Where to put the sign when solving absolute value problems

[curtisemartin]

This might need some discussion.

Remark 1.4.4 Remark 1.4.4
https://tbil.org/preview/precalculus/EQ4.html#EQ4-3-5 suggests rewriting $|ax+b| = c$ as $ax+b = c$ and $ax+b = -c$.

I think it would make better sense to write it as $ax+b = c$ and $-(ax+b)=c$.
Rationale:

• This uses the definition (?) given in Remark 1.4.1, where both signs are applied to the argument of the absolute value, more directly.
• It also provides, in my view, a more consistent way to handle the signs in inequalities regardless of the sense of the inequality. In Activity 1.4.7b, the inequality $|x-7| \le 2$ can be written as a compound inequality, although that can be a little mysterious. When the sense of the inequality is changed, as in Activity 1.4.8, the inequality $|x-7| \ge 2$ cannot be written as a compound inequality. I think both cases are handled more reliably by considering the two signs assigned to the argument of the absolute value. For example: $x-7 \ge 2$ and $-(x-7) \ge2$. This doesn't hide the change in sense when the minus sign is attended to in the second inequality.

[siwelwerd]

I think I agree! @AbbyANoble @tdegeorge @kathypinzon , I don't remember which one of y'all wrote this, but what do y'all think?

[AbbyANoble]

I don't remember who wrote this one (though it's not looking familiar so I don't think it was me?), but I prefer it the way it is because it models the words I would use to describe what's going on when you solve. So for like the beginning of all this we might look at $|x|=4$. What does absolute value mean? It's asking you to find the x-value(s) that are a distance of 4 from 0. That means we can move 4 to the right of zero, meaning $x=4$ or 4 to the left, meaning $x=-4$. We don't need the opposite of x to solve for x. We can think about the actual positions on the number line of 4 and -4. I think Remark 1.4.4 follows well after the scaffolding of Activity 1.4.3.

Regarding the inequalities, I still prefer it as it is to emphasize the potential locations after you travel a certain distance on the number line. For $|x-7| < 2$, I would talk about how that expression must be "close" to zero because you can't go more than two away. So, that x-7 is less than 2 and at the same time bigger than -2. That leads quickly to the inequalities $x-7<2$ and $x-7>-2$. For $|x-7| > 2$, then we know we have to be more than 2 units away from 0. So past 2 or before -2. That leads to $x-7>2$ and $x-7<-2$.

[AbbyANoble]

For what it's worth, I didn't do this section last semester so I haven't looked super closely. But if I did do it in the future, I would use my own notes if we made this change.

[curtisemartin]

It seems to me there are two viewpoints here: one I would call definition and the other I would call meaning. Honestly, I'm not even sure I have those labels correct. In any case, I think both provide useful insight. The definition idea I suggested provides a mechanical way of ensuring that the sign is handled reliably, while using the meaning of distance from zero provides a way of reasoning to the answer. Having both ideas in mind can actually help one check their answer.

[curtisemartin]

Maybe the trouble I am trying to avoid with my suggestion is the student who gets caught between the two ideas and applies the plus and minus signs to the right hand side of the inequality without the reasoning part, resulting in $x-7 < 2$ and $x-7<-2$. This kind of error seems to occur easily when doing work in one's head. Using the replacement of $|x-7|$ with $(x-7)$ and $-(x-7)$ adds a minor inconvenience of an additional operation that I think is fully compensated for by the increased reliability of getting the right answer. Gaining confidence with the reliability of their computations can then help students to appreciate the meaning of "close to" that occurs so frequently with absolute value in later mathematical work.

However the text ends up, I would still present both viewpoints. My original comment was only meant to say that Remark 1.4.1 seems immediately de-emphasized in favor of the view introduced in the first sentence of Activity 1.4.3. If the intent is to emphasize that perspective, perhaps that sentence should be given its own remark.