Is |x| + |x -1| = 1? (1) x ≥ 0 (2) x ≤ 1

This one is very tricky!

Is |x| + |x -1| = 1?
(1) x ≥ 0
(2) x ≤ 1

Q is \(|x| + |x -1| = 1\). Let's check when this equation holds true. We should consider three ranges (as there are two check points \(x=0\) and \(x=1\)):

A. \(x<0\) --> \(-x-x+1=1\) --> \(x=0\), but this solution is not valid as we are checking the range \(x<0\);

B. \(0\leq{x}\leq{1}\) -->\(x-x+1=1\) --> \(1=1\), which is true. That means that for ANY value from the range \(0\leq{x}\leq{1}\), equation \(|x| + |x -1| = 1\) holds true.

C. \(x>1\) --> \(x+x-1=1\) --> \(x=1\), but this solution is not valid as we are checking the range \(x>1\).

So we get that equation \(|x| + |x -1| = 1\) holds true ONLY in the range \(0\leq{x}\leq{1}\).

Statements:
(1) \(x\geq{0}\). Not sufficient, as \(x\) must be also \(\leq{1}\);
(2) \(x\leq{1}\). Not sufficient, as \(x\) must be also \(\geq{0}\);

(1)+(2) \(0\leq{x}\leq{1}\), exactly the range we needed. Sufficient.

Answer: C.

Bunuel, how do you know that the second range \(0\leq{x}\leq{1}\) includes both 0 and 1 and the first and second range don't include 0 and 1?