The crucial point with double inequalities is that you must have and .
Personally, I think the simplest way to solve complicated inequalities is to solve the associated equation first.
If [/tex]0= \sqrt{1- x^2}[/tex] then , .
If then , x= 0.
The whole point of that is that the three points, -1, 0, and 1 separate intervals where is less than 0, greater than 0, or does not exist. Try one value of x in each of the intervals x< -1, -1< x< 0, 0< x< 1, and x> 1.
For example, if x= -2< -1, then which is not a real number so does not satisfy the inequality. No value of x less than -1 satisfies the inequality.
If x= -1/2, between -1 and 0, then [tex]\sqrt{1- x^2}= \sqrt{1- 1/4}= \sqrt{3}{2}. which is positive and less than 1. Every value of x between -1 and 0 satisfies the inequality.
If x= 1/2, between 0 and 1, then [tex]\sqrt{1- x^2}= \sqrt{1- 1/4}= \sqrt{3}{2}. which is positive and less than 1. Every value of x between 0 and 1 satisfies the inequality.
Finally, if x= 2, between 0 and 1, then which is not a real number so does not satisfy the inequality. No value of x larger than 1 satisfies the inequality.
Since the numbers -1, 0, and 1 make the two sides equal, the solution set is .