Solve the inequality (|x − 2| − x) / (x² − 4) ≤ 0 and give the solution in interval notation.

Step-by-step solution

1. 1

   Exclude the points where the expression is undefined. A fraction is only defined when its denominator is non-zero. Here $x^2-4=(x-2)(x+2)$, so

   $x\ne 2\quad\text{and}\quad x\ne -2.$

   These two points can never belong to the solution set, no matter what the numerator does — a $\le$ sign never rescues a zero denominator.

2. 2

   Split the absolute value at its breakpoint. $|x-2|$ changes formula at $x=2$: it equals $x-2$ when $x\ge 2$ and $2-x$ when $x<2$. Because the whole inequality behaves differently on the two sides, solve it separately on each and then take the union. Do it algebraically rather than by testing sample numbers, so nothing is missed between test points.

3. 3

   Case 1: $x>2$. Here $|x-2|=x-2$, so the numerator collapses to a constant:

   $\frac{(x-2)-x}{x^2-4}=\frac{-2}{x^2-4}.$

   For $x>2$ the denominator $x^2-4$ is positive, so the fraction is negative for every such $x$. The whole ray $(2,+\infty)$ satisfies the inequality (with $x=2$ itself excluded from step 1).

4. 4

   Case 2: $x<2$. Here $|x-2|=2-x$, so the numerator is $2-2x=2(1-x)$ and

   $\frac{2(1-x)}{(x-2)(x+2)}\le 0.$

   The numerator is zero at $x=1$ and positive for $x<1$; the factor $x-2$ is negative throughout this case; the factor $x+2$ changes sign at $x=-2$.

5. 5

   Build the sign table for Case 2. Split the line at the critical values $-2$ and $1$:

   interval	$2(1-x)$	$x-2$	$x+2$	fraction
   $x<-2$	$+$	$-$	$-$	$+$
   $-2<x<1$	$+$	$-$	$+$	$-$
   $1<x<2$	$-$	$-$	$+$	$+$

   Only $-2<x<1$ makes the fraction negative. At $x=1$ the numerator is $0$ while the denominator is $-3\ne 0$, so the fraction equals $0$ — and $0\le 0$ is true, so $x=1$ is included.

6. 6

   Take the union of the two cases. Combining $(-2,1]$ from Case 2 with $(2,+\infty)$ from Case 1:

   $(-2,\,1]\ \cup\ (2,\,+\infty).$

   Spot-checking confirms it: at $x=0$ the value is $-0.5$ (true), at $x=1.5$ it is $+0.571$ (false), at $x=3$ it is $-0.4$ (true), and at $x=-2.5$ it is $+3.11$ (false).