Irrational & Rational numbers

GRE Quant Problem Solving
1

a is an irrational number, b is also an irrational number, and ∣a∣ ≠∣b∣. We can definitively conclude that a+b is an irrational number.
The answer is false , stating it is not true. As description it is given “
Root2 +(− root2 +1) -That indeed equals a rational number”

Please can someone explain this.

4

Can you say for sure that the statement is true?

5

This is a proof by contraction.
Let the irrationals be a = \sqrt{2}, b = 1 - sqrt(2)
It satisfies the condition |a| \neq |b|
a + b = sqrt(2) + 1 - sqrt(2) = 1
Since 1 is a rational, a + b is a rational.