In the solution to the problem following session 8, it reads
"We know that -1 is less than or equal to sin(x) which is less than or equal to 1."
I'm good with this, but then it reads:
"So it must be true that:"
-1/x is less than or equal to sin(x)/x which is less than or equal to 1/x.
I don't understand (algebraically) this could be true of all values of x. Can someone please explain?
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
If you begin with the inequality stated after "We know that," and multiply every term by 1/x, we get the inequality stated after "So it must be true that." We're just multiplying through by 1/x.
You might wonder how we can get away with this, for two reasons. One is the problem at x=0, and the solution acknowledges that we don't have a clear understanding of what happens there. The other is what happens when x turns negative, because multiplying an inequality by a negative number changes the direction of the inequality. That doesn't happen here, which I guess can be explained by the fact that sin(-x)=-sin(x).
Ok. So I understand the algebra now. Thanks!
However, I'm a little confused about the "x turns negative" situation. Can you possibly explain this further?
I probably caused unnecessary confusion with that remark, just thinking out loud about the general rule that when you multiply an inequality by a negative number, the inequality reverses direction. For example, x>4 implies that -x<-4. That made me wonder why we can multiply by 1/x through this inequality and have it work even when 1/x is negative. It doesn't cause a problem here so we don't have to worry about it.