What are the values of x that will satisfy the equation:
|x - 2| = |x - 3| + |x - 5|?

The CRITICAL POINTS are 2, 3 and 5.
These are the values where the expressions within the absolute values are equal to 0.
When x is to the right or left of these critical points, some of the expressions within the absolute values might be LESS than 0 -- unacceptable, since an absolute value must be nonnegative.
Implication:
If within a given range an expression will be less than 0, we must FLIP ITS SIGNS.

Case 1: x>5
Since none of the expressions is less than 0 in this range, no signs need to be flipped.
x-2 = x-3 + x-5
x-2 = 2x-8
6=x
x=6.

Case 2: 3<x<5
Since x-5<0 in this range, we have to flip the signs in this expression.
x-2 = x-3 + 5-x
x-2 = 2
x=4.

Case 3: 2<x<3
Since x-3<0 and x-5<0 in this range, we have to flip the signs in these expressions.
x-2 = 3-x + 5-x
x-2 = 8-2x
3x = 10
x = 10/3.
Since only values such that 2<x<3 are valid in this range, x=10/3 is not a valid solution.

Case 4: x<2
Since x-2<0, x-3<0 and x-5<0 in this range, we have to flip the signs in all 3 expressions.
Flipping the signs in all 3 expressions is equivalent to multiplying both sides of the equation by -1.
Since multiplying both sides of an equation by the same value does not change the equation, the solution when ALL of the signs are flipped (Case 4) will be the same as when NO signs are flipped (Case 1).
Since Case 4 will yield the same value for x as did Case 1, no need to solve Case 4.

The solutions of the equation are [spoiler]x=4 and x=6[/spoiler].

More practice:
https://www.beatthegmat.com/complex-abso ... 73453.html