Systems of linear inequalities in two unknowns

Given the system of linear inequalities $x + y < 5$ and $2x - y \ge 0$, determine which of the following points are solutions to the system: $A(2, 1)$, $B(0, 0)$, $C(3, 3)$ and $D(4, 0)$.

Represent the inequality $2x - 3y \le 6$ graphically. Then, state two points whose coordinates belong to the solution region and two points that do not.

You can do it on paper and send a photo with your answer.

Consider the following system of linear inequalities:

1. a) Estimate: Without drawing the axes or performing written calculations, observe the algebraic expressions and intuitively reason whether the points $P(2, 3)$ and $Q(4, 3)$ will belong to the feasible region. Justify your prediction by mentally estimating the sums $x+y$ and $2x+y$ for each point compared to the boundary values $6$ and $4$.
2. b) Check: Formally substitute the coordinates of $P$ and $Q$ into each of the inequalities of the system to verify whether your estimation was correct.
3. c) Assess: Graph the feasible region and visually confirm that the positions of $P$ and $Q$ match what you deduced.

From the shaded region in the figure, determine the linear inequality in two variables that defines it. Note that the boundary line passes through the points $(0, 2)$ and $(3, 0)$ and that the region includes the origin.

Represent the region of the plane that simultaneously satisfies the following constraints: $3x + y \ge 6$, $x \le 4$ and $y \le 3$.

To graphically solve the inequality $x - 2y > 0$, Cian carries out the following procedure in his notebook:

1. He draws the boundary line $x - 2y = 0$ as a solid line.

2. As a test point to find out which half-plane to shade, he chooses the origin $(0,0)$.

3. He substitutes into the inequality: $0 - 2 \times 0 > 0 \implies 0 > 0$, and reasons: "Since the inequality is false, the origin is not a solution; therefore, the solution is the half-plane where the origin is not located, and I include the boundary line because it is what separates both regions."

1. a) Identify and explain the two conceptual errors Cian made (one regarding the choice of the test point and the other regarding the type of line for the boundary).
2. b) Correct the exercise: indicate what type of line corresponds to the boundary, choose an appropriate test point that does not lie on the line (justifying your choice), and clearly determine which half-plane is the solution.

Find algebraically the coordinates of the vertices of the feasible region defined by the following system of inequalities: $x \ge 0$, $y \ge 0$ and $x + y \le 4$.

Check analytically whether the point $P(1, 2)$ belongs to the region defined by the system of inequalities: $2x + y \le 5$, $x - 3y \ge -6$ and $3x + 2y \ge 1$.

Graphically represent the system formed by the inequalities $x + 2y \ge 4$ and $x - y \le 1$, clearly shading the resulting feasible region.

You can do it on paper and send a photo with your answer.

Represent the solution set of the system formed by the inequalities $x + y \ge 2$ and $x + y \le 5$. Describe what type of geometric region is obtained.

The figure shows a polygonal region in the Cartesian plane representing the viable production combinations of two electronic components. Determine the system of inequalities that defines this region, given that its vertices are $A(1, 1)$, $B(5, 1)$, $C(4, 4)$ and $D(2, 3)$.

Calculate the area of the region of the plane defined by the following system of inequalities: $y \ge x + 2$, $y \ge -x + 2$, and $y \le 4$.

On an urban planning map, a triangular green area has been defined whose exact vertices are $A(0, 1)$, $B(4, 1)$, and $C(2, 5)$.

1. a) Obtain the general or implicit equation ($ax + by = c$) of each of the three lines containing the sides of the triangle (sides $AB$, $AC$, and $BC$).
2. b) Write a system of three linear inequalities with two variables whose feasible region matches exactly the interior and the sides of said triangle.
3. c) Justify how you decided the direction of each inequality ($\le$ or $\ge$) by verifying that the coordinates of an obvious interior point (such as the point $(2, 2)$) simultaneously satisfy the three formulated inequalities.

A bakery makes two types of cake, T1 and T2. Cake T1 requires $2$ hours of preparation and $1$ hour of baking, while cake T2 requires $1$ hour of preparation and $2$ hours of baking. A maximum of $10$ hours of preparation and $8$ hours of baking are available per day. Set up the system of inequalities that models this situation (considering that the quantities cannot be negative) and graphically represent the region of possible daily production quantities.

You can do it on paper and send a photo with your answer.

A farmer has $50$ hectares available to plant wheat and barley. Wheat seed costs $20$ euros per hectare and barley seed costs $30$ euros per hectare. The maximum budget for seeds is $1200$ euros. Furthermore, due to environmental regulations, the area of barley cannot exceed twice the area of wheat. Write the associated system of inequalities and find the vertices of the feasible region.

A refinery produces two types of alloys, A and B. For alloy A, $3$ kg of copper, $1$ kg of zinc and $2$ kg of tin are needed per unit. For alloy B, $1$ kg of copper, $2$ kg of zinc and $3$ kg of tin. A maximum of $300$ kg of copper, $200$ kg of zinc and $240$ kg of tin is available in the warehouse. Set up the system of inequalities describing the possible production of both alloys, represent the feasible region and calculate the coordinates of all its vertices.

Consider the region of the plane defined by the inequalities $x + 2y \ge 4$, $3x - y \le 6$, and $x - y \ge -2$. Prove algebraically that it is a bounded region and determine the coordinates of the vertices of this region.

Given the system of inequalities $x + y \le 6$, $2x + y \le 8$, $x \ge 0$ and $y \ge k$, discuss, in terms of the parameter $k$, the number of vertices of the feasible region. For which values of $k$ does the feasible region reduce to the empty set?

We want to determine whether any solution exists for the following system of inequalities:

To solve it, three students propose different approaches:

• Approach A (Geometric): Accurately draw the four boundary lines, shade each half-plane using test points, and find the common region.

• Approach B (Direct algebraic): Multiply the first inequality by $2$ and add it term by term to the second to analyze the resulting inequality.

• Approach C (Vertex calculation): Solve the $2 \times 2$ systems of linear equations by equating the lines in pairs to find the vertices of the feasible region.

1. a) Apply Approach B and explain what immediate and indisputable conclusion can be drawn about the feasible region.
2. b) Critically compare the three approaches: argue why Approach B is the most efficient and explain the waste of time or unnecessary difficulty faced by those who apply Approach A or Approach C (analyze the relative position of the boundary lines).
3. c) Justify in what type of systems of inequalities problems it would indeed be essential to apply Approach A or Approach C instead of Approach B.