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# Inequalities watch

1. Simple question relly but I can only get half of it...

Solve this inequality:
2/(x^2 - 1) > 1
I get that re-arranging to 0 > x^2 -3
x can be + or - square root 3

but also + or - 1 is a solution but how do you show this?
2. The solution to inequalities is normally a range of values, or several ranges.

You have to be careful not to multiply by something (like (x^2-1)) that may be either positive or negative.

So you could start by saying 2/(x^2-1) - 1 > 0 it helps to get a 0 on the RHS

Put everything over a common denominator; factorise numerator and denominator

Now it depends on your taught method. You can sketch. Or you can build a table of 'critical values' at which one factor changes sign. And then break x into those ranges, consider each one separately and think hard about boundary values.
3. Since the sign of (x^2-1) is not the same for different values of x, it's best not to multiply both sides by (x^2-1).

Instead, bring the 1 to the LHS and form one fraction:

This inequality shows that you should look at the behaviour of the fraction around the points -3,-1,1,3 (x<-3, -3<x<-1 etc.) to find the solutions.
4. ahhh of course!! Thanks a lot!!

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Updated: February 4, 2010
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