how do I get a better intuition for absolute value? authors always

What's your best guess?

>>7805574

Distances. It's literally non-negative distances.

You might fly from New York to Chicago in a straight line (1500 miles, say), and then back to Chicago from New York (1500 miles), but you wouldn't characterize either trip as "-1500 miles". Not under normal circumstances, anyway.

Any time you have some model, or case x of something (or change x to something) which is manifestly non-negative, it's time to start thinking about abs(x). It might not be necessary in the end, but that's where your head should start going.

There are certain elementary calc identities which entail absolute value, as your picture suggests. But of course this is a bit more abstract when the idea of absolute value itself is as simple as can be. I would hazard to tell you what you're really having a hard time with. It's not the notion of absolute value, but just how this-or-that train of thought in a calculus problem was developed.

>>7805574
Someone please tell me if I'm wrong.

In the last equation, there is an equality and one side is being multiplied by x squared.

x squared will always be positive, and the absolute value of cx will also always be positive. If one side is always positive, then the other will also be always positive. So you can just remove the absolute value because it changes nothing.

Is my reasoning correct? I have not yet seen these kind of problems on my own. Just a freshman baby.

>>7805574
in that particular case, both absolute values would "change signs" at the same spot (x=0) as a function of x, so you can safely work without absolute values.

>>7805599

Ignore my language flub in the second paragraph, no need to call me out on it. I was describing a round trip.

>>7805574
abs shows that an expression can throw negatives when whomever put it together doesn't want negatives there.

Like the above anon said with things like distance, working with temperature in Kelvin, time, and the like, often you don't use negatives.

As far as solving them goes, just remember that it's also a sign that you likely are going to have to solve it twice...once for the negative and once for the positive. Using abs also shows that you'll have a negative domain as well if you're looking for domain and range.

literally none of these replies helped me.