Fractional inequality with an absolute value ((|x|-1)/(x-2) ≤ 0)

Statement

Solve the inequality $\dfrac{|x|-1}{x-2}\le0$.

Solution

The signs of the numerator and of the denominator are studied separately.

• Numerator $|x|-1$: it vanishes at $x=\pm1$; it is negative for $-1<x<1$ and positive for $|x|>1$.
• Denominator $x-2$: it vanishes at $x=2$ (excluded value); negative for $x<2$, positive for $x>2$.

The fraction is $\le0$ where numerator and denominator have opposite signs, or where the numerator vanishes (with denominator $\neq0$):

Interval	$\lvert x\rvert-1$	$x-2$	fraction
$x\le-1$	$\ge0$	$<0$	$\le0$ ✓
$-1<x<1$	$<0$	$<0$	$>0$
$1\le x<2$	$\ge0$	$<0$	$\le0$ ✓
$x>2$	$>0$	$>0$	$>0$

Solution: $\boxed{x\le-1\ \lor\ 1\le x<2}$.

Links

Topics: Parabola
Concepts: Absolute value inequalities · Sign table · Absolute value
Skills: Reasoning by cases · Solving inequalities
Exercise type: Solving an inequality