How do I confirm that the constraint set wL + rK < Q is convex
Consider an arbitrary pair of vectors $(L_{1},K_{1})$ and $(L_{2},K_{2})$ that satisfy $wL_{i} + rK_{i} \leq Q$ for $i = 1, 2$. To show that constraint set is convex, we need to show that any convex combination of that (arbitrary) pair of vectors lies in the constraint set. Namely,
for all $\gamma \in [0,1]$ we have 

$$\begin{split}
w\left(\gamma L_{1} + (1-\gamma)L_{2}\right) + r\left(\gamma K_{1} + (1-\gamma)K_{2}\right) &amp;= \gamma \left(w L_{1} + r K_{1}\right) + \left(1-\gamma \right)\left(w L_{2} + r K_{2}\right)\\
&amp;\leq \gamma Q + (1-\gamma)Q = Q
\end{split}$$
Thus the vector, $(L_{3},K_{3})$, is also in the constraint set (where $L_{3} := \gamma L_{1} + (1-\gamma)L_{2}$, and $K_{3} := \gamma K_{1} + (1-\gamma)K_{2}$) and so the set is convex.

Your answer (or at least the rearranging of inequalities in your answer) is largely correct. However, it is important to note that you've shown this inequality for an arbitrary vector in the set and for an arbitrary convex combination and not for a particular one.