Algebraic Mess!

Algebra Level 5

The number of integral solution(s) the two inequalities

5<2x1<22αβγyα3+β3+γ3 \begin{aligned} -5 < & 2x-1 & < 2 \sqrt{2} \\ \alpha \beta \gamma \leq & y& \leq \alpha^3 + \beta^3 + \gamma^3 \\ \end{aligned}

are pp and qq respectively.

Given that

1α+1β+1γ=121α2+1β2+1γ2=94α+β+γ=2 \begin{aligned} \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{aligned}

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

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

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