Exercises: Linear inequalities and systems of linear inequalities

Represent on the plane the solution half-plane of the inequality $2x + 3y \leq 12$ and check algebraically whether the points $A(3, 2)$ and $B(2, 5)$ belong to it. What special feature does the point $A$ have with respect to the boundary line?

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

Check, without graphing it, whether the points $P(1, 4)$, $Q(-2, 0)$ and $R(0, -3)$ are solutions to the inequality $3x - y > -5$. Also indicate whether the line $3x - y = -5$ forms part of the solution and why.

The figure shows a solution half-plane bounded by a line that intersects the axes at two marked points, as shown in the figure.

Deduce the equation of the boundary line and write the inequality whose solution is the shaded half-plane. Is the boundary included in the solution?

Represent the set of points in the plane satisfying $y \geq 2x - 1$. Explain how you decide which half-plane to shade and whether the line $y = 2x - 1$ belongs to the solution or not.

True or false, giving reasons for your answer: "The solution to the inequality $3x - y < 6$ includes all the points on the line $3x - y = 6$".

Represent the solution region of the system formed by $x \geq 0$, $y \geq 0$ and $x + y \leq 5$. What geometric shape is obtained? State its vertices.

The shaded triangle in the figure is the solution to a system of three inequalities, as shown in the figure.

Which of the following systems is it? Justify your choice by ruling out the other options.

• a) $x \geq 1$, $y \geq 1$, $3x + 4y \leq 19$

• b) $x \leq 1$, $y \leq 1$, $3x + 4y \geq 19$

• c) $x \geq 1$, $y \geq 1$, $3x + 4y \geq 19$

• d) $x \geq 1$, $y \leq 1$, $3x + 4y \leq 19$

A sports clothing shop is preparing its spring order: let $x$ be the number of technical t-shirts and $y$ be the number of hoodies it will order. Due to warehouse space, it cannot receive more than $40$ garments in total, and the supplier requires it to order at least $10$ t-shirts. Write the system of inequalities that captures all these conditions.

Marcos needs to represent the solution half-plane of the linear inequality $3x - 2y \geq 0$. To find out which side of the boundary line he should shade, he decides to use the origin $(0,0)$ as a test point, arguing that "it is the fastest point to substitute". Marcos calculates: $3 \cdot 0 - 2 \cdot 0 = 0 \geq 0$, observes that the inequality holds, and concludes that he should shade the entire half-plane containing $(0,0)$. However, when checking his drawing against the solutions in the book, he realizes that he has not unambiguously determined which half-plane is the correct one.

1. a) Explain why the method followed by Marcos does not allow him to decide which half-plane is the solution, precisely identifying his conceptual error.
2. b) Propose an alternative test point that does allow solving the inequality without ambiguity, carry out the algebraic check, and indicate which half-plane corresponds to the solution (specifying whether the boundary line is part of it or not).

To represent and determine the solution half-plane of the inequality $4x - y < 8$, two 2nd-year Bachillerato students propose different approaches:

• Lucía proposes first isolating the variable $y$ to write the inequality as $y > 4x - 8$. From there, she draws the line $y = 4x - 8$ with a dashed line and states that, since $y$ is "strictly greater", the solution directly corresponds to all points located above the line, without the need to test any point.

• Sergio prefers not to isolate any variable. He draws the boundary line $4x - y = 8$ with a solid line, chooses the test point $(0,0)$, and checks that $4 \cdot 0 - 0 = 0 < 8$. Since the inequality is true, he shades the half-plane where the origin lies.

1. a) Point out what error Sergio makes in his graphical procedure and explain how to correct it.
2. b) Compare both methods: which one seems more efficient to you to avoid errors with inequality signs when working with negative coefficients? Justify your answer indicating which method you would choose and why.

Represent the solution region of the system formed by $x \geq 0$, $y \geq 0$, $x + y \leq 6$ and $x \leq 4$. Calculate the coordinates of all its vertices and state what type of polygon it is.

Given the system formed by $x \geq 0$, $y \geq 0$, $2x + y \leq 10$ and $x + 2y \leq 8$:

• a) Represent the solution region.

• b) Find its vertices by solving the necessary systems of equations.

A fitness coach plans her athletes' recovery using two types of shakes. Each type A shake provides $12$ g of protein and $8$ g of carbohydrates; each type B shake provides $6$ g of protein and $14$ g of carbohydrates. The daily guideline requires a minimum of $48$ g of protein and $56$ g of carbohydrates. Set up, without solving it, the system of inequalities that the daily numbers of shakes $x$ and $y$ of each type must satisfy.

