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Constraints on Inequation Representation

This activity belongs to the GeoGebra book GeoGebra Principia. However, not always does the algebraic equation allow GeoGebra to represent the corresponding inequations. As shown in the official manual , this representation is limited to the following cases:  
  • Polynomial inequalities in one variable, like x³ > x + 1
  • Quadratic inequalities in two variables, like x² + y² + x y < 4
  • Linear inequalities in one of the variables, like 2x > sen(y) or y < sqrt(x).
When we find the algebraic equation corresponding to XA – XB = k, we obtain the same as the one corresponding to XA + XB = k:   4 XB2 XA2 = (k² – XA2 – XB2)²   This equation reduces to a quadratic in two variables, allowing GeoGebra to represent its corresponding inequations.
  • Note: The common quadratic equation of the ellipse and hyperbola is nothing but the general equation of a conic a x² + b x y + c y² + d x + e y + f = 0, where the ellipse and the hyperbola differ only by the sign of the discriminant b² – 4 a c.
However, the algebraic equation corresponding to XA XB = k doesn't represent a conic, so GeoGebra can't represent the corresponding inequations. On the other hand, the algebraic equation corresponding to XA = k XB becomes a conic once again, allowing GeoGebra to represent the corresponding inequations.
Author of the construction of GeoGebra: Rafael Losada.