The Truth About Absolute Values

Just because a method works does not mean it is right. That's the lesson learned from our latest round of Algebra homework. Here's the offending problem:

2 | 4x - 1 | + 3 = 9

While working through the above absolute value formula, the positive side calculates correctly but the negative did not. Note that the blue ink in the photo is the original and notations, red ink is the failed attempt and green is the correct solution.

Besides memorizing the evaluation process, I've been trying to understand the theoretical concept that is taking place. What I concluded is
that the absolute value must be treated as a single idea. Even Google defines absolute as "not qualified or diminished in any way; total."

When multiplying out each element inside the absolute value, we mistakenly doubled the absolute value but it's equal on the opposite side remained untouched. This works with parenthesis () but those denote organization as opposed to undiminished value. By failing to understand the theory, blindly following a sequence of memorized steps gave us the wrong answer.

Isn't it the same way with God's moral law? We try so hard to apply rules to every little situation, developing complex patterns and justifications. The fundamental thought is that a = c and a = -c where a is the absolute value. So we should first reduce the function to it's bare comparison before solving for x. In life, we should also reduce the situation down to a bare comparison with God's absolute truth and then apply it our problem.

I also struggled to think of a real world situation where we would need to use this formula. Most online explanations failed to convert their illustration into a workable formula but I did find a nice explanation using body temperature and the standard deviation from normal.