Question 741468
Assuming k is any real number...

{{{(1 + kx)/(1 + x^2) < k}}} 
leads to
{{{(1 + kx)/(1 + x^2) - k < 0}}}

then

{{{(1+kx)/(1+x^2)-k((1+x^2)/(1+x^2)) < 0}}}

and combining them we get

{{{((1+kx)-k(1+x^2))/(1+x^2) < 0}}}

after simplifying the numerator...

{{{(1+kx-k-kx^2)/(1+x^2) < 0}}}

Notice that {{{1+x^2}}} is always positive, so to determine when the sign of the expression will be less than zero we only need to look at the numerator:

{{{-kx^2+kx+(1-k)}}}

using the quadratic formula to find x...

{{{x = (-k +- sqrt( k^2-4*(-k)*(1-k) ))/(2*(-k)) }}}

then...

{{{x = (-k +- sqrt( k^2-4*(-k+k^2)))/(-2k) }}}

then...

{{{x = (-k +- sqrt( k(4-3k) ))/(-2k) }}}

which only has a positive real value when

{{{4-3k<=0}}} and {{{k<=0}}}


or

{{{4-3k>=0}}} and {{{k>=0}}}

So we notice x is only real on the interval 

{{{0<=k<=4/3]}}}

This interval gives us boundary conditions to test. After testing (using a graphing utility) we see only the values k>4/3 satisfies the inequality for ALL values of x