Fractional inequality with a common denominator

When several fractions appear, everything is brought to a single fraction without ever cancelling the denominator. The zeros of the denominator remain points of non-existence.

Example — Fractional inequality with a common denominator

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

Typical mistake: taking the common denominator as one does with equations and then cancelling it. Never do this!

Correct procedure: bring to a single fraction without eliminating it: $\frac{1}{x-1}-\frac{1}{x+2} = \frac{(x+2)-(x-1)}{(x-1)(x+2)} = \frac{3}{(x-1)(x+2)}.$ The inequality becomes $\dfrac{3}{(x-1)(x+2)} > 0$. Since $3>0$, it is enough to study the sign of $(x-1)(x+2)$:

	$x<-2$	$x=-2$	$-2<x<1$	$x=1$	$x>1$
$x-1$	$-$	$-$	$-$	$0$	$+$
$x+2$	$-$	$0$	$+$	$+$	$+$
$(x-1)(x+2)$	$+$	$\nexists$	$-$	$\nexists$	$+$

The columns $x=-2$ and $x=1$ (marked with $\nexists$) are points of non-existence, because they made the original denominator zero.

Solution: $\boxed{x<-2 \;\text{or}\; x>1}$.

Links

Topics: Inequalities
Concepts: Fractional inequality · Point of non-existence · Sign analysis · Sign table
Skills: Solving inequalities · Simplifying