CAT 2024 Slot 3 QA Question 15: The number of distinct real values of $x$, satisfying the equation
$\max\{x,2\} - \min\{x,2\} = |x + 2| - |x - 2|$, is

Question

The number of distinct real values of $x$, satisfying the equation
$\max\{x,2\} - \min\{x,2\} = |x + 2| - |x - 2|$, is

Accepted Answer

The left side is the difference between the larger and smaller of $x$ and $2$, so it equals $|x - 2|$.

$|x - 2| = |x + 2| - |x - 2|$
$2|x - 2| = |x + 2|$

Now solve by ranges.

Case 1: $x \geq 2$
$2(x - 2) = x + 2$
$x = 6$

Case 2: $-2 \leq x 
$2(2 - x) = x + 2$
$x = \dfrac{2}{3}$

Case 3: $x 
$2(

The number of distinct real values of xxx, satisfying the equation max⁡{x,2}−min⁡{x,2}=∣x+2∣−∣x−2∣\max\{x,2\} - \min\{x,2\} = |x + 2| - |x - 2|max{x,2}−min{x,2}=∣x+2∣−∣x−2∣, is