I came across a relatively simple problem in one of the reference books whose answer surprised me. It is based on finding the range of numbers that can be taken by x so that the inequality is satisfied.
The question goes on like this:
Find the number of integers satisfying the inequality; (x^2+2)/(x^2-1) < -2
What I did to find the answer was to being the '-2' to LHS and then simply to get a result:
[(3x^2)/{(x+1)(x-1)}] < 0
Since for any Real value of x the numerator will be positive, I took it to the RHS and simplified the inequality further:
[1/{(x+1)(x-1)}] < 0
Using the method of wavy curve:
I get the possible range of values of x as x ∈ (-1,1). Hence the answer to the question asked terns out to be "only one integer which here is zero.".
But here comes the real problem, when "zero, 0" is substituted inn the place of x in the above inequality the answer turns out to be:
{2/(-1)} < -2
Hence, (-2) < (-2)
which is clearly not correct. What is the mistake I am doing or is there some other flaws in the method of wavy curve?
The numerator in $$\frac {3x^2}{(x-1)(x+1)}$ is not positive for all real values of x.
It is zero at $x=0$ where you need to pay a special attention.