Need help to solve this:

0<√1-x^2<1

answer is -1<x<1

How does that work?

Thanks.

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- March 5th 2010, 12:46 AManonymous_mathsDouble Inequalities
Need help to solve this:

0__<__√1-x^2__<__1

answer is -1__<__x__<__1

How does that work?

Thanks.

- March 5th 2010, 01:27 AMProve It
- March 5th 2010, 01:32 AMHaven
- March 5th 2010, 02:01 AManonymous_maths
yes, but that's different to the answer supplied of -1

__<__x__<__1 - March 5th 2010, 02:18 AMProve It
- March 5th 2010, 03:34 AMHallsofIvy
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 . - March 5th 2010, 07:22 AMKrizalid
it's true as long as

as for (1) fix and by squaring we get which holds for any thus the solution set for (1) is again and the final solution set it's just