I am having a hard time understanding how from
r^2 < |r|
we arrived at the conclusion that ---------------> -1<r<1
What approach is to be used in arriving at the conclusion above? An algebraic approach or an approach of plugging numbers will be more useful?
Think about it this way . . . most numbers when squared become larger. For example, 5 squared is 25. So in this case, we need to think about which numbers become smaller when squared. The first thought that should occur to you is fractions between 0 and 1. For example, 1/2 squared is 1/4.
Now, we may have one other possibility since the denominator has the absolute value of r. What if I square a number between -1 and 0? Well, the squaring will make that number positive, but it will still be smaller than the absolute value of that number (since that particular value will be positive). For example, -1/2 squared is 1/4, and the absolute value of -1/2 is 1/2. So these numbers fit too.
Personally, I think the knowledge that fractions between 0 and 1 become smaller when squared is all the information you need to solve, and then you can just test numbers from there to help you include -1 to 0. Much better than algebra!