1. When $-1 \leqslant x \leqslant 2$, the function $y=(k-2)x + 2|k|-1$ is always positive for $x$. Then the range of values for $k$ is
$$
\text { II. } 1 . k>\frac{5}{4} \text {. }
$$

From the problem, we have
$$
\left\{\begin{array} { l } 
{ 2 - k + 2 | k | - 1 > 0 , } \\
{ 2 k - 4 + 2 | k | - 1 > 0 , }
\end{array} \text { i.e., } \left\{\begin{array}{l}
2|k|-k+1>0, \\
2|k|+2 k-5>0 .
\end{array}\right.\right.
$$

When $k \geqslant 0$, we get $k>\frac{5}{4}$; when $k<0$, we also get $k>\frac{5}{4}$.