The integer ‘k’, for which the inequality  $x^2-2(3k-1)x+8k^2-7>0$ is valid for every x in R, is

$x^2-2(3k-1)x+8x^2-7>0\Rightarrow$ The roots are imaginary or (real and equal)
$\Rightarrow$  Discriminant $\le 0\Rightarrow 4{(3k-1)}^2-4\times 1\times (8k^2-7)\le 0\Rightarrow 9k^2-6k+1-8k^2+7\le 0$
$\Rightarrow k^2-5k+8\le 0\Rightarrow (k-4)(k-2)\le 0\Rightarrow k\in [2,4]$ .
The number of integer values of k is 3.