge 0$ (sign $+$ or zero), excluding $x=4$: $\boxed{-1\le x\le 2 \;\;\text{or}\;\; x > 4}$

Watch out: the interval $x>4$ is open on the left: at $x=4$ the fraction does not exist. It would be a mistake to write $x\ge 4$.

Links

Topics: Inequalities
Concepts: Fractional inequality · Point of non-existence · Sign analysis · Sign table
Methods: Sign analysis of fractional inequalities · Sign analysis of a product
Skills: Solving inequalities