so I understand how to calculate the 0.5 quantile of the given question. I calculate the CDF of x and then I multiply it to 0.5 But what if there's more than one function for multiple intervals? How can I calculate the 0.5 quantile of that? The blue is what I am trying to find the 0.5 quantile of. It's the CDF I found from the PDF.
The calculated cdf is not quite right. We have $F(x)$ for $x\le 0$, and $F(x)=\frac{x}{12}$ for $0\lt x\lt 2$. We also have $F(x)=\frac{1}{6}$ between $2$ and $5$. So far fine.
Between $x=5$ and $x=10$, we have $$F(x)=\Pr(X\le x)=\frac{1}{6}+2\cdot\frac{x-5}{12},$$ which is $\frac{x-4}{6}$. More generally, we can write for $x$ in this interval $$F(x)=F(5)+\int_5^x f(t)\,dt.$$
And finally $F(x)=0$ for $x\ge 10$.
The median ($0.5$ quantile) is located in the interval from $5$ to $10$. To find it, we solve $\frac{x-4}{6}=0.5$.