Exercises: Linear inequalities in two variables: systems

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.

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 $D(4, 0)$.

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.

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.

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 $x + y \le 4$.

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

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 $3x + 2y \ge 1$.

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.

To graphically solve the inequality $x - 2y > 0$, Mateo 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: $0 - 2 \cdot 0 > 0 \implies 0 > 0$, 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 Mateo 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.

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.

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.

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 $D(2, 3)$.

To transport $60$ tonnes of goods, a logistics company has large lorries with a capacity of $8$ tonnes and small lorries with a capacity of $5$ tonnes. Due to staffing constraints, no more than $10$ vehicles in total can be sent. Set up the system of inequalities that determines the number of lorries of each type that can be used and find analytically the vertices of the solution region.

Graphically represent the system of inequalities $\frac{x}{2} + \frac{y}{3} \le 1$, $x - \frac{y}{2} \ge -1$ and $y \ge 0$. Calculate the exact coordinates of the vertices of the resulting region.

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

A personal trainer plans a routine with strength exercises ($x$) and cardio exercises ($y$). The total workout time must be between $40$ and $60$ minutes. Each strength exercise lasts $5$ minutes and each cardio exercise lasts $10$ minutes. In addition, the number of strength exercises must be at least equal to the number of cardio exercises. Set up the system of inequalities and represent it graphically.

A vehicle rental company has cars and vans. The daily maintenance of a car costs $10$ euros and that of a van costs $20$ euros. The daily maintenance budget does not exceed $300$ euros. The car park has a maximum capacity for $20$ vehicles. Determine the feasible region and find the maximum number of vans that can be maintained if there are exactly $5$ cars.

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 municipal sports centre weekly organises adapted swimming sessions ($x$, in hours) and pilates for seniors ($y$, in hours). The city council sets the following management conditions:

• Between both activities, they cannot exceed $30$ hours per week in total.

• To meet the minimum demand from the senior group, at least $8$ hours per week of pilates must be scheduled.

• The reserved pool is only available for a maximum of $18$ hours per week for adapted swimming.

• By pedagogical criteria, the number of pilates hours must be at least half the number of hours dedicated to adapted swimming.

1. a) Set up the system of linear inequalities that models all the constraints (including the non-negativity ones) and interpret the meaning of each inequality.
2. b) Graph the feasible region and find the coordinates of all its vertices.
3. c) An association proposes scheduling $18$ hours of swimming and $8$ hours of pilates. Check whether that combination is feasible; if it is not, explain which condition it violates and what the minimum increase in pilates hours needed to make it feasible is, keeping the $18$ hours of swimming.

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.

To organise a concert, VIP tickets ($x$) and standard tickets ($y$) are sold. The maximum capacity of the venue is $5000$ people. For safety and profitability reasons, the number of standard tickets cannot exceed three times the number of VIP tickets. Furthermore, the organisers require that at least $1000$ VIP tickets are sold. Write the system of inequalities, find the vertices of the feasible region and determine algebraically whether it is possible to sell exactly $1500$ VIP tickets and $4000$ standard tickets.

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?

A nutritionist designs a diet based on two foods, X and Y. Each portion of food X provides $20$ g of protein, $10$ g of fat and $40$ g of carbohydrates. Each portion of Y provides $15$ g of protein, $20$ g of fat and $30$ g of carbohydrates. The diet must provide at least $120$ g of protein, no more than $100$ g of fat and exactly $240$ g of carbohydrates. Set up the associated system of inequalities and equations. Is it a two-dimensional region or a line segment? Justify your answer and find the endpoints of the solution set.

The image shows a quadrilateral feasible region modelling the constraints of a software engineering project (programming hours on the $X$-axis vs testing hours on the $Y$-axis). The vertices are $P(0, 4)$, $Q(2, 5)$, $R(5, 2)$ and $S(3, 0)$. Determine the system of inequalities that defines this region exactly.

A furniture factory produces chairs and tables. Each chair requires $2$ hours of cutting and $1$ hour of assembly. Each table requires $1$ hour of cutting and $3$ hours of assembly. The workers can dedicate a maximum of $40$ hours per week to cutting and $60$ hours to assembly. In addition, under a previous contract, at least $5$ tables must be manufactured weekly, and the number of chairs cannot be less than the number of tables. Set up the system of inequalities, plot it and find all the vertices of the feasible region.

Given the system of inequalities $2x + 3y \le 12$, $x - y \ge -1$ and $y \ge 0$, analyse how the area of the feasible region changes if the first constraint is modified to $2x + 3y \le 12 + k$, with $k > 0$. Express the area of the new region in terms of $k$, assuming that the geometric shape of the resulting polygon remains a triangle.

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.

Consider the system of linear inequalities depending on a real parameter $k$:

1. a) For $k = 1$, graph the feasible region, calculate the exact coordinates of all its vertices, and classify the resulting polygon according to its number of sides.
2. b) Analyze the role of the parameter: determine with justification for which values of $k \le 0$ the inequality $x + y \le 6$ becomes redundant (that is, any point in the first quadrant that satisfies $y - kx \le 2$ already automatically satisfies $x + y \le 6$, so that the feasible region becomes a triangle that does not depend on the line $x + y = 6$).
3. c) Is there any real value of $k$ for which the feasible region is the empty set or reduces to a single point? Justify your answer by analyzing whether the origin $(0,0)$ belongs to it and the behavior of the inequalities in a neighborhood of that point.

Topic Challenge. Energy and comfort audit in your home.

1. Real data: Check the temperature ($T$) and wind speed ($v$) for your town on the AEMET website, and the hourly price per kWh on Red Eléctrica de España (REE).

2. Single-variable modeling: Model allowable thermal comfort with an absolute value inequality, $|T - 21| \le 2,5$, and wind chill using a radical inequality: $\sqrt{v} + T \ge 18$. Next, formulate a quadratic inequality that bounds the hours of critical consumption and a rational inequality, such as $\frac{P(t)}{t - 1} \le 1,4$, to control the average unit cost. Solve them analytically.

3. Feasible region with GeoGebra: Let $x$ and $y$ be the hours of use of two high-consumption appliances. Set up a system of linear inequalities with two variables based on each appliance's power, a daily energy cap (kWh) and your daily budget. Find and shade the feasible region.

4. Reasoned proposal: Write a short report justifying the optimal time slots based on the calculated intervals and vertices.

Assessment criteria:

• Algebraic rigour in solving rational, radical, quadratic, and absolute value inequalities.

• Graphical precision in GeoGebra when demarcating the feasible region of the linear system.

• Consistent justification of the final proposal based on the real data collected.