About Solving the Following System of Inequalities

Question

I would like to solve the following system of inequalities for a range of $\alpha$:

\begin{align*} p_1 + p_4 + p_3 & = \alpha\tag1 \\ p_1 + p_2 + p_5 & = \alpha\tag2 \\ 1 - p_1 - p_4 - p_5 & = \alpha\tag3 \\ 0 \leq p_1 & \leq 1\tag4 \\ 0 \leq p_2 & \leq 1\tag5 \\ 0 \leq p_3 & \leq 1\tag6 \\ 0 \leq p_4 & \leq 1\tag7 \\ 0 \leq p_5 & \leq 1\tag8 \\ 0 \leq 1 - \sum_{i=1}^5p_i & \leq 1.\tag9 \end{align*}

What I have tried is to add and\or subtract scalar multiple of one equation from another, but it does not go any where. I do not know if there is any algorithm for solving this kind of question. I am hoping to get that $\alpha \in \left[\frac{1}{3},\frac{2}{3}\right]$, even though I am not completely sure if it is possible. Could someone please help me out? Thanks a lot in advance!

Answer 1

Some thoughts.

We have \begin{align*} \alpha - \frac13 &= - \frac13(p_1 + p_4 + p_3 - \alpha) - \frac13(p_1 + p_2 + p_5 - \alpha)\\[6pt] &\qquad - \frac13(1 - p_1 - p_4 - p_5 - \alpha) + \frac13 p_1 + \frac13 p_2 + \frac13 p_3\\[6pt] &\ge 0. \end{align*}

Also, we have \begin{align*} \frac23 - \alpha &= \frac13 (p_1 + p_4 + p_3 - \alpha) + \frac13(p_1 + p_2 + p_5 - \alpha)\\[6pt] &\qquad + \frac13(1 - p_1 - p_4 - p_5 - \alpha)\\[6pt] &\qquad + \frac13p_4 + \frac13 p_5 + \frac13(1 - p_1 - p_2 - p_3 - p_4 - p_5)\\[6pt] &\ge 0. \end{align*}