Algebraic Mess!

Algebra Level 5

The number of integral solution(s) the two inequalities

\[ \begin{eqnarray} -5 < & 2x-1 & < 2 \sqrt{2} \\ \alpha \beta \gamma \leq & y& \leq \alpha^3 + \beta^3 + \gamma^3 \\ \end{eqnarray} \]

are \(p\) and \(q\) respectively.

Given that

\[ \begin{eqnarray} \dfrac{1}{\alpha} + \dfrac{1}{\beta} + \dfrac{1}{\gamma} & = & \dfrac{1}{2} \\ \dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} + \dfrac{1}{\gamma^2} & = & \dfrac{9}{4} \\ \alpha + \beta + \gamma & = & 2 \\ \end{eqnarray} \]

What is the value of \(\lvert p-q\rvert?\)

This problem is an extension to the note: Solving Linear Inequalities from the set: Inequalities

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