The figure shows the solution region of a system of inequalities and four points marked on it, as shown in the figure.

Decide graphically which points are solutions to the system and confirm your answer by checking each point algebraically in the system $x + 2y \leq 10$, $3x + y \leq 15$, $x \geq 0$, $y \geq 0$.

Consider the system formed by $x \geq 0$, $y \geq 0$, $x + y \geq 4$ and $x - y \leq 2$.

• a) Represent the solution region and find its vertices.

• b) Reason, based on the graphical representation, whether the region is bounded or unbounded.

A student needs to represent the solution to $3x - 2y \geq 0$. Since the line $3x - 2y = 0$ passes through the origin, they test the point $(0, 0)$ and obtain $0 \geq 0$, concluding that "the solution is the entire plane". Find the error in their reasoning and explain how to correctly determine the solution half-plane in this case.

The figure shows a shaded triangular region whose boundaries are the lines $x = 2$, $y = 1$ and $x + 2y = 12$, as shown in the figure.

• a) Write the system of inequalities whose solution is that triangle.

• b) Calculate the coordinates of its three vertices.

The residents' association is organising its annual party and selling adult tickets ($x$) and child tickets ($y$). The hall has a maximum capacity of $300$ people; by agreement with the local council, at least $50$ adult tickets must be sold, and experience from previous years indicates that the number of children will not exceed twice the number of adults. Set up the corresponding system of inequalities and represent the region of possibilities.

A green courier company delivers parcels in a Low Emission Zone (LEZ) in Seville using two types of vehicles: cargo bikes ($x$) and electric vans ($y$). The operational constraints of the service are as follows:

• Each bicycle carries a maximum of $150\text{ kg}$ of cargo and requires $2\text{ m}^2$ of space at the logistics hub.

• Each van carries a maximum of $900\text{ kg}$ and takes up $10\text{ m}^2$ at the hub.

• The company must guarantee a combined daily carrying capacity of at least $5\,400\text{ kg}$ to fulfill its contracts.

• The total area available at the logistics hub to park these vehicles cannot exceed $70\text{ m}^2$.

• The municipal mobility ordinance requires that the number of cargo bikes be at least double the number of electric vans.

1. a) Model the situation by setting up the system of linear inequalities that define the company's operational constraints, making explicit the natural conditions on the variables.
2. b) If on a given day only $4$ electric vans are in operation ($y = 4$), deduce from the system what number of bicycles $x$ the company can use to satisfy all constraints simultaneously.

An agricultural cooperative in La Rioja produces two types of wine with Qualified Designation of Origin: Crianza ($x$, in thousands of liters) and Reserva ($y$, in thousands of liters). For the current season, it has a maximum of $40$ thousand liters of selected base wine and $60$ barrel-month equivalents for the aging process. Each thousand liters of Crianza consumes $1$ thousand liters of base wine and $1$ unit of aging; each thousand liters of Reserva consumes $1$ thousand liters of base wine and $2$ units of aging. Furthermore, commercial commitments require producing at least $10$ thousand liters of Crianza and at least $5$ thousand liters of Reserva.

1. a) Estimate: Before formally solving the system, look at the available resources. If the technical director proposes bottling $25$ thousand liters of Crianza ($x = 25$) and $18$ thousand liters of Reserva ($y = 18$), estimate with reasoning whether this combination has a chance of being feasible or if it will clearly exceed any of the capacities, justifying your intuition without solving the full system.
2. b) Check: Set up the system of inequalities that defines the feasible region and check algebraically whether the point $(25, 18)$ belongs to it or not. If not, indicate precisely which resource runs out and by how much it is exceeded.
3. c) If it is decided to keep the production of Reserva at $18$ thousand liters ($y = 18$), what is the maximum amount of Crianza that can be produced while fulfilling all constraints? Compare this value with your initial estimate.

Consider the system formed by $x \geq 0$, $y \geq 0$, $x + y \leq 4$ and $y \leq k$, where $k > 0$ is a parameter.

• a) Represent the region for $k = 6$ and for $k = 2$.

• b) Determine for which values of $k$ the solution region is a quadrilateral and for which it is a triangle, justifying your answer.

Represent the region of the plane defined by $|x| + |y| \leq 2$. To do so, reason quadrant by quadrant: rewrite the condition without modulus signs in each of the four quadrants and obtain the corresponding inequality. What shape is produced and what are its vertices?

