How to solve the system of inequalities when using topological proof to show that there are exactly 5 platonic solids?I have seen other answers explaining the topological proof up until the point of 1 / p + 1 / q &gt; 1 / 2 and p, q are greater than or equal to threeThen they proceed to say that the 5 platonic solids have the only values that satisfy these conditions. My question is how do I solve for these values?

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How to solve the system of inequalities when using topological proof to show that there are exactly 5 platonic solids?I have seen other answers explaining the topological proof up until the point of  1  /  p  +  1  /  q  &amp;gt;  1  /  2 and   p,   q are greater than or equal to threeThen they proceed to say that the 5 platonic solids have the only values that satisfy these conditions. My question is how do I solve for these values?

Kiana Cantu · Accepted Answer

We&#039;re looking for integer values of   p and   q that satisfy the three given equations   p  ≥  3,   q  ≥  3 and       1    p    +      1    q    &amp;gt;      1    2  .Suppose both   p and   q are not equal to   3. Since they are integers, they must both be greater or equal to   4. But if that&#039;s the case, then       1    p   and       1    q   are both less or equal to       1    4  , meaning that       1    p    +      1    q    ≤      1    4    +      1    4    =      1    2  , contradicting one of the equations.So to solve the equation either   p or   q must be equal to   3.Let&#039;s consider only the case where   p is equal to   3. Then we have       1    3    +      1    q    &amp;gt;      1    2  . Subtracting       1    3   from both sides yields       1    q    &amp;gt;      1    6  , which implies   q  &amp;lt;  6.So if   p is equal to   3,   q must be an integer with   3  ≤  q  &amp;lt;  6, so   q must be either   3 or   4 or   5.By a symmetrical argument, if   q is equal to   3,   p must be equal to   3 or   4 or   5.So we have five cases remaining:  (  p  =  3  ,  q  =  3  )  ,  (  p  =  3  ,  q  =  4  )  ,  (  p  =  3  ,  q  =  5  )  ,  (  p  =  4  ,  q  =  3  )  ,  (  p  =  5  ,  q  =  3  )  .It is now easy (and necessary) to check that all these five cases satisfy the three given equations.There isn&#039;t really a general method to do this kind of proof, it just requires finding a bunch of solutions, then a sufficient bunch of good arguments for why there aren&#039;t any more solutions.

prirodnogbk · Accepted Answer

Multiply both sides of the inequality by   2  p  q to get   2  q  +  2  p  &amp;gt;  p  q, which is equivalent to   (  p  −  2  )  (  q  −  2  )  &amp;lt;  4. Now   p  −  2 and   q  −  2 are integers   ≥  1, so it is easy to list the five possibilities:  (  p  −  2  ,  q  −  2  )  ∈  {  (  1  ,  1  )  ,  (  1  ,  2  )  ,  (  1  ,  3  )  ,  (  2  ,  1  )  ,  (  3  ,  1  )  }  ,which implies that  (  p  ,  q  )  ∈  {  (  3  ,  3  )  ,  (  3  ,  4  )  ,  (  3  ,  5  )  ,  (  4  ,  3  )  ,  (  5  ,  3  )  }  .

How to solve the system of inequalities when using topological proof to show that there are exactly

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