where Öx is the positive root.
The following solution is from Ang Zhi Ping, River Valley
High School, Singapore. Hyeyoun Chung, St Paul's Girls'
School, London and Yatir Halevi from Maccabim and Reut
High School, Israel also sent excellent solutions.
Taking Ö(x)=p, thus we are solving for p + 1/p < 4.
Multiplying both sides by the positive number p a quadratic
inequality is obtained, namely p2 + 1 < 4p or p2 - 4p + 1 < 0. To factorise the quadratic expression, we find the roots by
using the formula
p=
-b ±Ö(b2-4ac)2a
,
where a,
b and c are the coefficients of p2, p and the constant in the
quadratic expansion. The given inequality holds when
(p - (2 + Ö3))(p -(2 - Ö3)) < 0.
Knowing that (p -(2 + Ö(3)) < (p -(2 - Ö(3)) for all real
values of p, to make the product negative for all values of p it
follows that (p -(2 + Ö(3)) must be the negative factor and
(p -(2 - Ö(3)) the positive factor.
So, the intersection of both ranges p - (2 + Ö3) < 0 and p - (2 - Ö3) > 0 is found to be
2 - Ö3 < p < 2 + Ö3.
Substituting p = Öx, we eliminate the square root by
squaring the whole inequality, thus, we get the answer as: