[Algebra]
Problem: Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x|+k$ do not intersect.

Solution: Since each of these two graphs is symmetric about the $y$-axis (i.e. both are even functions), then we only need to find $k$ so that there are no points of intersection with $x \geq 0$.

So let $x \geq 0$ and consider the intersection between $y=2 x+k$ and $y=x^{2}-4$.

Equating, we have, $2 x+k=x^{2}-4$.

Rearranging, we want $x^{2}-2 x-(k+4)=0$ to have no solutions.



For no solutions, the discriminant is negative, i.e.

$$
\begin{aligned}
20+4 k & <0 \\
4 k & <-20 \\
k & <-5 .
\end{aligned}
$$

So $y=x^{2}-4$ and $y=2|x|+k$ have no intersection points when $k<-5$.



Final answer: (-\infty,-5)