Correct takeaways from absolute value data sufficiency

[StellaL608]

What is the value of y?

(1) 3| x^ 2 – 4| = y – 2

(2) |3 – y| = 11

(1) INSUFFICIENT: Since this equation contains two variables, we cannot determine the value of y. We can, however, note that the absolute value expression | x ^2 – 4| must be greater than or equal to 0. Therefore, 3| x 2 – 4| must be greater than or equal to 0, which in turn means that y – 2 must be greater than or equal to 0. If y – 2 > 0, then y > 2.

^^ In this answer explanation above, is the correct takeaway that you can use the definition of absolute value to make new inequalities? I broke down the step by step logic below in just slightly more detail so I can understand the inferences.

Absolute values must be positive
| x ^2 – 4| must be positive
| x ^2 – 4| must be greater than or equal to 0
3| x 2 – 4| must be greater than or equal to 0
3| x^ 2 – 4| = y – 2
So y – 2 must be greater than or equal to 0
If y – 2 >= 0, then y >= 2

Is this a "theoretical approach" or would you be able to do the same in a Problem Solving question?

[Sage Pearce-Higgins]

Another good absolute value question. I would say that a useful takeaway here is to remember that point from my last post: absolute values are always positive or zero. So statement (1) is just a complicated way of saying that "y - 2" is greater than or equal to 0. In answer to your question, yes, you can use absolute values to create new inequalities.