= Solution
Integrating the two linear pieces gives the <cumulative distribution function>
$$
F(x)=\begin{cases}
0,&x<1,\\(x-1)^2/12,&1\le x\le3,\\1-(7-x)^2/24,&3\le x\le7,\\1,&x>7.
\end{cases}
$$
The split probability is $F(3)=1/3$. Invert each branch to obtain the <method of inversion>:
$$
\boxed{X=\begin{cases}1+\sqrt{12U},&0\le U\le1/3,\\7-\sqrt{24(1-U)},&1/3<U\le1.\end{cases}}
$$
Both expressions equal three at the split. For an ideal uniform draw, monotonicity gives $\mathbb P(F^{-1}(U)\le x)=\mathbb P(U\le F(x))=F(x)$, proving the target law. \b[Inversion is preferable here]: its quantile is explicit, it uses one uniform and one square root per output, and has no rejected draws. Rejection remains useful for densities with no convenient inverse CDF.
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