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## Homework Statement

Find all x such that ##|x-1|+|x-2|>1##.

## Homework Equations

Definition of absolute value:

|x| = x if x ≥ 0.

|x| = -x if x ≤ 0.

## The Attempt at a Solution

I figured the most straightforward way if to do this case-wise:

Case 1: ##(x-1)>0 \wedge (x-2)>0## then

##(x-1) + (x-2) > 1 \implies x > 2.##

Case 2: ##(x-1)<0 \wedge (x-2)<0## then

## (1-x) + (2-x) > 1 \implies x<2.##

Case 3: ##(x-1)>0 \wedge (x-2)<0## then

## (x-1)+(1-x) > 1 \implies 0 >1. ##

Case 4: ##(x-1)<0 \wedge (x-2)>0## then

## (1-x) + (x+2) > 1 \implies 3 > 1. ##

Cases 3 and 4 are bothering me because x 'drops out.' Case 3 makes no sense and has me wondering if Case 4 even makes sense.