The lines $x + y = 6$, $y = x - 2$ and $y = 0$ bound a triangle in the plane, as shown in the figure.

• a) Calculate the vertices of the triangle.

• b) Write the system of three inequalities whose solution is precisely that triangle, checking that an interior point, such as $(4, 1)$, satisfies the three inequalities.

A natural cosmetics laboratory produces two creams, moisturising ($x$ jars daily) and repairing ($y$ jars daily). Each jar of moisturising cream requires $3$ minutes in the mixing machine and $2$ minutes in the packaging machine; each jar of repairing cream requires $2$ minutes of mixing and $4$ of packaging. The mixing machine can operate for $60$ minutes per day and the packaging machine for $64$ minutes per day, as summarised in the figure.

• a) Set up the system of inequalities for the problem.

• b) Represent the region of possible production quantities.

• c) Find its vertices and indicate the maximum daily production of moisturising cream possible if no repairing cream were manufactured that day.

A transport company defines its feasibility region in the Cartesian plane by means of a convex polygon whose vertices are $A(1, 2)$, $B(6, 2)$, $C(5, 5)$, and $D(1, 5)$. Due to operational and legal requirements:

• The boundary segments $AB$, $BC$, and $DA$ are part of the permitted solution (boundary included).

• However, the upper segment joining $D$ and $C$ corresponds to a critical pollutant emission limit that under no circumstances can be reached, so the points on this segment are excluded from the solution (boundary excluded).

1. a) Pose the inverse problem: determine the equations of the four lines that bound the polygon and write the system of linear inequalities that exactly defines this region, paying special attention to the use of strict or non-strict inequalities.
2. b) Justify algebraically, without relying on a visual approximation, whether the point $P(3; 4,8)$ and the point $Q(3, 5)$ belong to the feasibility region.
3. c) It is intended to add a new linear constraint of the form $x + y \leq k$ that reduces the region into a pentagon, without leaving out any of the vertices $A$, $B$, or $D$. Determine between which values the parameter $k$ must lie for this to happen and justify your answer by analyzing the positions of the vertices.

At a renewable energy evacuation node in Castilla y León, the grid injection of wind energy ($x$, in MWh) and solar photovoltaic energy ($y$, in MWh) is managed. The technical constraints of the node are given by:

Additionally, the electrical system operator establishes a constraint due to seasonal evacuation capacity: $x + y \leq k$, where $k$ is a positive real parameter ($k > 0$).

1. a) Determine the minimum value of $k$ for which there is at least one point satisfying all constraints simultaneously (that is, for the feasible region not to be empty). Justify which point on the plane determines this threshold.
2. b) Find the smallest value of $k$ from which the constraint $x + y \leq k$ becomes redundant (that is, it does not eliminate any point of the feasible region bounded by the other four constraints). Justify the result by calculating the coordinates of the vertex furthest in that direction.
3. c) An analyst claims that if $k = 35$ is set, the feasible region is a quadrilateral. Rigorously prove whether his claim is true or false by determining the number and coordinates of the vertices obtained for that value of $k$.

Topic challenge. How can you optimize a budget tailored to reality? You are going to plan the weekly purchase of two basic foods ($x$ and $y$) by applying linear constraints.

1. Phase 1 (Data collection): Visit the INE website (or a supermarket's digital catalogue) and record the price per kilo of two foods (for example, legumes and fruit), as well as a reference nutritional intake (such as grams of fiber or protein per $100\text{ g}$).

2. Phase 2 (Algebraic modeling): Translate the situation into a system of linear inequalities with at least four constraints: maximum weekly budget limit (for example, a maximum of $18,50\text{ €}$), minimum required nutritional intake, and minimum consumption quantities ($x \ge 0,5$, $y \ge 1,0$).

3. Phase 3 (Representation in GeoGebra): Enter the system into GeoGebra, shade the set of solutions (feasible region), and precisely identify the coordinates of all its vertices.

4. Phase 4 (Reasoned proposal): Prepare a brief one-page report with the data table, the screenshot of the graph, and the justified choice of two possible purchase combinations, comparing one located at a vertex with another in the interior of the region.

Assessment criteria:

• Precision in the algebraic formulation of the system and the use of units with real data.

• Correct delimitation of the feasible region and calculation of vertices in GeoGebra.

• Critical interpretation and mathematical justification of the proposed solutions in the report.