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Mathematics for Social SciencesMadrid 2026Question 3.2Free lesson

Linear programming: feasible region and minimum value

From each inequality to a half-plane, from half-planes to vertices, and why the optimum sits at a corner

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Problem statement

(2.5 marks) Consider the region $S$ of the plane bounded by the following constraints: $x \le 6$, $\;3y-2x \le 10$, $\;3y \ge 2-2x$, $\;x \le 10-y$, $\;y \ge x-6$.

3.2.a) (2.0 marks) Find the coordinates of the vertices of $S$ and sketch the region $S$.

3.2.b) (0.5 marks) We want to minimise three times $y$ minus half of $x$ over $S$. State the minimum value and the point of the region where it is reached.

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Mathematics for Social Sciences · Question 3.2. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.

What this lesson answers

  1. What is asked, and what can we predict without drawing?
  2. How do we turn an inequality into a piece of the plane?
  3. Which region satisfies all five at once?
  4. Why are vertices crossings of two lines, and which ones count?
  5. Why is the minimum at a vertex?
  6. Is there another route without sliding the line, and when is each one better?
  7. The answer and how to check it
  8. Common mistakes to avoid
  9. Exam tips
  10. If you want full marks

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