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baboosaa
I was learning my maths lesson. In an entrance preparation book i saw this question:
-x< x*x< 2x + 1
Bikerman
0<x<1+root2
infinisa
 Bikerman wrote: 0

Right answer, but I suspect baboosaa would like to know how one gets the answer:

First, we separate into 2 inequalities, both of which must be satisfied:
a) -x< x^2
b) x^2< 2x + 1

Starting with a), we have:
x^2 + x > 0
which factorises to x(x + 1) > 0
The graph of the left side is a a parabola, with +ve values outside the zeros
(Algebraically, between the 2 zeros, the terms have opposite sign and so negative product)
To find the zeros, use factorised form: x (x + 1) = 0
The zeros are x= 0 and x= -1
So a) is true if x < -1 OR x > 0

Similarly for b), we have:
x^2 - 2x - 1 < 0
Once again, the graph of the left side is a a parabola, with -ve values between the zeros
This time, we can't easily factorize, so must use the quadratic formula to find the zeros, which gives:
(2 ± √(8))/2 which simplifies to 1 ± √(2) (as √(8) = 2√(2))
So b) is true if 1 - √(2) < x < 1 + √(2)

Since a) and b) must both be true, and -1 < 1-√(2), we get
0 < x < 1 + √(2)
as per Bikerman

Hope this helps
Bikerman
I thought providing the answer in this case might stimulate the questioner to play around with it to see where it came from